If we wanted to calculate *a*50 with this recursive routine we would realize that the calculation time increases a lot because many of the calculations are recalculated and also there is a risk of overflowing the system stack due to a large amount of recursive *Application of Discrete Mathematics for Programming Discrete Mathematics Calculations DOI: http://dx.doi.org/10.5772/intechopen.102990*

calls. Fortunately, we can reprogram an iterative version of this function and it is the one shown below.

```
function y = recpentagI(n)
%Author: CRL
a0=1;
a1=5;
if (n==0)
   y=a0;
else
   if (n==1)
       y=a1;
   else
       an=0;
       for i=2:n
          an=2*a1-a0+3;
          a0=a1;
          a1=an;
       end
       y=an;
   end
end
end
```
We will test this routine by first executing it for the known value *a*5 ¼ 51, then for the value *a*30 whose value obtained with the recursive version was 1426 and finally, we will obtain the value *a*50 as well as the *a*100 value.

```
>> z=recpentagI(5)
z =
    51
>> z=recpentagI(30)
z =
    1426
>> z=recpentagI(50)
z =
    3876
>> z=recpentagI(100)
z =
    15251
```
The runtime improvement of the iterative version over the recursive version is truly impressive. However, the execution time of a routine can be further improved to carry out this calculation by resorting to discrete mathematics tools to solve the recurrence and implement a routine that the only thing that does is evaluate in the given position the mathematical formula obtained from the solution of the recurrence. Discrete mathematics provides us with many methods to solve non-homogeneous linear difference equations, such as the one we will solve, associated with recurrence relationships. This type of difference equations can be obtained by methods, such as the iteration of recurrence, the characteristic polynomial method, or the generating function method. In this subsection, we will resolve the recurrence by applying several of these methods for illustrative purposes. For more details on solving recurrences using these methods, we recommend consulting [3, 5–8].

#### **4.2 Solving a recurrence by iteration**

We start with the application of the iteration method of recurrence. For this purpose, we will use the third row of the recurrence relation (20). This equation would be the following

$$a\_n = 2a\_{n-1} - a\_{n-2} + \text{3.}\tag{21}$$

We apply Eq. (21) to the case of *n* � 1 which would give the following equation

$$
\mathfrak{a}\_{n-1} = 2\mathfrak{a}\_{n-2} - \mathfrak{a}\_{n-3} + \mathfrak{3}.\tag{22}
$$

We substitute 22 in 21 and obtain the following equation

$$a\_n = 3a\_{n-2} - 2a\_{n-3} + 2 \cdot 3 + 3. \tag{23}$$

We apply Eq. (21) to the case of *n* � 2 which would give the following equation

$$
\mathfrak{a}\_{n-2} = 2\mathfrak{a}\_{n-3} - \mathfrak{a}\_{n-4} + \mathfrak{3}.\tag{24}
$$

We substitute 24 in 23 and obtain the following equation

$$a\_n = 4a\_{n-3} - 3a\_{n-4} + \mathfrak{Z} \cdot \mathfrak{Z} + \mathfrak{Z} \cdot \mathfrak{Z} + \mathfrak{Z}.\tag{25}$$

We apply Eq. (21) to the case of *n* � 3 which would give the following equation

$$a\_{n-3} = 2a\_{n-4} - a\_{n-5} + 3.\tag{26}$$

We substitute 26 in 25 and obtain the following equation

$$a\_n = \mathfrak{Z}a\_{n-4} - 4a\_{n-5} + 4\cdot \mathfrak{Z} + \mathfrak{Z}\cdot \mathfrak{Z} + \mathfrak{Z}\cdot \mathfrak{Z} + \mathfrak{Z}.\tag{27}$$

Continuing with this procedure until reaching the base cases and noting that certain regularities appear, such as the presence of a sum of successive natural numbers, we arrive at the following expression

$$a\_n = n \cdot a\_1 - (n - 1) \cdot a\_0 + \mathbf{3} \cdot \sum\_{i=1}^{n-1} i. \tag{28}$$

Substituting *a*0 ¼ 1, *a*1 ¼ 5 and applying Gauss's formula to the summation in Eq. (28), we obtain the following solution

$$a\_n = \frac{3}{2} \cdot n^2 + \frac{5}{2} \cdot n + 1. \tag{29}$$

#### **4.3 Solving a recurrence by the characteristic polynomial method**

We can try another method of solving the recurrence to illustrate another tool provided by discrete mathematics. In this case, we will use the characteristic polynomial method. The difference equation that we are going to solve is linear inhomogeneous with coefficients, which makes it capable of being solved by this method. *Application of Discrete Mathematics for Programming Discrete Mathematics Calculations DOI: http://dx.doi.org/10.5772/intechopen.102990*

These methods are closely related to the methods of solving differential equations of the same type. The difference equation that we are going to solve is the following

$$a\_n - 2a\_{n-1} + a\_{n-2} = \mathfrak{Z}, a\_0 = \mathfrak{1}, a\_1 = \mathfrak{Z}.\tag{30}$$

The theory on the solution of equations in non-homogeneous linear differences says that the general solution is composed of a solution of the homogeneous equation plus a particular solution related to the non-homogeneous part. The homogeneous equation associated with Eq. (30) would be the following

$$
\mathfrak{a}\_n - 2\mathfrak{a}\_{n-1} + \mathfrak{a}\_{n-2} = \mathbf{0}.\tag{31}
$$

A characteristic polynomial is associated with the homogeneous Eq. (31) that is obtained by substituting a possible solution in the difference equation. Said possible solution has the form *an* ¼ *c* � *rn* where *c* is an arbitrary constant if we substitute said possible solution in 31 we obtain the following polynomial

$$r^2 - 2r + 1 = 0.\tag{32}$$

The roots of the characteristic polynomial 32 are *r*1 ¼ 1 and *r*2 ¼ 1, that is, they are real and repeated roots and since the solutions of a difference equation must be linearly independent, the form of the solution of the homogeneous difference equation would be the following

$$a\_n = c\_1 \cdot \mathbf{1}^n + c\_2 \cdot n \cdot \mathbf{1}^n. \tag{33}$$

Now we proceed to solve the particular equation whose solution must be linearly independent of the solutions of the associated homogeneous equation. We note that the right-hand side of the difference Eq. (30) is *f n*ð Þ¼ 3 which is equivalent to *f n*ð Þ¼ 3 � 1*n*. So the form of the particular solution would be the following

$$
\mathfrak{a}\_n = \mathbf{A} \cdot \mathfrak{n}^2. \tag{34}
$$

If we evaluate the equation in differences 30 the solution 34 we obtain the following expression

$$A \cdot n^2 - 2 \cdot A \cdot (n - 1)^2 + A \cdot (n - 2)^2 = 3 \cdot 1^n. \tag{35}$$

After algebraically simplifying Eq. 35 we deduce that *A* ¼ 3 2 , so the general solution would have the form

$$a\_n = c\_1 \cdot \mathbf{1}^n + c\_2 \cdot n \cdot \mathbf{1}^n + \frac{3}{2} \cdot n^2. \tag{36}$$

Applying on 37 *a*0 ¼ 1 we deduce that *c*1 ¼ 1. Applying on 37 *a*1 ¼ 5 we get *c*2 ¼ 5 2 In this way we obtain that the general solution is

$$a\_n = 1 + \frac{5}{2} \cdot n + \frac{3}{2} \cdot n^2. \tag{37}$$

We can verify that solution 37 coincides with solution 29. Finally, we can use this solution to implement a Matlab function to perform this calculation and obtain the value of an element of the sequence given the position. The Matlab function would be the following.

```
function y = recpentag(n)
% Author: Carlos Rodriguez Lucatero
y=1+(5/2)*n+(3/2)*(n^2);
end
```
For being convinced about the correctness of the solution, we can evaluate this Matlab function for the same values used before and we obtain the following results.

```
>> y = recpentag(5)
y =
    51
>> y = recpentag(10)
y =
    176
>> y = recpentag(30)
y =
    1426
>> y = recpentag(50)
y =
    3876
>> y = recpentag(100)
y =
    15251
```
#### **4.4 Solving a recurrence by generating function method**

Another powerful method of discrete mathematics to solve difference Eq. (19) associated with the recurrence 20 is that of ordinary generating functions. Said method consists, in the simplest case, in converting a numerical sequence into an infinite polynomial of a single variable whose coefficients are precisely the elements of the numerical sequence. The reason for doing this transformation is that the algebraic manipulation of these infinite polynomials is relatively simple. For this reason, we define below the concept of the ordinary generating function of a single variable.

**Definition** 1.1 Given a numerical sequence *a*0, *a*1, *a*2, … , *ak*, … the function

$$A(\mathbf{z}) = \sum\_{k \ge 0} a\_k \mathbf{z}^k. \tag{38}$$

It is called the ordinary generating function (OGF) of the sequence. The notation *zk* � �*A z*ð Þ will be used to refer to the coefficient *ak* of the k-th term of the infinite polynomial.

By means of the generating functions, we can map sequences of numbers that often are integers to power series. The coefficient of the n-th adding will, therefore, be related to the n-th element of the associated numerical sequence. For example, in the generating function P∞ *i*¼0*zi* ¼ 1 þ *z* þ *z*2 þ *z*3 þ *z*4 þ … , we observe that it is the geometric series that converges if ∣*z*∣<1 and can be expressed as 1 1�*z* . We can also observe that all the terms of the sum have a coefficient of 1, so this generating function is

*Application of Discrete Mathematics for Programming Discrete Mathematics Calculations DOI: http://dx.doi.org/10.5772/intechopen.102990*

associated with the numerical sequence 1, 1, 1, 1, … . On the other hand, since the generating functions are infinite polynomials, it is easy to apply the derivative operation to them. Thus, the derivative of the geometric series would be expressible as follows

$$\frac{d}{dx}\left(\frac{1}{\left(1-x\right)}\right) = \frac{1}{\left(1-x\right)^2} = \frac{d}{dx}\left(1+x+x^2+x^3+x^4+\dots\right) = 1+2x+3x^2+4x^3+\dots \tag{39}$$

As can be seen from 39, the derivative of the geometric series can be related to de succession of natural numbers.

It is also possible to do certain types of algebraic operations on ordinary generating functions that have an impact on the sequence of numerical values associated with the given generating function. For example, if I multiply the generating function of 1 ð Þ 1�*z* 2 by *z*, we obtain the following effect in the numerical sequence

$$\frac{z}{\left(1-z\right)^{2}} = 0 + z + 2z^{2} + 3z^{3} + 4z^{4} + \dots \leftrightarrow 0, 1, 2, 3, 4, 5, \dots \tag{40}$$

that is to say that we obtain a shift to the right in the sequence of numbers. If we apply the derivative to Eq. (40), we obtain the following relationship

$$\frac{d}{dz}\left(\frac{z}{\left(1-z\right)^{2}}\right) = \frac{d}{dz}\left(\mathbf{0} + \mathbf{z} + 2\mathbf{z}^{2} + 3\mathbf{z}^{3} + 4\mathbf{z}^{4} + \dots\right) = \frac{\mathbf{z}+\mathbf{1}}{\left(1-\mathbf{z}\right)^{3}}\tag{41}$$

$$= \mathbf{1} + \mathbf{2}^{2}\mathbf{z} + \mathbf{3}^{2}\mathbf{z}^{2} + 4\mathbf{z}^{3} + \dots$$

then we have

$$\frac{z+1}{\left(1-z\right)^{3}} = 1 + 2^{2}z + 3^{2}z^{2} + 4^{2}z^{3} + 5^{2}z^{4} + \dots \leftrightarrow 1^{2}, 2^{2}, 3^{2}, 4^{2}, \dots \tag{42}$$

We have exemplified the following relationships between numerical sequences and generating functions

$$1, 1, 1, 1, \dots \leftrightarrow \frac{1}{1 - z} \tag{43}$$

$$(1,2,3,4,\ldots)\to\frac{1}{\left(1-z\right)^2}.\tag{44}$$

$$(0,1,2,3,4,\ldots \leftrightarrow \frac{z}{\left(1-z\right)^2} \tag{45}$$

$$(1^2, 2^2, 3^2, 4^2, \dots \leftrightarrow \frac{z+1}{(1-z)^3} \tag{46}$$

$$(0^2, 1^2, 2^2, 3^2, 4^2, \dots \leftrightarrow \frac{z(z+1)}{(1-z)^3} \tag{47}$$

After all those examples, we can be convinced that it is possible to establish relationships between sequences of numbers and generating functions, as shown in **Table 5**.

#### **Table 5.**

*Table of sequences and ordinary generating functions.*

For more detailed information on recurrences that appear in the analysis of algorithms as well as generating functions, we recommend the excellent book [5].

After this brief summary on ordinary generating functions, we can solve the difference Eq. (19) using generating functions and that is just what we will do next. The difference equation to solve is

$$a\_n - 2a\_{n-1} + a\_{n-2} = \mathfrak{Z}, a\_0 = \mathfrak{1}, a\_1 = \mathfrak{Z}.\tag{48}$$

We know from definition 38 that a generating function is expressed mathematically as

$$A(\mathbf{z}) = \sum\_{k \ge 0} a\_k \mathbf{z}^k. \tag{49}$$

and we apply this operator to the difference Eq. (48) keeping the same summation index for the terms of the difference Eq. (48) obtaining the following expression

$$\sum\_{n\geq 2} a\_n z^n - 2\sum\_{n\geq 2} a\_{n-1} z^n + \sum\_{n\geq 2} a\_{n-2} z^n = 3\sum\_{n\geq 2} z^n. \tag{50}$$

Because of the index shift, the first summation is *A z*ð Þ� *a*0 � *a*1*z*. In the second term of the left hand of Eq. (50) the index of the coefficient and the power of *z* being different; so, we can factorize *z* and take into account the shift in the summation index and obtain the term 2*zAz* ð ð Þ� *a*0. The difference between the summation index and the power of *z* in the third term of the left hand of 50 can be arranged by factoring *z*2 and in that case, we obtain the term *z*2*A z*ð Þ. The right hand of Eq. (50) is a geometric

*Application of Discrete Mathematics for Programming Discrete Mathematics Calculations DOI: http://dx.doi.org/10.5772/intechopen.102990*

series but given the shift on the index of the summation, it is necessary to subtract the two first terms of the geometric series giving, as a result, the term 3 1 1�*z* � 1 � *z* � �. Then taking into account these remarks, we obtain the next equation

$$A(z) - \mathbf{1} - \mathbf{5}z - 2zA(z) - \mathbf{2}z + z^2A(z) = \mathbf{3} \left( \frac{\mathbf{1}}{\mathbf{1} - z} - \mathbf{1} - z \right). \tag{51}$$

Simplifying Eq. (51) we get the next equation

$$A(z)\left(\mathbf{1} - 2\mathbf{z} + z^2\right) - \mathbf{1} - \mathbf{3}z = \mathfrak{d}\left(\frac{z^2}{\mathbf{1} - z}\right). \tag{52}$$

The left hand of 53 can be algebraically simplified obtaining the following expression

$$A(z)(\mathbf{1}-z)^2 = \mathbf{1} + \mathbf{3}z + \mathbf{3}\left(\frac{z^2}{\mathbf{1}-z}\right). \tag{53}$$

Simplifying de left hand of Eq. (53) we get the equation

$$A(z) = \frac{1}{\left(1 - z\right)^2} + 3\frac{z}{\left(1 - z\right)^2} + 3\left(\frac{z^2}{\left(1 - z\right)^3}\right). \tag{54}$$

We apply the *zn* ½ � operator to Eq. (54) with the purpose of obtaining the coefficient of the nth addend of the sum that will correspond to the element in the position *n* of the sequence of numbers studied, arriving at the equation

$$a\_n = [z^n]A(z) = [z^n] \left(\frac{\mathbf{1}}{\left(\mathbf{1} - z\right)^2}\right) + \mathbf{3}[z^n] \left(\frac{z}{\left(\mathbf{1} - z\right)^2}\right) + \mathbf{3}[z^n] \left(\frac{z^2}{\left(\mathbf{1} - z\right)^3}\right). \tag{55}$$

and then to the final result

$$a\_n = (n+1) + 3n + 3\binom{n}{2} = 4n + 1 + \frac{3n^2}{2} - \frac{3n}{2} = 1 + \frac{5n}{2} + \frac{3n^2}{2}.\tag{56}$$

As you can see the expressions (29), (37), and (56) are the same.

#### **5. Modular arithmetical calculations**

In discrete mathematics courses at the undergraduate level, topics of modern algebra are addressed where the most elementary algebraic structures are defined, such as the group structure and the ring structure. The algebraic structure to which we are going to devote our attention in this section is the ring structure in the context of modular arithmetic.

