Μερική Διαφορική Εξίσωσις

Partial Differential Equation


Διαφορική Εξίσωση
Διαφορική Ανάλυση
Συνήθης Διαφορική Εξίσωση
Μερική Διαφορική Εξίσωση
Πρωτοτάξια Διαφορική Εξίσωση
Δευτεροτάξια Διαφορική Εξίσωση

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Πρότυπο:Equations Η ονομασία " Εξίσωση" σχετίζεται ετυμολογικά με την λέξη " ".


In mathematics, partial differential equations (PDE) are a type of differential equation, i.e., a relation involving an unknown function (or functions) of several independent variables and their partial derivatives with respect to those variables. Partial differential equations are used to formulate, and thus aid the solution of, problems involving functions of several variables; such as the propagation of sound or heat, electrostatics, electrodynamics, fluid flow, and elasticity. Seemingly distinct physical phenomena may have identical mathematical formulations, and thus be governed by the same underlying dynamic. They find their generalization in stochastic partial differential equations. Just as ordinary differential equations often model dynamical systems, partial differential equations often model multidimensional systems.

Introduction Edit

A partial differential equation (PDE) for the function $ u(x_1,...x_n) $ is of the form

$ F(x_1, \cdots x_n,u,\frac{\partial}{\partial x_1}u, \cdots \frac{\partial}{\partial x_n}u,\frac{\partial^2}{\partial x_1 \partial x_1}u, \frac{\partial^2}{\partial x_1 \partial x_2}u, \cdots ) = 0 \, $

F is a linear function of u and its derivatives if, by replacing u with v+w, F can be written as $ F(v)+F(w), $ and if, by replacing u with ku, F can be written as $ k \cdot F(u) $

If F is a linear function of u and its derivatives, then the PDE is linear. Common examples of linear PDEs include the heat equation, the wave equation and Laplace's equation.

A relatively simple PDE is

$ \frac{\partial}{\