This algebraic structure is often used, informally, in arithmetic courses to learn to add and multiply. Formally speaking we can say that this structure is composed of a set of objects that could be in the case of arithmetic the set of integers as well as two

operations which are addition and multiplication which is denoted as ð Þ , þ , � . The two operations must be closed, that is to say, that the results they give must belong to the same set from which the operands are taken. Additionally, the operations must respect certain properties that we will define below.

**Definition** 1.2 (Ring) Let be *R* nonempty set that has two closed binary operations denoted as þ y � . Then ð Þ *R*, þ , � it is a ring if ∀*a*, *b*,*c*∈ *R* the following conditions are met:

a) *a* þ *b* ¼ *b* þ *a* (commutative law of þ).

b) *a* þ ð Þ¼ *b* þ *c* ð Þþ *a* þ *b c* (Associative law of þ).

c) ∃*z*∈*R* such that *a* þ *z* ¼ *z* þ *a* ¼ *a*, ∀*a*∈*R* (existence of identity element for þ).

d) For each *a*∈*R*, ∃ *b*∈*R* such that *a* þ *b* ¼ *b* þ *a* ¼ *z* (existence of inverse under þ).

e) *a* � ð Þ¼ *b* � *c* ð Þ� *a* � *b c* (Associative law for �).

f) *a* � ð Þ¼ *b* þ *c a* � *b* þ *a* � *c* y ð Þ� *b* þ *c a* ¼ *b* � *a* þ *c* � *a* (Distributive laws for � over þ).

In order to simplify the notation, the operation *a* � *b* can be written as *ab*. The associative properties of both operations as well as the distributivity of one operation over the other can be generalized for the case of more than 2 operands.

The þ operation of the ring is commutative but the � operation is not commutative in the 1.2 definitions. We can apply the commutativity property to the � operation and we would obtain the commutative ring structure. Formally the definition of a commutative ring is as follows.

**Definition** 1.3 (Commutative ring) Let be *R* a nonempty set having two closed binary operations denoted as þ and � . Then ð Þ *R*, þ , � is a ring if ∀*a*, *b*,*c*∈ *R* meet the following conditions:

a) If *ab* ¼ *ba*, ∀*a*, *b*∈*R* then *R* is a commutative ring.

b) It is said that the ring *R* has no proper divisors of zero if for any *a*, *b*∈ *R*, *ab* ¼ *z* ) *a* ¼ *z*∨*b* ¼ *z*.

c) If an element *u* ∈*R* such that *u* 6¼ *z* and *ua* ¼ *au* ¼ *a*, ∀*a*∈*R* we say that *u* is a unit or identity element of multiplication and that *R* is a ring with unity.

Another interesting property that can be added to the list of ring properties is the existence of multiplicative inverse. The definition is the following.

**Definition** 1.4 (Ring with multiplicative inverse) Let *R* be a ring with the unit *u* if *a*, *b*∈*R* and *ab* ¼ *ba* ¼ *u* then *b* is called inverse multiplicative of *a* and *a* is called a unit of *R*. (An element *b* is also a unit of *R*).

Taking into account the existence of the unit property, we can add it to the ring commutative ring and obtain a commutative ring with unity.

**Definition** 1.5 (Commutative ring with unity) Let be *R* a commutative ring with unity. Then:

a) *R* is said to have an integer domain if *R* has no proper divisors of zero.

b) *R* is said to be a field if each non-zero element of *R* is a unit.

Rings can have subsets that satisfy the properties of a ring as defined in 1.2 in which case we are talking about subring structures. These algebraic structures can have additive cancelation properties as well as multiplicative cancelation properties and therefore will have symmetric or inverse elements of addition as

well as multiplication. Once we have understood the theoretical part of the rings we are better positioned to tackle the construction and use of finite rings specials and fields. We will start by presenting some results of the modular arithmetic needed for the de notion of operations in modular rings. These notions will be explained below.

#### **5.1 Integers mod** *n*

In this section, we will review some notions of modular arithmetic that usually appear in introductory courses of the Theory of Numbers and that allow us to understand important concepts, such as divisibility, primality, the greatest common divisor operation as well as the algorithm of Euclid to calculate it efficiently. Let us start with establishing the meaning of some mathematical symbols that appear in this context. The expression *d*∣*a* translates into words like d divides a to what is the same as ∃*k*∈ such that *a* ¼ *kd* and *d* is called divisor of *a*. This means that if we divide *a* by *d*, the remainder will be 0. From here we can make the following properties and definitions.

**Definition** 1.6 Every integer divides 0. That is ∀*x*∈ , *x*∣0.

**Proposition** 1.7 If *a*> 0 and *d*∣*a* then ∣*d*∣≤∣*a*∣.

**Proposition** 1.8 *d*∣*a* if and only if �*d*∣*a*.

**Definition** 1.9 A number *x* is said to be composite if it has divisors other than 1 and *x*. **Example** 1.10 The number 39 is composite since 3∣39 is true and 3 6¼ 1 is true as like 3 6¼ 39.

**Example** 1.11 The divisors of 24 are 1, 2, 3, 4, 6, 8, 12, 24 f g. The 1 and 24 are known as trivial divisors.

**Definition** 1.12 The set of divisors of a number *a* that are neither 1 nor *a* are called factors of *a*.

**Definition** 1.13 A number prime is here the one that has no factors.

**Example** 1.14 The numbers 1, 3, 5, 7, 11, 13 f g are prime.

**Definition** 1.15 A number that is not prime is said to be a composite number.

**Theorem** 1.16 (Division Theorem) For any *a*, *n*∈ there exist *q*,*r*∈ únique such that *a* ¼ *qn* þ *r* where 0 ≤*r*<*n*.

*q* ¼ ⌊*a n*⌋ is called the quotient and *r* ¼ *a*mod*n* is called the remainder. Then *n*∣*a* if *a*mod*n* is equal to 0. From this, in conjunction with theorem 16, it can be stated that *a* ¼ ⌊*a n*⌋*n* þ *a*mod*n* or equivalently that *a* � ⌊*a n*⌋*n* ¼ *a*mod*n*.

**Definition** 1.17 If ð Þ¼ *a*mod*n* ð Þ *b*mod*n* then we say that *a* � *b*mod*n* if they have the same remainder *a* and *b* when divided by *n* is to say that *a* � *b*mod*n*⇔*n*∣ð Þ *b* � *a* .

When it is true that ð Þ¼ *a*mod*n* ð Þ *b*mod*n* we will denote it as *a* � *b*mod*n* and the equivalence classes that this relation generates will be expressed as ½ � *a n* ¼

f g *a* þ *kn*j*k*∈ . In general, we can say the congruence relation modulo some number *n* partitions to the set of integers in equivalence classes by the remainder left when divided by a number *n* which we express as ½ � *n* ¼ ½ � *a n* : 0≤ *a*≤ *n* � 1 .

#### **5.2 Common divisors and greatest common divisor**

Let us start by defining the concept of a common divisor. **Definition** 1.18 *d* is a common divisor of the numbers *a* and *b* if *d*∣*a* and *d*∣*b* hold. Common divisors have the following properties. **Proposition** 1.19 *d*∣ð Þ *a* þ *b* and *d*∣ð Þ *a* � *b* .

**Proposition** [1.20] If *d*∣*a* and *d*∣*b* then *d*∣ð Þ *ax* þ *by* ∀*x*, *y*∈. That is, *d* divides every linear combination of *a* and *b*.

**Proposition** 1.21 If *a*∣*b* then ∣*a*∣≤∣*b*∣ or *b* ¼ 0.

**Proposition** 1.22 If *a*∣*b* and *b*∣*a* hold then *a* ¼ þ*b* or *a* ¼ �*b*.

**Definition** 1.23 The greatest common divisor of *a* and *b* denoted as *gcd a*ð Þ , *b* .

(greatest common divisor) is: gcdð Þ¼ *a*, *b* max f g *d* : *d*j*ayd*j*b* .

**Example** 1.24 *gcd*ð Þ¼ 24, 30 6, *gcd*ð Þ¼ 5, 7 1, *gcd*ð Þ¼ 0, 9 0.

**Proposition** 1.25 If *a* and *b* are not both equal to 0, then *gcd a*ð Þ , *b* is an integer between 1 and min ð Þ j*a*j, j*b*j .

**Definition** 1.26 *gcd*ð Þ¼ 0, 0 0.

**Definition** 1.27 *a* and *b* are relatively prime if *gcd a*ð Þ¼ , *b* 1.

**Proposition** 1.28 *gcd a*ð Þ¼ , *b gcd b*ð Þ , *a* .

**Proposition** 1.29 *gcd a*ð Þ¼ , *b gcd*ð Þ �*a*, *b* .

**Proposition** 1.30 *gcd a*ð Þ¼ , *b gcd*ð Þ j*a*j, j*b*j .

**Proposition** 1.31 *gcd a*ð Þ¼ , 0 ∣*a*∣.

**Proposition** 1.32 *gcd a*ð Þ¼ , *ka a*, ∀*k*∈ .

**Theorem** 1.33 If *a* and *b* are any integers and are not both equal to 0 then *gcd a*ð Þ , *b* is the smallest positive element of the set f g *ax* þ *by* : *x*, *y* ∈ of linear combinations of *a* and *b*.

**Proof:** Let *s* be the smallest positive element of the set of such linear combinations of *a* and *b*, then *s* ¼ *ax* þ *by* for some *x*, *y*∈. Let *q* ¼ ⌊*a s* ⌋. The equation *a*mod*n* ¼ *a* � ⌊*a n*⌋ implies *a*mod*s* ¼ *a* � *qs* ¼ *aq ax* ð Þ¼ þ *by a*ð Þþ 1 � *qx b*ð Þ �*qy* , so *s*mod*s* is a linear combination of *a* and *b*. Since *a*mod*s*< *s* then *a*mod*s* ¼ 0 since *s* is the smallest value that is a linear combination of *a* and *b*. The above means that *s*∣*a*. Reasoning in a similar way we can prove that *s*∣*b*. Then *s* is a common divisor of *a* and *b*, and it also holds that *gcd a*ð Þ , *b* ≥*s* and like *d*∣*a* and *d*∣*b* implies that *d*∣ð Þ *ax* þ *by* , so as a consequence of all this *gcd a*ð Þ , *b* ∣*s* since *gcd a*ð Þ , *b* divides any linear combination of *a* with *b* and *s* is a linear combination of *a* with *b*. But *gcd a*ð Þ , *b* ∣*s* and *s* >0 imply that *gcd a*ð Þ , *b* <*s*. Combining that is satisfied simultaneously that *gcd a*ð Þ , *b* >*s* and that *gcd a*ð Þ , *b* <*s* we can conclude that *gcd a*ð Þ¼ , *b s*.

**Corollary** 1 For any *a*, *b*∈ , if *d*∣*a* and *d*∣*b* then *d*∣*gcd a*ð Þ , *b* .

**Proof:** If *d*∣*a* and *d*∣*b* then ∀*x*, *y* ∈*d*∣ð Þ *ax* þ *by* and since *gcd a*ð Þ , *b* is a linear combination of *a* and *b* we can conclude. □.

**Corollary** 2 ∀*a*, *b*, *n*∈ and *n* ≥0 *gcd an* ð Þ¼ , *bn ngcd a*ð Þ , *b* .

**Corollary** 3 ∀*a*, *b*, *n*∈ and *a*, *b*, *n* ≥0 if *n*∣*ab* and *gcd a*ð Þ¼ , *n* 1 then *n*∣*b*. **Theorem** 1.34 ∀*a*, *b*∈þ, *gcd a*ð Þ¼ , *b gcd b*ð Þ , *a*mod*b* . **Proof:**

• (demo sketch).

• We first prove that *gcd a*ð Þ , *b* ∣*gcd b*ð Þ , *a*mod*b* .

• Let *d* ¼ *gcd a*ð Þ , *b* then *d*∣*a* and *d*∣*b*.

• We know that ð Þ¼ *a*mod*b a* � *qb* where *q* ¼ ⌊*a b*⌋.

• Since ð Þ *a*mod*b* is a linear combination of *a* and *b*.

• from the above and the fact that *d*∣*a* and *d*∣*b* this implies that *d*∣ð Þ *ax* þ *by* then *d*∣*b* and *d*∣ð Þ *a*mod*b* where *gcd a*ð Þ , *b* ∣*gcd b*ð Þ , *a*mod*b* .

• As a second part, it is proved in a similar way that *gcd b*ð Þ , *a*mod*b* ∣*gcd a*ð Þ , *b* .

• If both *gcd a*ð Þ , *b* ∣*gcd b*ð Þ , *a*mod*b* and *gcd b*ð Þ , *a*mod*b* ∣*gcd a*ð Þ , *b* hold we can conclude that ∀*a*, *b*∈þ, *gcd a*ð Þ¼ , *b gcd b*ð Þ , *a*mod*b* .

*Application of Discrete Mathematics for Programming Discrete Mathematics Calculations DOI: http://dx.doi.org/10.5772/intechopen.102990*

Theorem 1.34 provides us with the mathematical elements to be able to define a recursive algorithm for the calculation of the Greatest Common Divisor is the following.

Euclid(a,b) 1) if b=0 2) then return a 3) else Euclid(b, a mod b) The following is an example of how Euclid's algorithm works. **Example** [1.35] *We will calculate the gcd*ð Þ 30, 21 *applying the Euclidean algorithm*

*that I have just explained. The execution steps are displayed in* **Table 6**.

#### **5.3 Modular rings**

**Definition** 1.36 Let *n* ∈þ, *n*>1. For *a*, *b*∈, tenths that *a* is congruent to *b* modulo *n* and is written as *a* � *b*ð Þ mod*n* , if *n*∣ð Þ *ab* , or equivalently *a* ¼ *b* þ *kn* for some *k in*.

**Example** 1.37 For example:

a) 17 � 2 mod5 ð Þ. b) �7 � �49 mod6 ð Þ.

**Theorem** 1.38 The congruence modulo *n* is an equivalence relation . **Proof:** (Do it as an exercise).

Since an equivalence relation on a set produces a partition of that set, then for *n* ≥2 the congruence relation modulo *n* produces a partition of set in the following *n* equivalence classes:

$$[0] = \{\dots, -2n, -n, 0, n, 2n, \dots\} = \{0 + n\mathbf{x} | \mathbf{x} \in \mathbb{Z}\}$$

$$[1] = \{\dots, -2n + 1, -n + 1, 1, n + 1, 2n + 1, \dots\} = \{1 + n\mathbf{x} | \mathbf{x} \in \mathbb{Z}\}$$

$$[2] = \{\dots, -2n + 2, -n + 2, 2, n + 2, 2n + 2, \dots\} = \{2 + n\mathbf{x} | \mathbf{x} \in \mathbb{Z}\}$$

$$\vdots$$

$$[n - 1] = \{\dots, -n + 1, -1, n - 1, 2n - 1, 3n - 1, \dots\} = \{(n - 1) + n\mathbf{x} | \mathbf{x} \in \mathbb{Z}\}$$

By the division algorithm, we know that ∀*t*∈, *t* ¼ *qn* þ *r* where 0≤*r*< *n*, so *t*∈ ½ �*r* or also means that ½�¼*t* ½ �*r* . We use to denote 0 f g ½ �, 1½ �, 2½ �, … ½ � *n* � 1 or when there is no ambiguity we also use 'in *n* ¼ f g 0, 1, 2, … , *n* � 1 .

We can define closed operations addition and pipeline on the set of equivalence classes of *n* as ½ �þ *a* ½ �¼ *b* ½ � *a* þ *b* and ½ � *a cdot b*½ �¼ ½ � *a* ½ �¼ *b* ½ � *ab* . **Example** [1.39] If *n* ¼ 7 then 2½ �þ ½ �¼ 6 ½ �¼ 8 ½ � 1 and 2½ �� ½ �¼ 6 ½ �¼ 12 ½ � 5 .

**Table 6.**

*Execution steps of the Euclid's algorithm.*

Before accepting the definitions of the modular operations ½ �þ *a* ½ �¼ *b* ½ � *a* þ *b* and ½ � *a* ½ �¼ *b* ½ � *ab* we must convince ourselves that they are well defined, that is, if ½ �¼ *a* ½ �*c* and ½ �¼ *b* ½ � *d* then ½ �þ *a* ½ �¼ *b* ½ �þ*c* ½ � *d* and ½ � *a* ½ �¼ *b* ½ �*c* ½ � *d* . We will prove that these operations are independent of the choice of elements within a class. So, then ½ �¼ *a* ½�)*c a* ¼ *c* þ *sn* for some *s*∈ and ½ �¼ *b* ½ �)*d b* ¼ *d* þ *tn* for some *t*∈. Then *a* þ *b* ¼ ð Þþ *c* þ *sn* ð Þ¼ *d* þ *tn* ð Þþ *c* þ *d* ð Þ *s* þ *t n* so that ð Þ� *a* þ *b* ð Þ *c* þ *d* mod*n* that is ½ �¼ *a* þ *b* ½ � *c* þ *d* .

In the same way *ab* ¼ ð Þ *c* þ *sn* ð Þ¼ *d* þ *tn cd* þ ð Þ *sd* þ *ct* þ *stn n* so that *ab* � *cd*mod*n*, that is ½ �¼ *ab* ½ � *cd* . This leads us to establish the following theorem.

**Theorem** 1.40 For *n* ∈þ, *n*>1 under the closed binary operations just defined *n* is a commutative ring with the unit 1½ �.

**Proof:** The proof is left as an exercise for the student. What would have to be done for this is to verify that the ring properties hold for the definition of the addition and multiplication operations on *n* and the properties of the ring ð Þ , þ , � .

Before continuing with more theoretical results, let us see the tables of the operations of þ and � for 5 and 6. The 5 operation tables are displayed in **Table 7** and operation tables corresponding to 6 ring are shown in **Table 8**.

We can see in the tables of the þ and � operations of 5 all elements other than 0 have an inverse element, which is why this is a field. In the case of the tables of 6, as opposed to those of 5, only 1 and 5 are elements with inverse (bf units) and 2, 3, 4 are proper divisors of 0. If we obtained the corresponding tables for 9, we could see that 3 � 3 ¼ 3 � 6 ¼ 0, that is there are also proper divisors of 0 which means that it is not enough for *n* to be an odd number for the set *n* to be a field. The latter leads us to establish the following theorem.

**Theorem** 1.41 *n* would be a field if and only if *n* is a prime number.

**Proof:** Let *n* be a prime, and suppose 0 <*a*<*n*. Then the gcdð Þ¼ *a*, *n* 1 and since the gcdð Þ *a*, *n* is a linear combination of *a* and *n* that is to say that ∃*s*, *t*∈*n*, *as* þ *tn* ¼ 1. Therefore *as* � 1 mod ð Þ *n* i.e. ½ � *a* ½�¼*s* ½ � 1 , which implies that *a* is an element with inverse (unit) which makes *n* a field. Conversely, if *n* is not a prime then it can be represented as a product of two numbers, that is *n* ¼ *n*1*n*2 where 1< *n*1, *n*2 <*n* and if

#### **Table 7.**

*Operations tables of the ring* 5, þ , � *.*

*Application of Discrete Mathematics for Programming Discrete Mathematics Calculations DOI: http://dx.doi.org/10.5772/intechopen.102990*

#### **Table 8.**

*Operations tables of the ring* ð Þ 6, þ , � *.*

½ � *n*1 6¼ 0 so as ½ � *n*1 6¼ 0 but ½ � *n*1 ½ �¼ *n*2 0 means that *Zn* is not an integer domain since it has proper divisors of 0 and therefore cannot be a field.

In 6 the 5½ � has an inverse (it is a unit) and on the other hand, 3½ � is a proper divisor of 0. It is useful and necessary to be able to detect which elements have an inverse (ie they are units) when *n* is composite. For this, the following theorem is established.

**Theorem** 1.42 In *n*, ½ � *a* has an inverse (it is a unit) if and only if gcdð Þ¼ *a*, *n* 1. **Proof:** If gcdð Þ¼ *a*, *n* 1 the proof will be the same as the theorem??. In the reverse direction, let ½ � *a* ∈*n* and ½ � *a* �1 ¼ *s*. So ½ � *a* ½�¼*s* 1, so then *as* � 1 mod ð Þ *n* and *as* ¼ 1 þ *tn* for some *t*∈ but 1 ¼ *as* þ *n*ð Þ) �*t* gcdð Þ¼ *a*, *n* 1.

In the following example, we illustrate the application of Euclid's algorithms to obtain multiplicative inverses of elements of a modular field-type group.

**Example** 1.43 Find 25 ½ ��1 in 72. Since gcd 25, 72 ð Þ¼ 1 Euclid's gcd algorithm we get the following:

$$72 = 2(25) + 22, 0 < 22 < 25$$

$$25 = 1(22) + 3, 0 < 3 < 22$$

$$22 = 7(3) + 1, 0 < 1 < 3$$

since 1 is the l0 ast non-zero remainder, we have

$$\mathbf{1} = \mathbf{2}\mathbf{2} - 7(\mathbf{3}) = \mathbf{2}\mathbf{2} - 7[\mathbf{25} - \mathbf{22}] = -7(\mathbf{25}) + 8(\mathbf{22}) = -7(\mathbf{25}) + 8[\mathbf{72} - \mathbf{2(25)}] = 8(\mathbf{72}) - \mathbf{23}(\mathbf{25})$$

but

1 ¼ 8 72 ð Þ� 23 25 ð Þ) 1 � �ð Þ 23 þ 72 ð Þ 25 ð Þ mod72 , so then 1½ �¼ ½ � 49 ½ � 25 then ½ �¼ 49 ½ � 25 �1 in 72.

For a more detailed exposition of the previous results in modular arithmetic, greatest common divisor and Euclid's algorithm, it is recommended to consult these bibliographical references [3, 6, 9, 10].

After this brief overview of the properties of the algebraic ring structure as well as how to work with it in the context of modular arithmetic, we are ready to discuss the advantage of applying results from discrete mathematics to perform calculations of modular arithmetic efficiently. As already explained in the previous paragraphs of this subsection, we can, from the tables of the modular ring in which we are working, obtain all the multiplicative inverses of the ring as well as all the proper divisors of zero in the event that the ring does not be a field. We can do this by generating the sum and product tables of the ring and going through the multiplication table detecting a box where the value is equal to 1 in the case of multiplicative inverses, or that the box is equal to 0 in the case of proper divisors of zero. This procedure for obtaining the proper divisors of zero could be carried out with the following Matlab program.

```
function Fz = FacPropZero(n)
   % Modulo n [S, P] = TabOpMod (n)
   % Search on the table P all those factors whose product is equal to 0
   % CRL 29/sep/2021
   [S,P] = TabOpMod(n);
   [r,c] = size(P);
   k=1;
   for i=1:r
      for j=1:r
          if (P(i,j)==0)
            Fz{1,k}=[i-1,j-1];
            k=k+1;
          end
       end
   end
   end
   The procedure for obtaining the ring elements that have multiplicative inverse
could be carried out with the following Matlab program.
```

```
function U = unitsZn(n)
% Modulo n [S,P] = TabOpMod(n)
% Search on the table P all those factors whose product is equal to 1
% CRL 22/sep/2021
[S,P] = TabOpMod(n);
[r,c] = size(P);
k=1;
for i=1:r
    for j=1:r
         if (P(i,j)==1)
            U{1,k}=[i-1,j-1];
            k=k+1;
         end
    end
end
end
```
*Application of Discrete Mathematics for Programming Discrete Mathematics Calculations DOI: http://dx.doi.org/10.5772/intechopen.102990*

The results of the execution of the Matlab function for obtaining the list of elements of the modular ring that have multiplicative inverse for the case of *n* ¼ 5 and *n* ¼ 6 are the following:

```
>> U=unitsZn(5)
U =
  1�4 cell array
    {1�2 double} {1�2 double} {1�2 double} {1�2 double}
>> U{1,1}
ans =
      1 1
>> U{1,2}
ans =
    2 3
>> U{1,3}
ans =
    3 2
>> U{1,4}
ans =
    4 4
>> U=unitsZn(6)
U =
    1�2 cell array
       {1�2 double} {1�2 double}
>> U=unitsZn(6)
U =
    1�2 cell array
       {1�2 double} {1�2 double}
>> U{1,1}
ans =
    1 1
>> U{1,2}
ans =
    5 5
```
The above Matlab routines look good and bad from an algorithmic perspective. The good side is that since it is a direct translation of the operations table generation of a modular ring, the programming of the routines is relatively simple. The negative aspect is in the fact that the size of the arrays to generate the tables can become very large as the number *n* with respect to which the module is taken grows a lot, that is, the memory occupied will be of the order of *O n*2 ð Þ. It is at this point that the properties and results of discrete mathematics in the subject of modular arithmetic come to the rescue and allow us to obtain more efficient algorithms for obtaining proper divisors of zero as well as the list of elements of the modular ring that have an inverse multiplicative. The efficient algorithm for obtaining the list of elements of a modular ring with multiplicative inverse and the respective Matlab implementation will make use of discrete mathematics results, such as Euclid's algorithm for obtaining the greatest common divisor, as well as the version extension of this, and that will be called by a routine that solves linear modular equations of the type *ax* ¼ *b*mod*n*. The multiplicative inverse *x* of a number *a* modulo *n* will correspond to the solution of the modular linear equation *ax* ¼ 1mod*n*. Based on theorem 34, we can implement the Euclid's algorithm in Matlab as follows.

```
function y = mcdEuclides(a,b)
% Recursive function for the GCD using Euclid's algorithm
% The first parameter musy be bigger than the second parameter
if a < b
  temp=a;
  a=b;
  b=temp;
end
if b == 0
  y=a;
else
    y=mcdEuclides(b, mod(a,b));
end
end
```
Euclid's algorithm can be extended to obtain the coefficients *a* and *b* of *x* and *y* in the relation *d* ¼ *gcd a*ð Þ¼ , *b ax* þ *by*. This will be used for obtaining multiplicative inverses of a modular ring. The Matlab implementation of the extended Euclid algorithm is as follows.

```
Function [d,x,y] = mcdEuclidesExtend(a,b)
% Extended GCD Euclid's algorithm
if a < b
  temp=a;
  a=b;
  b=temp;
end
if b == 0
  d=a;
  x=1;
  y=0;
else
  [d1,x1,y1]=mcdEuclidesExtend(b, mod(a,b));
  d=d1;
  x=y1;
  y=x1-floor(a/b)*y1;
end
end
```
As already mentioned, obtaining the elements of a modular ring that have a multiplicative inverse can be reduced to the problem of calculating the elements *x* that satisfy the solution to the modular equation *ax* ¼ 1mod*n*. The Matlab implementation of the linear modular equation solver will be the following.

```
Function S = ModLinEqSolv(a,b,n)
% Modular linear equation solver
% That have the form ax=b mod n
% Author: Carlos Rodriguez Lucatero 29/sep/2021
[d,x,y] = mcdEuclidesExtend(a,n);
if (mod(b,d)==0)
   x0=mod((y*(b/d)),n);
   for i=0:d-1
       S(i+1)=mod(x0+(i*(n/d)),n);
   end
```
*Application of Discrete Mathematics for Programming Discrete Mathematics Calculations DOI: http://dx.doi.org/10.5772/intechopen.102990*

else S=-1; end end

If we call this routine with the parameter *b* ¼ 1, it would give us the result of the value of *x*, which is the inverse of *a* in the relation *ax* ¼ *b*mod*n*, if it exists, or it would return �1 to indicate that the element *a* of the modular ring does not have a multiplicative inverse. The following routine calls the linear modular equation solver routine to get the list of elements of the modular ring that have an inverse.

```
Function L = unitsZnV3(n)
% Get the list of units of a modular ring
% mod n calling the routine that solves linear modular equations
% Autor: Carlos Rodriguez Lucatero 14/Ene/2022
k=1;
for i=1:n
   S = ModLinEqSolv(i,1,n);
   if (S �=-1)
      L(1,k)=S;
   end
   k=k+1;
end
if (length(L)==0)
   L=-1;
end
end
```
The elements of the list in 0 correspond to the inverse of the element that does not have an inverse and therefore we would be talking about dividing elements proper to zero of the modular ring in question. The elements in the list that are not null are the inverse of the element of the modular ring represented by the position in the list. For instance, in ð Þ 5, þ , � , we have that 2 � 3 ¼ 1 and 4 � 4 ¼ 1.

>> L = unitsZnV3(5) L = 1324 >> L = unitsZnV3(6) L = 1 000 5

### **6. Conclusions**

In this chapter, we address some calculation problems that take place in discrete mathematics and how we can use the theoretical results provided by discrete mathematics itself to be able to carry out these calculations more efficiently. In this chapter, we were able to verify the great utility of the use of generating functions to carry out some calculations more efficiently. Some interesting applications of the use of generating functions to count graphs with some property can be found in refs. [11, 12]. These counting techniques can later be applied to the famous probabilistic method [13–15]. Generating functions are also very useful for counting the number of possible partitions of integers. An excellent text that addresses this interesting topic is [16].

It is true that computers have not stopped increasing their storage capacity as well as the speed of their processing units. However, these resources are not infinite and there are even calculation-intensive problems in topics, such as combinatorics or modular arithmetic that could quickly exhaust these calculation resources. That is why it is worth taking advantage, when possible, of the theoretical results that discrete mathematics itself provides, to efficiently carry out the calculations that it requires. For this, we take two specific topics of discrete mathematics where that is possible. One is the obtainment of elements in arbitrary positions of a numerical sequence and the other topic was the obtainment of elements with multiplicative inverse of the elements that are proper divisors of zero in a modular ring. We illustrate this by programming functions in Matlab. I hope that the chapter will convince you of the goodness of taking advantage of the mathematical results offered by discrete mathematics to perform efficient calculations in the area of discrete mathematics.

#### **Acknowledgements**

I would like to thank the Universidad Autonoma Metropolitana, Unidad Cuajimalpa for the support they gave me to make and publish this book chapter.

#### **Author details**

Carlos Rodriguez Lucatero Universidad Autónoma Metropolitana Unidad Cuajimalpa, CDMX, Mexico

\*Address all correspondence to<EMAIL_ADDRESS>
© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Application of Discrete Mathematics for Programming Discrete Mathematics Calculations DOI: http://dx.doi.org/10.5772/intechopen.102990*

#### **References**

[1] Rodríguez-Lucatero C. The Moser's formula for the division of the circle by chords problem revisited. 2017. Available from: https://arxiv.org/abs/1701.08155v1

[2] Feller W. An Introduction to Probability Theory and its Applications. Vol. I. New York, USA: Wiley and Sons Inc; 1968. pp. 52-53

[3] Cormen TH, Leiserson CE, Rivest RL, Stein C. Introduction to Algorithms. 3d ed. Massachusetts, USA: The MIT Press; 2009

[4] Miller CD, Heeren VE, Hornsby J. Matematica, razonamiento y aplicaciones. USA: Pearson; 2013

[5] Sedgewick R, Flajolet P. An Introduction to the Analysis of Algorithms, Second Printing. USA: Addison-Wesley; 2001

[6] Grimaldi RP. Discrete and Combinatorial Mathematics: An Applied Introduction. 3rd ed. USA: Addison-Wesley; 1994

[7] Graham RL, Knuth DE, Patashnik O. Concretel Mathematics, 6th Printing. USA: Addison-Wesley; 1990

[8] Wilf HS. Generatingfunctionology. 3th ed. Massachusetts, USA: A. K. Peters Ltd.; 2006

[9] Hardy GH, Wright EM. Introduction to the Theory of Numbers. 5th ed. Oxford, UK: Oxford Science Publications, reprinted; 1998

[10] Vinográdov I. Fundamentos de la Teora de números. URSS: Moscu: Editorial MIR; 1977

[11] Harary F, Palmer EM. Graphical Enumeration. New York, NY, USA; London, UK: Academic Press; 1973

[12] Rodríguez-Lucatero C. Combinatorial Enumeration of Graphs. Rijeka: IntechOpen; 2019

[13] Erdös P. Graph theory and probability. Canadian Journal of Mathematics. 1959;**11**:34-38

[14] Alon N, Spencer JH. The Probabilistic Method. 2nd ed. New York Wiley-Interscience; 2000

[15] Rodríguez-Lucatero C, Alarcón L. Use of enumerative combinatorics for proving the applicability of an asymptotic stability result on discretetime SIS epidemics in complex networks. MDPI Mathematics Open access Journal. 2019;**7**(1). DOI: 10.3390/math7010030

[16] Andrews GE. In: Rota GC, editor. The Theory of Partitions Encyclopedia of Mathematics and its Applications. Vol. 2. USA: Addison-Wesley; 1976

#### **Chapter 5**

## A Criticality Study of Fast Critical Experimental Benchmarks Using MCNP Code to Qualifying Different Evaluations

*Sanae El Ouahdani, Hamid Boukhal, El Mahjoub Chakir, Ahmed Gaga, Houda Elyaakoubi, Mustapha Makhloul, Abdelaziz Ahmed, Abdessamad Didi and Mohamed Bencheikh*

#### **Abstract**

In this chapter we present our MCNP modeling, concerning fast critical experimental benchmarks, about qualifying our libraries of cross-sections deduced from the evaluations ENDF/B-VII, JEFF-3.1, JENDL-3.3, JENDL-4 processed by the code NJOY. The benchmarks analyzed are characterized by simple geometries which help to have a precise calculation. In our neutron calculation, we used the MCNP code (version 5), the reference code for the neutron transport calculation with the Monte Carlo method. It is also very efficient for criticality calculation. The cross-section data for all the isotopes that make up the material of the studied benchmarks are processed in ACE format at 300 K temperature using the NJOY 99.9 modular system. A detailed comparison of the criticality results of our simulation was carried out to highlight the influence of these evaluations on the keff calculations.

**Keywords:** benchmark, MCNP code, NJOY code, ENDF/B-VII, multiplication factor, criticality

#### **1. Introduction**

The effective multiplication factor keff is an important parameter in the design, control and safety of reactors. For safety considerations, the keff is desired to be very close to one throughout the core life. The calculation of the keff is rather a complicated problem due to the contributions of different physical phenomena related to the neutron's population change. That is why it is important to validate any reactor calculation tool and any nuclear data library with an accurate prediction of this parameter.

The main objective of the present work is to perform the qualification and analysis of the most recent nuclear data libraries available to the scientific community, in particular; ENDF/B-VII [1], JENDL-4.0 [2], JENDL-3.3 [3], and JEFF-3.1 [4], to check the accuracy of cross-section libraries for the criticality calculations. For this objective

a set of critical fast benchmarks highly enriched uranium and with 233U and with 239Pu fuel rods were used to consider as closely as possible all types of geometries to simulate the criticality coefficient of interest. The continued energy cross sections necessary for the present work were processed by the NJOY system (version 99.9, update 364) [5] in the ACE format. The analysis and the interpretation of the results were reinforced by a comparison study of the parameter with the experimental values excerpt from the literature [6]. These experiments have already been analyzed by the Monte Carlo code MCNP using the American nuclear data ENDF/B-V continuous [6] and other codes.

The first part of the paper is reserved to explain the methodology and the materials used. The materials used are the MCNP code and the NJOY code and the JANIS code. The second part cites the characteristics of the different benchmarks selected for the present study. In the third part, we develop our results obtained about the simulation of the keff parameter and their interpretations concerning the qualification of the used libraries. We finished with a conclusion.

#### **2. Methods and materials**

#### **2.1 MCNP code**

The MCNP code [7] (Monte-Carlo N Particle transport), is a code that deals with the transport of neutrons, photons and electrons or coupled/photon/electron by the Monte-Carlo method, including the possibility of calculating the values clean for critical systems. The code deals with an arbitrary three-dimensional configuration with materials in geometric cells delimited by surfaces.

The code takes into account the processed cross-sections, for neutrons, all reactions, which are proposed in particular data evaluations (for example, ENDF/B-VI), thermal neutron scattering which is treated both by the model of the free gas and S (α, β), for photons the code takes into account incoherent and coherent diffusions, the possibility of fluorescence emission after the photoelectric effect, absorption in the production of pairs with a local broadcast of annihilation radiation. It can also treat the braking radiation emitted. In this way, MCNP is qualified as a three-dimensional, continuous energy code, thus it has been proven to simulate physical phenomena correctly. The series of important features that make MCNP very flexible and easy to use code is that it includes a powerful general source, criticality source, surface source, geometry and output pointing plotters, a rich collection of variance reduction techniques, the desired result structure called "tally" and has a large collection of cross-section data.

#### **2.2 NJOY code**

The NJOY nuclear data processing code [5] is a system developed at the Los Alamos laboratory in the USA since 1974. It is a modular code allowing, from the assessments of so-called basic nuclear data, to create specific or multigroup parameters (multigroup cross sections, fission spectra, etc…) because the information contained in these files is, such as it cannot be, exploited directly by the various transport codes MCNP, WIMS, APPOLO, EPRI-CELL… etc. The role of the NJOY system is to process this information and make it usable by these codes. The data processed by this system are then stored in files in a standardized ENDF (Evaluated Nuclear Data File) format.

*A Criticality Study of Fast Critical Experimental Benchmarks Using MCNP Code to Qualifying… DOI: http://dx.doi.org/10.5772/intechopen.102449*

#### **2.3 JANIS**

The enormous amount of data stored in the standard ENDF format files as well as the different versions or evaluations do not always allow easy access to the information desired by the user for a particular application. JANIS (Java-based Nuclear Information Software) [8] is a program designed to facilitate the visualization and manipulation of nuclear data. It was developed by the "OECD Nuclear Energy Agency", the "CSNSM-Orsay" and the University of Birmingham as an extension of the JEF-PC program. The main objective of this program is to allow the user to access the numerical values and the graphic representation of the various data without any prior information on the ENDF format. It gives maximum flexibility for the comparison of different types of nuclear data.

#### **3. The fast critical benchmarks**

#### **3.1 The benchmarks**

The benchmarks are fixed points of reference used to test the results of modeling and theoretical calculations and to validate nuclear data.

There are two types of benchmarks:

#### **3.2 Characteristics of the fast benchmarks used**

The benchmarks analyzed cover different and simple geometries (spherical, cylindrical, and parallelepiped), with or without reflector, and concern the three main fissile nuclei 235U, 233U, 239Pu in metallic form.

Fast benchmarks use a fast neutron spectrum that covers the energy range greater than 100 keV, and are therefore characterized by very high fission and capture percentages in the fast energy domain.

By way of example, **Tables 1**–**3** give the average percentages of the flux as well as the fission and capture rates in the following three energy intervals [6]:

**Table 1.**

*Average percentages of flux in the three energy intervals.*

#### **Table 2.**

*Average percentages of fissions caused by neutrons in the three energy fields.*

#### **Table 3.**

*Average percentages of neutron capture in the three energy domains.*

#### **3.3 Description of the fast benchmarks studied**

As we mentioned before, to qualify our cross-section libraries as well as the modeling method, we have chosen a series of critical fast experimental benchmarks which cover different geometries and relate to the three main fissile nuclei 235U, 239Pu and 233U. These benchmarks are derived from the International Handbook of Critical Benchmarks published by the nuclear energy agency AEN [6].

#### *3.3.1 Fast benchmarks highly enriched in U-235 (HEU-MET-FAST)*

We have processed a series of 20 highly enriched benchmarks known with HEU-MET-FAST that is chosen carefully with simple geometries. It includes GODIVA, TOPSY, FLATTOP and HEU-MET-FAST-xxx.

**HEU-MF-001**: GODIVA (1950–1959, LANL, USA), sphere containing metallic uranium highly enriched in the isotope 235U (93.71% wt\*).

\*wt = mass fraction.

**HEU-MF-002**: TOPSY-8 (1950, LANL, USA), assemblages of different geometry (depending on the case) containing uranium highly enriched in the 235U isotope (93.55% wt) reflected by natural uranium, (6 cases).

**HEU-MF-003**: ORALLOY (1950, LANL, USA), spherical assemblies containing metallic uranium highly enriched in 235U (93.5% wt), reflected by reflectors of different types and thicknesses depending on the case (12 cases): seven spheres are reflected by 5.08, 7.62, 10.16, 12.7, 17.78, 20.32, 27.94 cm of natural uranium, four

*A Criticality Study of Fast Critical Experimental Benchmarks Using MCNP Code to Qualifying… DOI: http://dx.doi.org/10.5772/intechopen.102449*

spheres are reflected by 4.826, 7.366, 11.43, 16.51 cm of Tungsten carbon, one sphere is reflected by 20.32 cm of nickel.

**HEU-MF-028**: FLATTOP-25 (1964–1966, LANL, USA), a sphere containing metallic uranium highly enriched in the 235U isotope (93.24% wt) reflected by natural uranium.

#### *3.3.2 Fast benchmarks in U-233 (U233-MET-FAST)*

**U233-MF-001:** JEZEBEL-23 (1961, LANL, USA), sphere containing metallic uranium highly enriched in the 233U isotope (98.11% wt).

**U233-MF-002:** (1958, LANL, USA), sphere containing metallic uranium highly enriched in the 233U isotope (50.59% wt) reflected by a layer of 235 U, (2 cases, in both cases the mass varies critical).

**U233-MF-003:** (1958, LANL, USA), sphere containing metallic uranium highly enriched in the 233U isotope (98.89% wt) reflected by natural uranium, (2 cases, in both cases the critical mass varies).

**U233-MF-004:** (1958, LANL, USA), sphere containing metallic uranium highly enriched in the 233U isotope (98.2% wt) reflected by tungsten, (2 cases, in the 2 cases varies the critical fatigue and the reflector thickness).

**U233-MF-005:** (1958, LANL, USA), sphere containing metallic uranium highly enriched in the 233U isotope (98.2% wt) reflected by beryllium, (2 cases, varies the critical mass).

**U233-MF-006:** FLATTOP-23 (1964, LANL, USA), sphere containing metallic uranium highly enriched in the 233U isotope (98.13% wt) reflected by natural uranium.

#### *3.3.3 Fast benchmarks in Pu-239 (Pu-MET-FAST)*

**Pu-MF-001:** JEZEBEL-39 (1950, LANL, USA), metallic plutonium sphere enriched in the 239Pu isotope (95.17%), (4.5 at% 240Pu, 1.02 wt% Ga), without a reflector.

**Pu-MF-002:** JEZEBEL-40 (1964, LANL, USA), metallic plutonium sphere enriched in the 239Pu isotope (20.1 at% 240Pu, 1.01 wt% Ga), without a reflector.

**Pu-MF-005:** (1958, LANL, USA), metallic plutonium sphere enriched in the 239Pu isotope (94.76%), reflected by tungsten.

**Pu-MF-006:** FLATTOP-39 (1964–1966, LANL, USA), metallic plutonium sphere highly enriched in the 239Pu isotope (94.84% wt) reflected by natural uranium.

**Pu-MF-008:** THOR (1960–1961, LANL, USA), metallic plutonium sphere highly enriched in the 239Pu isotope (94.54% wt), reflected by thorium.

**Pu-MF-009:** (1960, LANL, USA) plutonium metallic sphere highly enriched in the 239Pu isotope (94.8% wt), reflected by aluminum.

**Pu-MF-010:** DELTA-PHASE (1958, LANL, USA): metallic plutonium sphere highly enriched in the 239Pu isotope (94.76% wt), reflected by natural uranium.

**Pu-MF-011:** ALPHA-PHASE (1968, LANL, USA), metallic plutonium sphere highly enriched in 239Pu (94.4% wt) reflected by light water.

**Pu-MF-018:** DELTA-PHASE (1958, LANL, USA), metallic plutonium sphere highly enriched in the 239Pu isotope (94.7% wt) reflected by beryllium.

**Pu-MF-023:** (1962, VNIIEF, Russia), metallic plutonium sphere highly enriched in the 239Pu isotope (98.19%), reflected by the graphite.

**Pu-MF-024:** (1964, VNIIEF, Russia), metallic plutonium sphere highly enriched in the 239Pu isotope (98.19%), reflected by polyethylene.

**Pu-MF-025:** (1964, VNIIEF, Russia), metallic plutonium sphere highly enriched in the 239Pu isotope (98.19%), reflected by stainless steel (1.55 cm).

**Pu-MF-026:** (1962, VNIIEF, Russia), metallic plutonium sphere highly enriched in the 239Pu isotope (98.19%), reflected by stainless steel (11.9 cm).

**Pu-MF-027:** (1965, VNIIEF, Russia), metallic plutonium sphere highly enriched in the 239Pu isotope (89.66%), reflected by polyethylene.

**Pu-MF-028:** (1965, VNIIEF, Russia), spherical assembly in metallic plutonium highly enriched in the 239Pu isotope (89% wt) reflected by stainless steel.

**Pu-MF-029:** (1965, VNIIEF, Russia), spherical assembly in metallic plutonium highly enriched in the 239Pu isotope (88% wt), without a reflector.

**Pu-MF-030:** (1965, VNIIEF, Russia), spherical assembly in metallic plutonium highly enriched in the 239Pu isotope (88% wt) reflected by the graphite.

**Pu-MF-031:** (1965, VNIIEF, Russia), spherical assembly in metallic plutonium highly enriched in the 239Pu isotope (88% wt) reflected by polyethylene.

**Pu-MF-032:** (1965, VNIIEF, Russia), spherical assembly in metallic plutonium highly enriched in the 239Pu isotope (88% wt) reflected by stainless steel.

#### **4. Results and interpretations**

For the calculation of the keff parameter, we used the MCNP code based on the Monte Carlo method. The Monte Carlo method solves the transport equation in integral form. The latter is based on the random selection of several variables and after the estimation of their mathematical expectation which is equivalent to the value of the physical quantity sought. It simulates the history of each neutron through the different interactions it can have in the media where it propagates.

In the present calcul, we simulated 1500 cycles of 30,000 neutrons each, the first 50 cycles are used to ensure the homogeneity of the source distribution. With this number of simulated stories, all keff results are obtained with a standard deviation between +/− 9 and +/− 12 pcm.

### **5. Case of fast benchmarks highly enriched in 235U**

The experimental values obtained for the various Benchmarks concerning the effective multiplication factor keff, as well as the average deviations from experience are shown and compared to the experience in **Figures 1** and **2**.

**Figure 1** represents the variation of keff according to the cases for the fast benchmarks very highly enriched in 235U, from this figure we notice that for the majority of the cases studied, the ENDF/B-VII and JEFF-3.1 evaluations give results that are in good agreement with experience. The average deviation from experience is in the order of 0.42% for ENDF/B-VII and 0.39% for the JEFF-3.1 evaluation: all the libraries keep the same difference between themselves and the same behavior for the benchmarks reflected by natural uranium, except for the benchmarks from HEU-MF-008 to HEU-MF-011 which are reflected by tungsten carbide and HEU-MF-012 reflected by nickel. We also note that the JENDL-3.3 evaluation underestimates the criticality in most cases, with an average deviation from experience equal to 0.6%. However, there is a marked improvement when upgrading the evaluation from JENDL-3.3 to JENDL-4. Although, we still have an underestimation compared to the other evaluations of JENDL-3.3 and JENDL-4. We notice an overestimation of keff for all the evaluations

*A Criticality Study of Fast Critical Experimental Benchmarks Using MCNP Code to Qualifying… DOI: http://dx.doi.org/10.5772/intechopen.102449*

**Figure 1.** *keff depending on the case for the fast benchmarks very highly enriched in 235U.*

**Figure 2.** *The |C-E|/E ratio of keff for each evaluation.*

concerning the benchmarks reflected by the Tungsten carbide which contains the carbon the problem probably stems from a poor underestimation of the carbon capture cross-sections especially in the energy interval of 5 keV to 5 MeV of the capture cross-section where JENDL-4 over estimates ENDF/B-VII and JEFF-3.1.

#### **5.1 Fast benchmarks in U-233**

**Figures 3** and **4** represent the variation of keff according to the cases for the fast benchmarks in 233U isotope as well as the average deviations from the keff experiment.

**Figure 3.** *keff depending on the case for the fast benchmarks in 233U.*

#### **Figure 4.** *The |C-E|/E ratio of keff for the fast benchmarks in 233U.*

From **Figures 3** and **4** we find that the best results of criticality are given by JENDL-4 with a deviation from the experience of 0.16%, secondly, we find ENDF/B-VII with a deviation by compared to the experience of 0.26% we note an improvement during the transition from JENDL-3.3 to JENDL-4. We also notice an overestimation of JEFF-3.1 of the criticality with a deviation from the experience of 0.39%.

#### **5.2 Fast benchmarks in Pu-239**

**Figures 5** and **6** represent the variation of keff according to the cases for the fast benchmarks in 239Pu isotope as well as the average deviations from the experience.

In **Figures 5** and **6**, the variation of keff, shows that the process based on ENDF/ B-VII and JEFF-3.1 gives results that are in good agreement with the experimental values, the deviations from the experiment are 0.34% and 0.33% respectively, with

*A Criticality Study of Fast Critical Experimental Benchmarks Using MCNP Code to Qualifying… DOI: http://dx.doi.org/10.5772/intechopen.102449*

**Figure 5.** *keff depending on the case for the fast benchmarks in Pu-239.*

the exception of JENDL-3.3 and JENDL-4 which have deviations from criticality greater than the other libraries of 0.39% and 0.38% respectively. Apparently, JENDL-3.3 gives keffs far from 1 compared to other libraries in the Pu-MF-006 and Pu-MF-008 and Pu-MF-010 benchmarks as we notice that the problem is corrected in JENDL-4, and JENDL-4 far from 1 compared to other libraries in benchmarks Pu-MF-026, Pu-MF-28 and Pu-MF-32, We note that there is a deterioration during the transition from JENDL-3.3 to JENDL-4 in these three benchmarks. At the three benchmarks Pu-MF-11, PU-MF-27 and PU-MF-31 all the libraries have criticality estimates.

The Pu-MF-26, 28 and 32 benchmarks use stainless steel as a reflector, so the JENDL-4's underestimation of criticality compared to other libraries and due to the overestimation of JENDL-4 to other libraries in the cross-section of carbon capture.

**Figure 6.** *The |C-E|/E ratio of keff in the case of fast benchmarks in Pu-239.*

### **6. Conclusions**

In this work we were able to model rapid critical benchmarks using the main fissile nuclei which are, the 235U, the 233U, and the 239Pu, we previously generated crosssections using the NJOY code, these cross-sections come from the main evaluations ENDF/B-VII, JEFF-3.1, JENDL-3.3 and JENDL-4.

The Monte Carlo calculation that we carried out consisted in determining the keff parameter, the difference between the calculation and the experiment depends mainly on the type of evaluation used as well as the fissile core of the benchmarks considered this difference remains acceptable all the same. So that we can say, that our results are in good agreement with those obtained experimentally.

### **Author details**

Sanae El Ouahdani1 \*, Hamid Boukhal2 , El Mahjoub Chakir3 , Ahmed Gaga1 , Houda Elyaakoubi2 , Mustapha Makhloul2 , Abdelaziz Ahmed4 , Abdessamad Didi5 and Mohamed Bencheikh6

1 Polydisciplinary Faculty, LRPSI Laboratory, Physics Department, Sultan Moulay Slimane University, Beni Mellal, Morocco

2 Faculty of Sciences, ERSN, Abdelmalek Essaadi University, Tetouan, Morocco

3 Faculty of Sciences, LHESIR, Ibn Tofail University, Kenitra, Morocco

4 Faculty of Lawder, Physics Department, University of Abyan, Abyan, Yemen

5 National Center for Energy Sciences and Nuclear Techniques, Rabat, Morocco

6 Faculty of Sciences and Technologies, Physics Department, Mohammedia Hassan II University of Casablanca, Mohammedia, Morocco

\*Address all correspondence to<EMAIL_ADDRESS>
© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*A Criticality Study of Fast Critical Experimental Benchmarks Using MCNP Code to Qualifying… DOI: http://dx.doi.org/10.5772/intechopen.102449*

#### **References**

[1] Chadwick MB et al. Nuclear data sheets. 2006;**107**(12):2931-3060

[2] The JEFF-3.1 Nuclear Data Library. JEFF Report 21. ISBN 92-64-02314-3

[3] Shibata K et al. Japanese evaluated nuclear data library version 3 revision-3: JENDL-3.3. Journal of Nuclear Science and Technology. 2002;**39**:1125

[4] Shibata K et al. JENDL-4.0: A new library for nuclear science and engineering. Journal of Nuclear Science and Technology. 2011;**48**(1):1-30

[5] NJOY 99.9. A Code System for Producing Pontwise and Multigroup Neutron and Photon Cross Sections for ENDF/B, Evaluated Nuclear Data. New Mexico: Los Alamos National Laboratory. 1999

[6] International Handbook of Evaluated Criticality Safety Benchmark Experiments. NEA/NSC/DOC (95). Vol. I-VII. 2003

[7] MCNPXTM user's manual. Version 2.7.0. LA-CP-11-00438. Manuel/Los Alamos National Laboratory; 2011

[8] JANIS 3.0 user's guide. 2007

#### **Chapter 6**

## Applications of Fuzzy Set and Fixed Point Theory in Dynamical Systems

*Praveen Kumar Sharma, Shivram Sharma, Jitendra Kaushik and Palash Goyal*

#### **Abstract**

This chapter shall discuss various applications of fixed-point theory and fuzzy set theory. Fixed point theory and fuzzy set theory are very useful tools that are applicable in almost all branches of mathematical analysis. There are many problems that cannot be solved by applying the concept of other existing theories but can be solved easily by using the concept of fuzzy set theory and fixed point theory. So here in this chapter, we shall introduce the fuzzy set theory and fixed point theory concerning their applications in existing branches of science, engineering, mathematics, and dynamical systems.

**Keywords:** fixed point, fuzzy set, dynamical systems, stability, fuzzy differential equations, integral equations

#### **1. Introduction**

Fixed point theory is an area of mathematics linked to functional analysis and topology that is still in its infancy. Fixed point theory is an important subject in the fast-growing domains of nonlinear analysis and nonlinear operators. It is a relatively new scientific area that is developing rapidly. In topics as diverse as differential equations, topology, economics, game theory, dynamics, optimal control, and functional analysis, fixed points and fixed point theorems have always been important theoretical tools. Furthermore, with the development of accurate and efficient techniques for computing fixed points, the concept's relevance for applications has expanded dramatically, making fixed point methods a vital weapon in the arsenal of the applied mathematician.

Set theory, general topology, algebraic topology, and functional analysis are just a few of the major fields of mathematics that give natural settings for fixed point theorems. Approximation theory, potential theory, game theory, mathematical economics, theory of differential equations, and other disciplines use fixed point theorems to solve problems in approximation theory, potential theory, game theory, mathematical economics, and so on. It is possible to evaluate various problems from science and engineering using fixed point approaches when one is concerned with a system of differential/integral/functional equations. This method is particularly beneficial when dealing with control system issues and the idea of elasticity.

Fixed point theorems are the most important tools for proving the existence and uniqueness of solutions to various mathematical models (differential, integral, and partial differential equations, variational inequalities, and so on), which represent phenomena arising in multiple fields such as steady-state temperature distributions, chemical reactions, Neutron transport theories, economic theories, epidemics, and fluid flow. They are also employed to look at the difficulty of determining the best central for these systems.

Let *F* : *X* ! *X* represent a function on the set *X*. A point *x*∈ *X* is called a fixed point of *F* if *F x*ð Þ¼ *x*, that is, a point, which remains invariant under the transformation *F*, is called a fixed point, and fixed point theorems are theorems that deal with the attributes and existence of fixed points. If *F* is a function defined on the real numbers as *F x*ð Þ¼ *x* þ 2, then it has no fixed points since *x* is never equal to *x* þ 2 for any real number.

Let *F* : ½ �! 0, 1 ½ � 0, 1 be defined by *x=*10 then *F* ð Þ¼ 0 0. Hence 0 is a fixed point of *F*.

Poincare [1] was the first to establish the fixed-point theory in 1886. He arrived at the first result on a fixed point using a continuous function.

Browder [2] proved the following useful theorem in 1912. Browder's fixed point theorems are fundamental in fixed point theory and its applications. Browder's fixedpoint theorem states, "If *C* is a unit ball in *En* (Euclidean n-dimensional space) and *T* : *C* ! *C* a continuous function. Then *T* has a fixed point in *C*, or *Tx* ¼ *x* has a solution."

The Particular Case of this theorem on the real line can be stated in the following way.

Let *T* : ½ �! 0, 1 ½ � 0, 1 be a continuous function. Then *T* has a fixed point. Schauder proved the following theorem for compact maps.

Let *X* be a Banach space and let *C* be a closed, bounded subset of *X:* Let *T* : *C* ! *C* be a compact map. Then *T* has at least one fixed point in *C*.

This theorem is important in the numerical treatment of equations in analysis.

Banach [3] investigated the concept of contraction type mappings in metric space in 1922. Using the condition of contraction mapping, he established an interesting conclusion in metric space. "Every contraction mapping of a complete metric space into itself has a unique fixed point," according to the Banach contraction principle.

A contraction mapping is continuous, but a continuous map is not necessarily a contraction.

For example, a translation map *T* : *R* ! *R* is defined by *Tx* ¼ *x* þ *p*, *p*>0, is continuous but not contraction.

The Banach contraction principle has a lot of uses, but it has one major flaw: It requires the function to be consistent throughout the space. Kannan [4] proved the improved conclusion in fixed point theory to avoid this flaw. The Banach contraction principle has a lot of uses, but it has one major flaw: It requires the function to be consistent throughout the space. Kannan [5] proved the improved conclusion in fixed point theory to avoid this flaw. He proved that "let *F* be a self-mapping of complete metric space *X* satisfying the following inequality

*d Fx* ð Þ , *Fy* ≤ *α* ½ � *d x*ð Þþ , *Fx d y*ð Þ , *Fy for all x*, *y*∈ *X*, 0<*α* < 1*=*2*:*

Then *F* has a unique fixed point."

#### *Applications of Fuzzy Set and Fixed Point Theory in Dynamical Systems DOI: http://dx.doi.org/10.5772/intechopen.105830*

Jungck [6] generalized Banach's fixed point theorem by proving a common fixed point theorem for commuting maps. The Banach fixed point theorem has many applications, but it has one flaw: The definition necessitates function continuity.

In 1982, Sessa [7] refined Jungck's result and proposed the concept of weakly commuting mappings in metric space, demonstrating that "two commuting mappings also commute weakly, but two weakly commuting mappings are not certainly commuting."

Jungck [8] achieved a breakthrough when he declared the new concept of "compatibility" of mappings and demonstrated its utility in achieving a common fixed point of mappings. Every weak commutative pair of mappings is compatible, according to Jungck [9], but the opposite does not have to be true. In his study [10], Singh points out that commutativity does not entail the presence of a series of points that satisfy the compatibility criterion.

Jungck et al. [11] introduced the concept of compatible mappings of type (A) in 1993 and proved some common fixed point theorems.

Common fixed-point theorems for Mann-type iterations are useful for common fixed point theorems and their applications to the best approximation.

In 1999, Popa [12] proved some fixed point theorems for compatible mappings satisfying an implicit relation. Some other results of Popa have been targeted by many authors and common fixed point theorems in different spaces on using implicit relations.

The notion of convex metric spaces was initially introduced by Takahashi [13]. He and others gave some fixed point theorems for non-expansive mappings in convex metric spaces.

In the metric space setting, the strict contractive condition does not ensure the existence of a common fixed point unless the space is assumed compact or the tough conditions are replaced by stronger conditions as in [14, 15]. In 1986, Jungck [8] introduced the notion of compatible mappings. This concept was frequently used to prove the existence of theorems in common fixed point theory. However, the study of common fixed points of non-compatible mappings is also very interesting.

In an attempt to study fixed points of non-self-mappings, Assad and Kirk [16] gave sufficient conditions for such mappings to have a fixed point by proving a result for multivalued mappings in convex metric spaces. Naimpally et al. [17] proved fixed point theorems in convex metric spaces.

Gähler [18] introduced the concept of 2-metric space and further studied 2-metric and other spaces in [19, 20]. He defined 2-metric space as a real-valued function of a point triples on a set X, whose abstract properties were suggested by the area function in Euclidean space. It is natural to expect 3-metric space, which is indicated by the volume function.

In 1998, Pant [21] introduced the notion of R-weakly commuting maps and point-wise R-weakly commuting maps in metric spaces. He has observed that two self-maps on a metric space can fail to be point-wise R-weakly commuting only if they possess a coincidence point at which they do not commute.

The systematic study of fixed points of multivalued mappings had been started with the work of Nadler [22] in 1969, who proved that any multivalued contractive mapping of a complete metric space X into the family of a closed bounded subset of X has a fixed point. Ciric [23] was the first to prove the most general fixed point theorem for a generalized multivalued contraction mapping.

Naimpally et al. [1] obtained some interesting results on fixed point and coincidence point theorems for a hybrid of multivalued and single-valued maps satisfying a contraction condition.

In 1942, Menger [24] suggested associating a distribution function in place of a distance function to any two points in metric space and introduced the concept of probabilistic metric space under statistical metric space.

Sehgal [25] initiated the study of contraction mapping on probabilistic metric space in 1966. Sehgal and Bharucha-Reid [26] proved the Banach contraction principle for probabilistic metric space, stating that "A contraction mapping on a complete probabilistic metric space has a unique fixed point."

Due to the various paradigmatic changes in science and mathematics, many changes also took place in the concept of uncertainty. One such change is the concept of uncertainty which is at the stage of transition from the traditional view to the modern view and is characterized chiefly by the theories of the uncertainty of probability theory.

An important point in the evolution of the modern concept of uncertainty was the publication of a seminal paper by Zadeh [27], a computer scientist university of California; the U.S.A. was the first who introduce the concept of fuzzy set theory in his seminal paper as a new way to represent vagueness in our everyday life. Zadeh introduced a theory whose objects fuzzy sets are sets with boundaries that are not precise. The membership in a fuzzy set is not a matter of affirmation or denial but rather a degree. This concept is being used and found to be more appropriate in solving problems of all disciplines.

The concept of fuzzy sets was initially introduced by Zadeh [27] in 1965 and has caused great interest among pure and applied mathematicians. It has also raised enthusiasm among engineers, biologists, psychologists, and economists.

Let X be a set; we define a fuzzy set X as a map *M* : *X* ! *I* ¼ ½ � 0, 1 . It is to be remarked that fuzzy sets can be regarded as a generalization of characteristic functions taking values between 0 and 1 (including 0 and 1), and the characteristic function on a set *X* is the constant mapping.

In classical set theory, a subset *A* of a set *X* can be defined by its characteristic function *χA*ð Þ¼ *x* 0, *if x* ∉ *A* and *χA*ð Þ¼ *x* 1, if *x*∈ *A:*

The mapping may be represented as a set of ordered pairs f g ð Þ *x*, *χA* ð Þ *x* with exactly one ordered pair present for each element of *X*. The first element of the ordered pair is an element of the set *X*, and the second is its value {0, 1}. The value 0 is used to represent non-membership, and the value 1 is used to describe the membership of the element *A*. The truth or falsity of the statement, *x* is in *A*, determined by the ordered pair. The statement is true if the second element of the ordered pair is 1, and the statement is false if it is 0. Similarly, a fuzzy subset *A* of a set *X* can be defined as a set of ordered pairs f g ð Þ *x*, *χA*ð Þ *x* : *x*∈*X* , each with the first element from *X* and the second element from the interval 0, 1 ½ � with exactly one ordered pair present for each component of *X*. This defines a mapping *μA* between elements of the set X and the values in the interval 0, 1 ½ �. That is, *μA* : *X* ! ½ � 0, 1 *:*

The value 0 represents complete non-membership, the value 1 represents complete membership, and the values in between represent intermediate degrees of membership.

The fuzzy set theory has an application in neural network theory, robotics, reliability, stability theory, mathematical programming, modeling theory, engineering sciences, medical sciences, image processing, control theory, communication, etc.

In the last three decades, fuzzy mathematics's development and rich growth were tremendous. A large number of authors studied applications of fuzzy set theory in different engineering branches.

Zimmermann and Sebastian [28, 29] defined knowledge base system design and intelligent system design support. Tsourveloudis et al. [30] defined machine flexibility and established a result.

For fuzzy mathematics, we refer to Bedard [31], Butnariu [32], Grabiec [33], and Weiss [34].

#### **2. Main results/discussion/application of fuzzy set and a fixed point in a dynamical system**

In this chapter, our main aim is to give an application of fuzzy sets and fixed points in a dynamic system.

**A dynamical system** is one in which a function explains the time dependency of a point in an ambient space or one in which something evolves with time. For instance, mathematical models that represent the swinging of a clock pendulum, water flow in a conduit, the quantity of fish in a lake each spring, population expansion, and so on.

A dynamic system can be described over either discrete time steps or a continuous timeline.

**Discrete-time dynamical system-**

$$\mathbf{x}\_t = F(\mathbf{x}\_{t-1}, t) \tag{1}$$

This type of model is called a difference equation, a recurrence equation, or an iterative map (if the right-hand side is not dependent on *t*)

**Continuous-time dynamical system-**

$$\frac{d\mathbf{x}}{dt} = F(\mathbf{x}, t) \tag{2}$$

This type of model is called a differential equation.

In both cases, *xt* or *x* is the system's state variable at time *t*, which may take a scalar or vector value. *F* is a function that determines the rule by which the system changes its state over time.

Dynamical systems are often modeled by differential equations.

So, here we discuss an application of fuzzy Laplace transforms to solve differential equations/fuzzy differential equations:

Fuzzy differential equations (FDEs) are a natural technique to model dynamic systems under uncertainty. One of the most basic FDEs, first-order linear fuzzy differential equations, can be found in many applications. Chang and Zadeh [35] were the first to present the fuzzy derivative notion. The concept of FDEs was used in the analysis of fuzzy dynamical problems by Kandel and Byatt [4, 36]. FDEs and their fuzzy beginning and boundary value problems are solved using the fuzzy Laplace transform approach. Fuzzy Laplace transforms make it easier to solve an FDE by converting it to an algebraic problem.

Operational calculus is an important area of applied mathematics that involves switching from calculus operations to algebraic operations on transforms. The fuzzy Laplace transform approach is practically the essential functional method for engineers. The fuzzy Laplace transform also benefits by directly addressing difficulties, fuzzy initial value problems without identifying a general solution, and non-homogeneous differential equations without first solving the corresponding homogeneous equation.

There are a number of mathematicians who studied and developed several approaches to study FIVP [37–41]. They initially used H-Differentiability for fuzzyvalued functions. Using this concept, they looked at the existence and uniqueness of the solution of FIVP [37, 41, 42]. This concept has a drawback: The fuzzy solution behaves quite differently from the crisp solution. Bede B and Gal SG [43] introduced a new idea called the strongly generalized differentiability, and it was studied and used for solving FIVPs in [44–47]. This concept allows us to overcome the drawback mentioned above. So, we use this differentiability concept to find out the solution to FIVP in this chapter.

The fuzzy Laplace transform method solves the problems of FDEs and corresponding fuzzy initial and boundary values. This method solves FIVPs/FDEs directly and gives the complete solution without determining the complementary and particular solution (one can refer to [44, 45, 48–51]. This chapter uses this technique to solve FIVPs/FDEs/ODEs.

We need some definitions and theorems given under the following to solve the fuzzy differential equation by the fuzzy Laplace transform.

**Definition 2.1.** ([46]). Let *f x*ð Þ be a continuous fuzzy-value function. Suppose that *f x*ð Þ⊙*e*�*px* is improper fuzzy Riemann integrable on 0, ½ Þ ∞ , then Ð ∞ 0 *f x*ð Þ⨀ *e*�*pxdx* is called fuzzy Laplace transforms and is denoted as

$$\begin{aligned} L[f(\mathbf{x})] &= \int\_0^\infty f(\mathbf{x}) \mathsf{C} e^{-p\mathbf{x}} d\mathbf{x} = \left( \int\_0^\infty \underline{f}(\mathbf{x}, r) e^{-p\mathbf{x}} d\mathbf{x}, \int\_0^\infty \overline{f}(\mathbf{x}, a) e^{-p\mathbf{x}} d\mathbf{x} \right) \\ &= \left( l \left[ \underline{f}(\mathbf{x}, a) \right], l \left[ \overline{f}(\mathbf{x}, a) \right] \right), \end{aligned}$$

$$\begin{aligned} \text{where } & \left[\underline{f}(\mathfrak{x}, a)\right] = \left[\prescript{\text{ess}}{f}(\mathfrak{x}, a)e^{-p\mathfrak{x}}d\mathfrak{x} \text{ , } l\left[\overline{f}(\mathfrak{x}, a)\right] = \int\_{0}^{\infty} \bar{f}(\mathfrak{x}, a)e^{-p\mathfrak{x}}d\mathfrak{x} \\ \text{(Or) } L\left[f(\mathfrak{x})\right] = \left(l\left[\underline{f}(\mathfrak{x}, a)\right], l\left[\overline{f}(\mathfrak{x}, a)\right]\right) \end{aligned}$$

**Theorem 2.2.** Chalco-Cano et al. [46] Let *f* : *R* ! *E* be a fuzzy valued function and denote *f t*ðÞ¼ *f t*ð Þ , *α* , �*f t*ð Þ , *α* � �, for each *α*∈½ � 0, 1 . Then

1. If *f* is (i)-differentiable, then *f t*ð Þ , *α* and �*f t*ð Þ , *α* are differentiable functions and *f* 0 ðÞ¼ *t f*0 ð Þ *t*, *α* , �*f* 0 ð Þ *t*, *α* � �

2. If *f* is (ii)-differentiable, then *f t*ð Þ , *α* and �*f t*ð Þ , *α* are differentiable functions and *f* 0 ðÞ¼ *t* �*f* 0 ð Þ *t*, *α* , *f*0 ð Þ *t*, *α* � �

**Formulae 2.3.** Consider the fuzzy initial value problem

$$\begin{cases} \mathcal{Y}'(t) = f(t, \mathcal{Y}(t)) \\ \mathcal{Y}(\mathbf{0}) = \left( \underline{\mathcal{Y}}(\mathbf{0}, a), \bar{\mathcal{Y}}(\mathbf{0}, a) \right), \mathbf{0} < a \le 1. \end{cases}$$

Where *f* : *R*þ � *E* ! *E* is a continuous fuzzy mapping.  Consider the initial value problem

$$\begin{cases} \quad \mathcal{Y}'(t) = \mathcal{y}(t)\mathbf{0} \le t \le T \\\quad \mathcal{Y}(\mathbf{0}) = \left(\underline{\mathcal{Y}}(\mathbf{0}, a), \bar{\mathcal{Y}}(\mathbf{0}, a)\right). \end{cases}$$

by using the fuzzy Laplace transform method, we have

*L y*0 ½ �¼ ð Þ*t L yt* ½ � ð Þ , and *L y*0 ½ �¼ ð Þ*t* Ð ∞ 0 *y*0 ð Þ*t* ⨀*e*�*ptdt* in (i)-differentiable then by using Case (i), we have *L y*0 ½ �¼ ð Þ*t* ð Þ *sL y t* ½ � ð Þ ⊖*y*ð Þ 0

Therefore, *Ly t* ½ �¼ ð Þ *sL y t* ½ �Þ ð Þ ⊖*y*ð Þ 0

$$l[\bar{\boldsymbol{y}}(t,a)] = \operatorname{s\,l}\left[\underline{\boldsymbol{y}}(t,a)\right] - \underline{\boldsymbol{y}}(0,a)$$

$$l\left[\underline{\boldsymbol{y}}(t,a)\right] = \operatorname{s\,l}[\bar{\boldsymbol{y}}(t,a)] - \bar{\boldsymbol{y}}(0,a) \quad \dots \text{\,} \tag{3}$$

Hence, the solution of system (3) is:

$$l\left[\bar{\boldsymbol{\eta}}(t,a)\right] = -\bar{\boldsymbol{\eta}}(0,a)\left(\frac{\mathfrak{s}}{\mathfrak{s}^2 - 1}\right) + \underline{\boldsymbol{\eta}}(0,a)\left(-\frac{1}{\mathfrak{s}^2 - 1}\right)$$

$$l\left[\underline{\boldsymbol{\eta}}(t,a)\right] = -\underline{\boldsymbol{\eta}}(0,a)\left(\frac{\mathfrak{s}}{\mathfrak{s}^2 - 1}\right) + \bar{\boldsymbol{\eta}}(0,a)\left(-\frac{1}{\mathfrak{s}^2 - 1}\right)$$

Thus

$$\bar{\jmath}(t,a) = -\bar{\jmath}(0,a)l^{-1}\left[\left(\frac{s}{s^2-1}\right)\right] + \underline{\jmath}(0,a)l^{-1}\left[\left(-\frac{1}{s^2-1}\right)\right]$$

$$\underline{\jmath}(t,a) = -\underline{\jmath}(0,a)\,l^{-1}\left[\left(\frac{s}{s^2-1}\right)\right] + \bar{\jmath}(0,a)\,l^{-1}\left[\left(-\frac{1}{s^2-1}\right)\right]$$

Finally, we have:

$$\begin{aligned} \bar{\chi}(t,a) &= e^{-t} \left( \frac{\underline{\chi}(0,a) - \bar{\jmath}(0,a)}{2} \right) - e^{t} \left( \frac{\underline{\wp}(0,a) + \bar{\jmath}(0,a)}{2} \right) \\ \underline{\chi}(t,r) &= e^{-t} \left( \frac{-\underline{\wp}(0,a) + \bar{\jmath}(0,a)}{2} \right) - e^{t} \left( \frac{\underline{\wp}(0,a) + \bar{\jmath}(0,a)}{2} \right) \end{aligned}$$

If *y*0 ð Þ*t* in (ii)-differentiable, then by using Case II, we have *L y*0 ½ �¼� ð Þ*t* ð Þ *y*ð Þ 0 ⊖ð Þ �*sL y t* ½ � ð Þ . Therefore, *Ly t* ½ �¼ � ð Þ ð Þ *y*ð Þ 0 ⊖ð Þ �*sL y t* ½ � ð Þ

$$l[\bar{\boldsymbol{y}}(t,a)] = \boldsymbol{s} \, l[\bar{\boldsymbol{y}}(t,a)] - \bar{\boldsymbol{y}}(0,a)$$

$$l\left[\underline{\boldsymbol{y}}(t,a)\right] = \boldsymbol{s}l\left[\underline{\boldsymbol{y}}(t,a)\right] - \underline{\boldsymbol{y}}(0,a) \quad \dots \text{ } \tag{4}$$

Hence, the solution of system (4) is:

$$l[\bar{\jmath}(t,a)] = -\bar{\jmath}(0,a)\left(\frac{1}{1+s}\right)$$

$$l\left[\underline{\jmath}(t,a)\right] = -\underline{\jmath}(0,a)\left(\frac{1}{1+s}\right)$$

Thus

$$\bar{\jmath}(t,a) = -\bar{\jmath}(0,a)l^{-1}\left(\frac{1}{\mathbf{1}+s}\right)$$

$$\underline{\jmath}(t,a) = -\underline{\jmath}(0,a)l^{-1}\left(\frac{\mathbf{1}}{\mathbf{1}+s}\right)$$

Finally, we have:

$$
\bar{\jmath}(t,a) = -\bar{\jmath}(\mathbf{0},a)e^{-t}
$$

$$
\underline{\jmath}(t,a) = -\underline{\jmath}(\mathbf{0},a)e^{-t}
$$

**Remark 2.5**. By following the above procedure, the solution of fuzzy IVP with initial conditions can be obtained. The solution of simultaneous fuzzy linear differential equations and fuzzy BVPs can also be obtained.

Now, we discuss the application of fixed points in existence and the uniqueness of the solution of the ordinary differential equation.

IVP's existence and uniqueness can be easily established using the fixed point technique. Banach fixed point theorem can be applied to derive the existence and *Applications of Fuzzy Set and Fixed Point Theory in Dynamical Systems DOI: http://dx.doi.org/10.5772/intechopen.105830*

uniqueness of the solution of an initial value problem. The function used in IVP satisfies the Lipschitz condition.

**Definition 2.6**.  A mapping *F* : *X* ! *X* is called a contraction if k k *Fx* � *Fy* ≤*α*k k *x* � *y* , ∀*x*, *y*∈*X*, for some *α* <1

Example-. If *F x*ð Þ¼ *x* 2, then *F* is a contraction for *α* ¼ 1 2

**Definition 2.7**. Banach Fixed-Point Theorem (or Banach Contraction Principle):

If *F* : *X* ! *X* is a contraction, then *F* has a unique fixed point; say it is *x*1. *Fx*1 ¼ *x*1 Further, the sequence f g *xn* defined by *xn* ¼ *Fxn* ð Þ ∀*n* ¼ 1, 2, 3, … , converges to the unique fixed point *x*1 of *F*.

**Definition 2.8**. Generalized Banach contraction principle:

If *Fn* is contraction, *F* : *X* ! *X* is a Banach space for *n*≥ 1, then *F* has a unique fixed point

**Theorem 2.9.** Consider an Initial Value Problem (IVP)

$$\frac{d\mathbf{y}}{d\mathbf{x}} = f(\mathbf{x}, \mathbf{y}), \mathbf{y}(\mathbf{x\_0}) = \mathbf{y\_0}$$

Let *f x*ð Þ , *y* be a continuous function defined on a domain *D* ⊆*R*2 . Let f be Lipschitz continuous concerning y on D. Then, there exists a unique solution to the IVP on an interval j j *x* � *x*0 ≤*h*, where *h* ¼ min *a*, *b M* � �, *M* ¼ *Max f x* j j ð Þ , *y*

ð Þ *x*, *y* ∈*R*, and *R* ¼ ð Þ *x*, *y* : j j *x* � *x*0 ≤*a*, *y* � *y*0 � � � �≤*b* � �⊂ *D*

Further, the unique solution can be computed from the successive approximation scheme *yn*þ1ð Þ¼ *x y*0 þ Ð *x x*0 *f t*, *yn*ð Þ*t* � �*dt*, *y*0ðÞ¼ *t y*0 ð Þ ∀*n* ¼ 0, 1, 2, …

**Proof:** The solvability of IVP follows

If the integral equation *y x*ð Þ¼ *y*0 þ Ð *x x*0 *f t*ð Þ , *y t*ð Þ *dt* is solvable, let *X* ¼ *C x*½ � 0, *x*1 set of all continuous functions defined on ½ � *x*0, *x*1 *:* Define k k*x* ¼ sup t∈½ � x0, *x*1 j j *x t*ð Þ ð Þ *X*, k k*:* is

complete normed linear space.

Define

$$\boldsymbol{y}(\mathbf{x}) = \boldsymbol{y}\_0 + \int\_{\mathbf{x}\_0}^{\mathbf{x}} \boldsymbol{f}(t, \boldsymbol{y}(t)) dt \tag{5}$$

Define an operator *F* : *C x*½ �! 0, *x*1 *C x*½ � 0, *x*1 by ð Þ *Fy* ð Þ¼ *x y*0 þ Ð *x x*0 *f t*ð Þ , *y t*ð Þ *dt*.

If *F* has a fixed point, that is, there exists a *y* such that *y* ¼ *Fy*, then the fixed point *y* is a solution to the integral equation (5).

Now, we will show *Fn* Is contraction for some large *n*.

$$(F\mathfrak{y})(\mathfrak{x}) = \mathfrak{y}\_0 + \int\_{\mathfrak{x}\_0}^{\mathfrak{x}} f(t, \mathfrak{y}(t)) dt$$

Let *y*1, *y*2 ∈*C x*½ � 0, *x*1 , then

$$\left| \left( F\mathbf{y}\_1 \right)(\mathbf{x}) - \left( F\mathbf{y}\_2 \right)(\mathbf{x}) \right| = \left| \int\_{\mathbf{x}\_0}^{\mathbf{x}} \left( f(t, \mathbf{y}\_1(t) - f(t, \mathbf{y}\_2(t))) dt \right) \right|$$

$$\leq a \int\_{\mathbf{x}\_0}^{\mathbf{x}} \left| \mathbf{y}\_1(t) - \mathbf{y}\_2(t) \right| dt \tag{6}$$

$$\begin{aligned} \leq a \int\_{x\_{0t}}^{x} \sup\_{t \in [x\_{0, \infty}]} & \left| y\_1(t) - y\_2(t) \right| dt \\ \leq a \int\_{x\_0}^{x} & \left|| y\_1(t) - y\_2(t) \right|| dt \\ \leq a(x - x\_0) \left|| y\_1 - y\_2 \right|| \\ \left| \left( F^2 \mathcal{Y}\_1 \right)(x) - \left( F^2 \mathcal{Y}\_2 \right)(x) \right| &= \left| F(F\mathcal{Y}\_1)(x) - F(F\mathcal{Y}\_2)(x) \right| \\ \leq a \int\_{x\_0}^{x} & \left| F\mathcal{Y}\_1(t) - F\mathcal{Y}\_2(t) \right| dt \\ \leq a^2 \int\_{x\_0}^{x} & (x - x\_0) \left|| \mathcal{Y}\_1 - \mathcal{Y}\_2 \right|| dt \\ \leq \frac{a^2}{2} (x - x\_0)^2 \left|| \mathcal{Y}\_1 - \mathcal{Y}\_2 \right|| \end{aligned} \tag{7}$$

$$\left| \left| \left( F^2 \mathcal{V}\_1 \right) - \left( F^2 \mathcal{V}\_2 \right) \right| \right| \le \frac{\alpha^2}{2!} \left( \varkappa\_1 - \varkappa\_0 \right)^2 \left\| \mathcal{V}\_1 - \mathcal{V}\_2 \right\|^2$$

$$\left| \left| \left( F^3 \boldsymbol{\uprho}\_1 \right) - \left( F^3 \boldsymbol{\uprho}\_2 \right) \right| \right| \le \frac{\alpha^3}{3!} (\boldsymbol{\upkappa}\_1 - \boldsymbol{\upkappa}\_0)^3 ||\boldsymbol{\uprho}\_1 - \boldsymbol{\uprho}\_2||$$

$$- - -$$

$$\left| \left( F^n \boldsymbol{\uprho}\_1 \right) - \left( F^n \boldsymbol{\uprho}\_2 \right) \right| \le \frac{\alpha^n}{n!} (\boldsymbol{\upkappa}\_1 - \boldsymbol{\upkappa}\_0)^n ||\boldsymbol{\uprho}\_1 - \boldsymbol{\uprho}\_2||\tag{8}$$

$$\mathcal{Y}\_{n+1}(\mathbf{x}) = \mathcal{Y}\_0 + \int\_{\mathbf{x}\_0}^{\mathbf{x}} f\left(t, \mathcal{Y}\_n(t)\right) dt$$

$$\mathcal{Y}\_1(\mathbf{x}) = \mathbf{1} + \int\_1^{\mathbf{x}} f\left(t, \mathcal{Y}\_0(t)\right) dt$$

$$\mathcal{Y}\_1(\mathbf{x}) = \int\_1^{\mathbf{x}} \frac{2}{t} dt$$

$$\mathcal{Y}\_1(\mathbf{x}) = 2 \log \left(\mathbf{x}\right)$$

*Applications of Fuzzy Set and Fixed Point Theory in Dynamical Systems DOI: http://dx.doi.org/10.5772/intechopen.105830*

$$\begin{aligned} \mathcal{y}\_2(\mathbf{x}) &= \mathbf{1} + \int\_1^\mathbf{x} f(t, \mathbf{y}\_1(t)) dt \\ \mathcal{y}\_2(\mathbf{x}) &= \int\_1^\mathbf{x} 4 \frac{\log t}{t} dt \\ \mathcal{y}\_2(\mathbf{x}) &= \int\_1^\mathbf{x} 4 \frac{\log t}{t} dt \\ \mathcal{y}\_2(\mathbf{x}) &= 2(\mathbf{x}^2 - 1) \end{aligned}$$

$$\begin{aligned} \mathcal{y}\_3(\mathbf{x}) &= \int\_1^\mathbf{x} f(t, \mathcal{y}\_2(t)) dt \\ \mathcal{y}\_3(\mathbf{x}) &= \mathbf{1} + \int\_1^\mathbf{x} f(t, \mathcal{y}\_2(t)) dt \\ \mathcal{y}\_3(\mathbf{x}) &= \mathbf{1} + \int\_1^\mathbf{x} \frac{4(t^2 - 1)}{t} dt \\ \mathcal{y}\_3(\mathbf{x}) &= \mathbf{1} + 4 \int\_1^\mathbf{x} \left(t - \frac{1}{t}\right) dt \\ \mathcal{y}\_3(\mathbf{x}) &= \mathbf{1} + 2 \left(\mathbf{x}^2 - 1\right) - 4 \left(\log \mathbf{x}\right) \\ \mathcal{y}\_3(\mathbf{x}) &= \left(2\mathbf{x}^2 - 1\right) - 4 \left(\log \mathbf{x}\right) \end{aligned}$$

On keep continuing this process up to the nth time and then taking *n* ! ∞ we have

*y x*ð Þ¼ *x*2

**Example 2.11**. Consider the IVP *dy dx* ¼ *y*, *y*ð Þ¼ 0 1 Here *f x*ð Þ¼ , *y y*, *x*0 ¼ 0, *y*0 ¼ 1

$$\mathcal{Y}\_{n+1}(\mathbf{x}) = \mathcal{Y}\_0 + \int\_{\mathbf{x}\_0}^{\mathbf{x}} f\left(t, \mathcal{Y}\_n(t)\right) dt$$

$$\mathcal{Y}\_1(\mathbf{x}) = \mathbf{1} + \int\_0^{\mathbf{x}} \mathcal{Y}\_0(t) dt$$

$$\mathcal{Y}\_1(\mathbf{x}) = \mathbf{1} + \int\_0^{\mathbf{x}} \mathbf{1} dt = \mathbf{1} + \mathbf{x}$$

$$\mathcal{Y}\_2(\mathbf{x}) = \mathbf{1} + \int\_0^{\mathbf{x}} \mathcal{Y}\_1(t) dt = \mathbf{1} + \mathbf{x} + \frac{\mathbf{x}^2}{2}$$

$$\mathcal{Y}\_3(\mathbf{x}) = \mathbf{1} + \int\_0^{\mathbf{x}} \mathcal{Y}\_2(t) dt = \mathbf{1} + \mathbf{x} + \frac{\mathbf{x}^2}{2} + \frac{\mathbf{x}^3}{6}$$

$$\vdots$$

$$\mathcal{Y}\_n(\mathbf{x}) = \mathbf{1} + \mathbf{x} + \frac{\mathbf{x}^2}{2} + \frac{\mathbf{x}^3}{3} + \dots + \frac{\mathbf{x}^n}{n}$$

$$\mathcal{Y}\_n(\boldsymbol{\kappa}) = \mathbf{1} + \boldsymbol{\varkappa} + \frac{\boldsymbol{\varkappa}^2}{2!} + \frac{\boldsymbol{\varkappa}^3}{3!} + \dots \\ + \frac{\boldsymbol{\varkappa}^n}{n!} = \sum \frac{\boldsymbol{\varkappa}^n}{n!}$$

Taking *n* ! ∞ , then we have *yn*ð Þ! *x ex* Thus *y x*ð Þ¼ *ex* Is a solution for a given IVP.

Stability of the solution: In these questions/examples, the functions *f x*ð Þ¼ , *y* 2*y x* and *f x*ð Þ¼ , *y y* satisfies the Lipschitz condition concerning the initial condition *y*0 ¼ 1, so the answer is stable concerning initial data.

#### **3. Conclusion**

Here in this chapter, we discussed the application of a fuzzy set and a fixed point in dynamic systems. We also tried to discuss applications of fuzzy sets and fixed points in other directions. We gave a solution of fuzzy ordinary differential equations with initial conditions by the fuzzy Laplace transform method and provided a solution to the existence and uniqueness problem of ODE of the first order by the fixed point technique. We solved examples of existence and uniqueness problems and checked the stability of the solution also.

#### **Author details**

Praveen Kumar Sharma1 \*, Shivram Sharma2 , Jitendra Kaushik3 and Palash Goyal4

1 Department of Mathematics, SVIS, Shri Vaishnav Vidyapeeth Vishwavidyalaya, Indore, M.P., India

\*Address all correspondence to<EMAIL_ADDRESS>
© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Applications of Fuzzy Set and Fixed Point Theory in Dynamical Systems DOI: http://dx.doi.org/10.5772/intechopen.105830*

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#### **Chapter 7**

## Study of a Dynamical Problem under Fuzzy Conformable Differential Equation

*Atimad Harir, Said Melliani and Lalla Saadia Chadli*

#### **Abstract**

The notion of inclusion by generalized conformable differentiability is used to analyze fuzzy conformable differential equations (FCDE). This idea is based on expanding the class of conformable differentiable fuzzy mappings, and we use generalized lateral conformable derivatives to do so. We'll see that both conformable derivatives are distinct and that they lead to different FCDE solutions. The approach's utility and efficiency are demonstrated with an example.

**Keywords:** fuzzy fractional differential equation, conformable fractional derivative, fuzzy number

#### **1. Introduction**

Aubin and Cellina [1] established the notion of differential inclusions systemically. They looked at the existence and qualities of differential inclusion solutions of the form [2].

$$u'(t) \in \Phi(u(t)) \quad or \quad u'(t) \in \Phi(t, u(t)). \tag{1}$$

In this paper we will consider the conformable fractional differential equation.

$$\begin{aligned} u^{(q)}(t) &= \Phi(t, u(t)) \\ u^{\kappa}(0) &\in [u\_0]^{\kappa}, \quad \kappa \in [0, 1] \end{aligned} \tag{2}$$

where *t*∈ ð Þ 0, *a* and *u*0 is a fuzzy number. *u*ð Þ*q* is the conformable fractional derivative of *u* of order *γ* ∈ð � 0, 1 [3–5]. There are numerous options for defining a fuzzy fractional derivatives and, as a result, see [6–9], studying Eq. (2). [10–16] constructed the generalized derivative of a set value function and investigated it, while [17–20] explored the generalized conformable fractional derivative.

The objective of this research is to see if fuzzy solutions exist using conformable differential inclusion, using the generalized conformable differentiability concept

This idea is based on expanding the class of differentiable fuzzy mappings, and we use lateral conformable derivatives to do so. We will see that both derivatives are different and they lead us to different solutions from an FCDE.

#### **2. Preliminaries**

We'll go through a few definitions now that will come in handy later in the paper. Let us start with a definition. RF the class of fuzzy subsets of the real axis f g *η* : R ! ½ � 0, 1 satisfying the following properties:

i. *η* is normal,

Then RF is called the space of fuzzy numbers [21].

If *η* is a fuzzy set, we define ½ � *η κ* ¼ f g *x*∈Rj*η*ð Þ *x* ≥*κ* the *κ*-level sets of *η*, with 0<*κ* ≤1. Also, if *η*∈RF then *κ*-cut of *η* denoted by ½ � *η κ* ¼ *ηκ* 1, *ηκ* 2 � �*:*

For *η*, *ν*∈RF and *λ*∈ R the sum *η* þ *ν* and the product *λη* are defined by ∀*κ* ∈½ � 0, 1 ,

$$[\eta + \nu]^\kappa = \left[\eta\_1^\kappa + \nu\_1^\kappa, \eta\_2^\kappa + \nu\_2^\kappa\right],\tag{3}$$

$$[\lambda\eta]^\kappa = \lambda[\eta]^\kappa = \begin{cases} \left[\lambda\eta\_1^\kappa, \lambda\eta\_2^\kappa\right], & \lambda \ge 0;\\ \left[\lambda\eta\_2^\kappa, \lambda\eta\_1^\kappa\right], & \lambda < 0, \end{cases} \tag{4}$$

Let *d* : RF � RF ! Rþ∪f g0 by the following equation [22].

$$d(\eta, \nu) = \sup\_{\kappa \in [0, 1]} d\_{\varPi}(\left[\eta\right]^{\kappa}, \left[\nu\right]^{\kappa}), \text{ for all } \eta, \nu \in \mathbb{R}\_{\mathcal{F}}, \tag{5}$$

$$=\sup\_{\kappa\in[0,1]}\max\left\{|\eta\_1^{\kappa}-\nu\_1^{\kappa}|,\ |\eta\_2^{\kappa}-\nu\_2^{\kappa}|\right\}\tag{6}$$

where *dH* is the Hausdorff metric.

The following properties are well-known see [22, 23]: ∀ *η*, *ν*, *ω*, *ρ* ∈RF and *λ*∈R.

$$\begin{aligned} d(\eta + \omega, \nu + \omega) &= d(\eta, \nu) \quad \text{and} \quad d(\eta, \nu) = d(\nu, \eta), \\ d(\lambda \eta, \lambda \nu) &= |\lambda| d(\eta, \nu), \\ d(\eta + \nu, \omega + \rho) &\le d(\eta, \omega) + d(\nu, \rho). \end{aligned} \tag{7}$$

And Rð Þ F , *d* is a complete metric space.

**Theorem 1.** [1, 22] *Let η* : ½ �! 0, *a* RF *and* ½ � *η*ð Þ*t κ* ¼ *ηκ* 1ð Þ*t* , *ηκ* 2ð Þ*t* � � *be Seikkala differentiable. Then, ηκ* 1ð Þ*t and ηκ* 2ð Þ*t are differentiable and*

$$\left[\eta'(t)\right]^\kappa = \left[\left(\eta\_1^\kappa\right)'(t), \ \left(\eta\_2^\kappa\right)'(t)\right], \ \kappa \in [0, 1]. \tag{8}$$

*Study of a Dynamical Problem under Fuzzy Conformable Differential Equation DOI: http://dx.doi.org/10.5772/intechopen.105904*

**Definition 1.** [24] *Let η* : ½ �! 0, *a* RF *.* Ð *c b η*ð Þ*t dt*, *b*,*c*∈ ½ � 0, *a is the fuzzy integral, defined by*

$$
\left[\int\_{b}^{c} \eta(t)dt\right]^{\kappa} = \left[\int\_{b}^{c} \eta\_{1}^{\kappa}(t)dt, \int\_{b}^{c} \eta\_{2}^{\kappa}(t)dt\right],\tag{9}
$$

for all 0 ≤*κ* ≤1. In [24], if *η* : ½ �! 0, *a* RF is continuous, it is fuzzy integrable.  Fuzzy conformable differentiability and integral**

The funcion Φ : ½ �! *a*, *b* F is called fuzzy function. The *κ*-level representation of fuzzy function Φ given by ½ � Φð Þ*t κ* ¼ *ϕκ* 1ð Þ*t* , *ϕκ* 2ð Þ*t* � �, ∀*t*∈½ � *a*, *b* , ∀*κ* ∈½ � 0, 1 .

**Definition 2.** [17] *Let* Φ : ð Þ! 0, *a* F *be a fuzzy function. γth order*" *fuzzy conformable derivative" of* Φ *is defined by*

$$T\_{\mathcal{I}}(\Phi)(t) = \lim\_{\varepsilon \to 0^{+}} \frac{\Phi(t + \varepsilon t^{1-\gamma}) \ominus \Phi(t)}{\varepsilon} = \lim\_{\varepsilon \to 0^{+}} \frac{\Phi(t) \ominus \Phi(t - \varepsilon t^{1-\gamma})}{\varepsilon}. \tag{12}$$

for all *t*> 0, *γ* ∈ ð Þ 0, 1 . Let Φð Þ*γ* ð Þ*t* stands for *Tγ* ð Þ Φ ð Þ*t* . Hence

$$\Phi^{(\gamma)}(t) = \lim\_{\varepsilon \to 0^{+}} \frac{\Phi(t + \varepsilon t^{1-\gamma}) \ominus \Phi(t)}{\varepsilon} = \lim\_{\varepsilon \to 0^{+}} \frac{\Phi(t) \ominus \Phi(t - \varepsilon t^{1-\gamma})}{\varepsilon}. \tag{13}$$

If Φ is *γ*-differentiable in some 0, ð Þ *a* , and lim *t*!0þ Φð Þ*γ* ð Þ*t* exists, then

$$\Phi^{(\mathcal{r})}(\mathbf{0}) = \lim\_{\mathfrak{t} \to \mathbf{0}^+} \Phi^{(\mathcal{r})}(\mathbf{t}) \tag{14}$$

and the limits (in the metric d). **Remark 1**. [26]

— *If* Φ *is γ-differentiable then the multivalued mapping* Φ*κ is γ-differentiable for all κ* ∈½ � 0, 1 *and*

$$T\_{\gamma} \Phi\_{\kappa} = \left[ \Phi^{(\gamma)}(t) \right]^{\kappa},\tag{15}$$

where *Tγ*Φ*κ* is denoted from the conformable fractional derivative of Φ*κ* of order *γ*.

— ½ � *η κ* ⊖ ½ � *ν κ* , *κ* ∈½ � 0, 1 *does not imply the existence of Hukuhara difference (Hdifference) η* ⊖ *ν:*

**Theorem 3.** [26]

Let Φ : ð Þ! 0, *a* F , ½ � Φð Þ*t κ* ¼ *ϕκ* 1ð Þ*t* , *ϕκ* 2ð Þ*t* � �, *κ* ∈½ � 0, 1 *.*

i. *If* Φ *is γ*ð Þ1 *-differentiable, then ϕκ* 1ð Þ*t and ϕκ* 2ð Þ*t are γ-differentiable and*

$$\left[\Phi^{\left(\mathcal{I}\_{(1)}\right)}(t)\right]^{\kappa} = \left[\left(\phi\_1^{\kappa}\right)^{(\mathcal{I})}(t), \left(\phi\_2^{\kappa}\right)^{(\mathcal{I})}(t)\right] \tag{16}$$

ii. *If* Φ *is γ*ð Þ2 *-differentiable, then ϕκ* 1ð Þ*t and ϕκ* 2ð Þ*t are γ-differentiable and*

$$\left[\Phi^{\left(\mathcal{I}\_{\left(\mathcal{I}\right)}\right)}(t)\right]^{\kappa} = \left[\left(\phi\_2^{\kappa}\right)^{\left(\mathcal{I}\right)}(t), \left(\phi\_1^{\kappa}\right)^{\left(\mathcal{I}\right)}(t)\right].\tag{17}$$

**Theorem 4.** [17] *Let γ* ∈ð � 0, 1 :*.*

i. *If* Φ *is* ð Þ1 *-differentiable and* Φ *is γ*ð Þ1 *-differentiable, then*

$$T\_{\mathcal{I}\_{(1)}}\Phi(t) = t^{1-\gamma}D\_1^1\Phi(t)\tag{18}$$

ii. *If* Φ *is* ð Þ2 *-differentiable and* Φ *is γ*ð Þ2 *-differentiable, then*

$$T\_{\mathcal{I}\_{(2)}}\Phi(t) = t^{1-\gamma}D\_2^1\Phi(t)\tag{19}$$

**Theorem 5.** *If* Φ : ð Þ! 0, *a* F *is γ-differentiable then it is continuous.*

*Proof.* Denote Φ*κ*ðÞ¼ *t ϕκ* 1ð Þ*t* , *ϕκ* 2ð Þ*t* � �, *κ* ∈ ½ � 0, 1 *:* Then *ϕκ* 1ð Þ*t* and *ϕκ* 2ð Þ*t* are continuous at *t*0, so Φ is continuous at *t*0*:*

If *ε*> 0 and *κ* ∈½ � 0, 1 , we have:

$$\mathbb{E}\left[\Phi\left(t\_0 + \epsilon t\_0^{1-\gamma}\right) \ominus \Phi(t\_0)\right]^\kappa = \left[\phi\_1^\kappa\left(t\_0 + \epsilon t\_0^{1-\gamma}\right) - \phi\_1^\kappa(t\_0), \phi\_2^\kappa\left(t\_0 + \epsilon t\_0^{1-\gamma}\right) - \phi\_2^\kappa(t\_0)\right]^\kappa$$

Dividing and multiplying by *ε*, we have:

$$\mathbb{E}\left[\Phi\left(t\_0 + \varepsilon t\_0^{1-\gamma}\right)\ominus\Phi(t\_0)\right]^\kappa = \left[\frac{\phi\_1^\kappa\left(t\_0 + \varepsilon t\_0^{1-\gamma}\right) - \phi\_1^\kappa(t\_0)}{\varepsilon} \cdot \varepsilon, \frac{\phi\_2^\kappa\left(t\_0 + \varepsilon t\_0^{1-\gamma}\right) - \phi\_2^\kappa(t\_0)}{\varepsilon} \cdot \varepsilon\right]^\kappa$$

Similarly, we obtain:

$$\left[\Phi(t\_0)\ominus\Phi\left(t\_0-\epsilon t\_0^{1-\gamma}\right)\right]^\kappa = \left[\frac{\phi\_1^\kappa(t\_0)-\phi\_1^\kappa(t\_0-\epsilon t\_0^{1-\gamma})}{\varepsilon}\cdot\varepsilon,\frac{\phi\_2^\kappa(t\_0)-\phi\_2^\kappa(t\_0-\epsilon t\_0^{1-\gamma})}{\varepsilon}\cdot\varepsilon\right]^\kappa$$

*Study of a Dynamical Problem under Fuzzy Conformable Differential Equation DOI: http://dx.doi.org/10.5772/intechopen.105904*

Then

$$\begin{aligned} \lim\_{\varepsilon \to 0^{+}} \left[ \Phi \left( t\_{0} + \varepsilon t\_{0}^{1-\gamma} \right) \ominus \Phi(t\_{0}) \right]^{\kappa} &= \quad \left[ \lim\_{\varepsilon \to 0^{+}} \frac{\phi\_{1}^{\kappa} \left( t\_{0} + \varepsilon t\_{0}^{1-\gamma} \right) - \phi\_{1}^{\kappa}(t\_{0})}{\varepsilon} \cdot \lim\_{\varepsilon \to 0^{+}} \varepsilon, \right. \\ & \quad \lim\_{\varepsilon \to 0^{+}} \frac{\phi\_{2}^{\kappa} \left( t\_{0} + \varepsilon t\_{0}^{1-\gamma} \right) - \phi\_{2}^{\kappa}(t\_{0})}{\varepsilon} \cdot \lim\_{\varepsilon \to 0^{+}} \varepsilon \end{aligned}$$

Similarly, we obtain:

$$\begin{aligned} \lim\_{\varepsilon \to 0^{+}} \left[ \Phi(t\_0) \ominus \Phi \left( t\_0 - \varepsilon t\_0^{1-\gamma} \right) \right]^\kappa &= \quad \left[ \lim\_{\varepsilon \to 0^{+}} \frac{\phi\_1^\kappa(t\_0) - \phi\_1^\kappa \left( t\_0 - \varepsilon t\_0^{1-\gamma} \right)}{\varepsilon} \cdot \lim\_{\varepsilon \to 0^{+}} \varepsilon \right] \\ &\lim\_{\varepsilon \to 0^{+}} \frac{\phi\_2^\kappa(t\_0) - \phi\_2^\kappa \left( t\_0 - \varepsilon t\_0^{1-\gamma} \right)}{\varepsilon} \cdot \lim\_{\varepsilon \to 0^{+}} \varepsilon \end{aligned}$$

Let *h* ¼ *εt* 1�*γ* 0 *:* Then

$$\lim\_{h \to 0^+} [\Phi(t\_0 + h) \ominus \Phi(t\_0)]^\kappa = \left[ \left( \phi\_1^\kappa \right)^{(\gamma)}(t\_0) \cdot \mathbf{0}, \left( \phi\_2^\kappa \right)^{(\gamma)}(t\_0) \cdot \mathbf{0} \right]^\kappa$$

Similarly, we obtain:

$$\lim\_{h \to 0^+} [\Phi(t\_0 - h)]^\kappa = [\Phi(t\_0)]^\kappa$$

which implies that

$$\lim\_{h \to 0^+} [\Phi(t\_0 + h)]^\kappa = [\Phi(t\_0)]^\kappa$$

Similary, we obtain:

$$\lim\_{h \to 0^+} [\Phi(t\_0 - h)]^\kappa = [\Phi(t\_0)]^\kappa$$

Hence, Φ is continuous at *t*0. □ **Remark 2.** *If* Φ : ð Þ! 0, *a* F *is γ-differentiable and* Φð Þ*γ for all γ* ∈ð � 0, 1 *is continuous, then we denote* Φ ∈*C*1 ð Þ ð Þ 0, *a* , F *.*

**Theorem 6.** *Let γ* ∈ð � 0, 1 *and if* Φ, Ψ : ð Þ! 0, *a* F *are γ-differentiable and λ*∈ *then*

i.

$$T\_{\mathcal{I}}(\Phi + \Psi)(t) = T\_{\mathcal{I}}(\Phi)(t) + T\_{\mathcal{I}}(\Psi)(t) \tag{20}$$

ii.

$$T\_{\gamma}(\lambda \Phi)(t) = \lambda T\_{\gamma}(\Phi)(t). \tag{21}$$

*proof.* We present the details only for case ið Þ, since the other case is anlogous. Since Φ is *γ*ð Þ1 -differentiable it follows that Φ *t* þ *εt* 1�*γ* ð Þ ⊖ Φð Þ*t* exists i.e. there exists *u*1 *t*, *εt* 1�*γ* ð Þ such that

$$\Phi(t + \epsilon t^{1-\gamma}) = \Phi(t) + u\_1(t, \epsilon t^{1-\gamma}) \tag{22}$$

Analogously since Ψ is *γ*ð Þ1 -differentiable there exists *v*1 *t*, *εt* 1�*γ* ð Þ such that

$$
\Psi(t + \epsilon t^{1-\gamma}) = \Psi(t) + \nu\_1(t, \epsilon t^{1-\gamma}),
$$

and we get

$$
\Phi(t + \epsilon t^{1-\gamma}) + \Psi(t + \epsilon t^{1-\gamma}) = \Phi(t) + \Psi(t) + u\_1(t, \epsilon t^{1-\gamma}) + v\_1(t, \epsilon t^{1-\gamma})\tag{23}
$$

that is the H-difference

$$\left(\Phi\left(t+\varepsilon t^{1-\gamma}\right)+\Psi\left(t+\varepsilon t^{1-\gamma}\right)\right)\ominus\left(\Phi(t)+\Psi(t)\right)=u\_1\left(t,\varepsilon t^{1-\gamma}\right)+v\_1\left(t,\varepsilon t^{1-\gamma}\right)\tag{24}$$

By similar reasoning we get that there exist *u*2 *t*, *εt* 1�*γ* ð Þ and *v*2 *t*, *εt* 1�*γ* ð Þ such that

$$\begin{aligned} \Phi(t) &= \Phi\left(t - \epsilon t^{1-\gamma}\right) + \nu\_2\left(t, \epsilon t^{1-\gamma}\right), \\ \Psi(t) &= \Psi\left(t - \epsilon t^{1-\gamma}\right) + \nu\_2\left(t, \epsilon t^{1-\gamma}\right) \end{aligned}$$

and so

$$\left( (\Phi(t) + \Psi(t)) \right) = \left( \Phi\left(t - \varepsilon t^{1-\gamma} \right) + \Psi\left(t - \varepsilon t^{1-\gamma} \right) \right) + u\_2\left(t, \varepsilon t^{1-\gamma} \right) + v\_2\left(t, \varepsilon t^{1-\gamma} \right)$$

that is the H-difference

$$\left(\left(\Phi(t) + \Psi(t)\right)\ominus\left(\Phi\left(t - \epsilon t^{1-\gamma}\right) + \Psi\left(t - \epsilon t^{1-\gamma}\right)\right) - \mathfrak{u}\_2\left(t, \epsilon t^{1-\gamma}\right) + \mathfrak{v}\_2\left(t, \epsilon t^{1-\gamma}\right) \tag{25}$$

We observe that

$$\lim\_{\varepsilon \to 0^{+}} \frac{u\_1(t, \varepsilon t^{1-\gamma})}{\varepsilon} \quad = \lim\_{\varepsilon \to 0^{+}} \frac{u\_2(t, \varepsilon t^{1-\gamma})}{\varepsilon} = \Phi^{(\gamma)}(t) \quad \text{and}$$

$$\lim\_{\varepsilon \to 0^{+}} \frac{v\_1(t, \varepsilon t^{1-\gamma})}{\varepsilon} \quad = \lim\_{\varepsilon \to 0^{+}} \frac{v\_2(t, \varepsilon t^{1-\gamma})}{\varepsilon} = \Psi^{(\gamma)}(t).$$

Finally, by multiplying (24) and (25) with 1 *ε* and passing to limit with lim *ε*!0þ we get that Φ þ Ψ is *γ*ð Þ1 -differentiable and *Tγ* ð Þ Φ þ Ψ ðÞ¼ *t Tγ*ΦðÞþ*t Tγ*Ψð Þ*t* The case when Φ and Ψ are *γ*ð Þ2 -differentiable is similar to the previous one. □

**Definition 3.** *Let* Φ ∈*C*ð Þ ð Þ 0, *a* , F ∩ *L*1 ð Þ ð Þ 0, *a* , F , *Define the fuzzy fractional integral for γ* ∈ ð � 0, 1 *:*

$$I\_{\boldsymbol{\gamma}}(\Phi)(t) = I\_1(t^{r-1}\Phi)(t) = \int\_0^t \frac{\Phi}{s^{1-\boldsymbol{\gamma}}}(s)ds,\tag{26}$$

where the integral is the usual Riemann improper integral.

**Theorem 7.** *Tγ Iγ* ð Þ Φ ð Þ*t , for t*≥ 0*, where* Φ *is any continuous function in the domain of Iγ. Proof.* Since Φ is continuous, then *Iγ* ð Þ Φ ð Þ*t* is clearly conformable differentiable. Hence,

*Study of a Dynamical Problem under Fuzzy Conformable Differential Equation DOI: http://dx.doi.org/10.5772/intechopen.105904*

$$\begin{aligned} \left[T\_{\boldsymbol{\gamma}}I\_{\boldsymbol{\gamma}}(\Phi)(t)\right]^{\kappa} &= \left[t^{1-\gamma}\frac{d}{dt}I\_{\boldsymbol{\gamma}}(\Phi)(t)\right]^{\kappa} \\ &= \left[t^{1-\gamma}\frac{d}{dt}\int\_{0}^{t}\frac{\phi\_{1}^{\kappa}(\boldsymbol{\chi})}{\boldsymbol{x}^{1-\gamma}}d\boldsymbol{x},t^{1-\gamma}\frac{d}{dt}\int\_{0}^{t}\frac{\phi\_{2}^{\kappa}(\boldsymbol{\chi})}{\boldsymbol{x}^{1-\gamma}}d\boldsymbol{x}\right] \\ &= \left[t^{1-\gamma}\frac{\phi\_{1}^{\kappa}(t)}{t^{1-\gamma}},t^{1-\gamma}\frac{\phi\_{2}^{\kappa}(t)}{t^{1-\gamma}}\right] \\ &= [\Phi(t)]^{\kappa} \end{aligned}$$

□

**Theorem 8.** *Let γ* ∈ð � 0, 1 *and* Φ *be γ-differentiable in* ð Þ 0, *a and assume that the conformable derivative* Φð Þ*γ is integrable over* ð Þ 0, *a . Then* ∀ *s*∈ð Þ 0, *a we have.*

$$\Phi(\mathfrak{s}) = \Phi(\mathfrak{a}) + I\_{\mathfrak{I}} \Phi^{(\mathfrak{I})}(\mathfrak{t}) \tag{27}$$

*Proof.* Let *γ* ∈ð � 0, 1 and *κ* ∈½ � 0, 1 be fixed. We will demonstrate this.

$$\Phi\_{\kappa}(\mathfrak{s}) = \Phi\_{\kappa}(\mathfrak{a}) + I\_{\mathcal{I}} \Phi\_{\kappa}^{(\mathcal{I})} \tag{28}$$

where Φð Þ*γ κ* is conformable derivative of Φ*κ*, the equation is then obtained by applying Theorems 3 and 4.

$$\begin{aligned} \Phi\_{\kappa}(s) &= \Phi\_{\kappa}(a) + I\_{\mathcal{I}} \Phi\_{\kappa}^{(\mathcal{I})} \\ &= \Phi\_{\kappa}(a) + I\_{\mathcal{I}} \left(t^{1-\gamma} \Phi\_{\kappa}^{\prime}\right), \end{aligned}$$

by (26) we have

$$\begin{aligned} \Phi\_{\kappa}(s) &= \Phi\_{\kappa}(a) + I\_{\gamma} \left( t^{1-\gamma} \Phi\_{\kappa}' \right) \\ &= \Phi\_{\kappa}(a) + \int\_{0}^{s} t^{\gamma - 1} \left( t^{1-\gamma} \Phi\_{\kappa}' \right) \end{aligned}$$

So

$$\Phi\_{\kappa}(\mathfrak{s}) = \Phi\_{\kappa}(\mathfrak{a}) + \int\_{0}^{\mathfrak{s}} \Phi\_{\kappa}' \tag{29}$$

where Φ0 *κ* is the derivative of Φ*κ*, For a fuzzy mapping, the (29) is likewise true Φ : ð Þ! 0, *a* F . In [1], the equality (28) now follows Theorem (8).

#### **4. Solutions via conformable differential inclusions**

We consider the fuzzy conformable differential equation.

$$u^{(\gamma)}(t) = \Phi(t, u(t)), \quad u(0) = u\_0, \ \gamma \in (0, 1], \tag{30}$$

where Φ : ð Þ� 0, *a* RF ! RF is generated from a continuous function using Zadeh's extension principle. *ψ* : ð Þ� 0, *a* R ! R*:*

Let Φð Þ *t*, *u* can be calculated at the level, i.e., ∀ *κ* ∈½ � 0, 1 .

$$[\Phi(t, u)]^\kappa = \psi(t, [u]^\kappa).$$

for all *t*∈ð Þ 0, *a* , *u* ∈RF and *κ* ∈½ � 0, 1 *:* We interpret the fuzzy initial value problem (30) as a set of differential inclusions, following Diamonds [7, 10].

$$\left(\boldsymbol{v}^{(\boldsymbol{\eta})}\right)^{\kappa}(t) = \boldsymbol{\psi}(t, \boldsymbol{v}^{\kappa}(t)), \quad \boldsymbol{v}^{\kappa}(\mathbf{0}) \in [\boldsymbol{u}\_{0}]^{\kappa} \tag{31}$$

The reachable sets, under reasonable assumptions.

$$\Upsilon\_{\kappa}(t) = \{v^{\kappa}(t) \mid v^{\kappa} \text{ is a solution } \text{ of} \ (\mathfrak{A}1)\},$$

are *κ*-cuts of a fuzzy set *u t*ð Þ, which we call a solution of (30). If we assume that the solutions to the initial value problem are unique, *v*ð Þ*γ κ* ðÞ¼ *t ψ t*, *vκ* ð Þ ð Þ*t* , *vκ*ð Þ¼ 0 *v*0, it follows that ϒ*κ*ðÞ¼ *t* ½ � *w*1ð Þ*t* , *w*2ð Þ*t* , where.

$$\begin{aligned} \left(w^{(\boldsymbol{\eta})}\right)\_1(t) &= \boldsymbol{\psi}(t, w\_1(t)), \quad w\_1(0) = u\_{01}^\kappa \quad \text{and} \\ \left(w^{(\boldsymbol{\eta})}\right)\_2(t) &= \boldsymbol{\psi}(t, w\_2(t)), \quad w\_2(0) = u\_{02}^\kappa \end{aligned}$$

**Theorem 9.** *The fuzzy solution and solution by differential inclusions solution using the first form are equivalent if ψ is nondecreasing with respect to the second argument.*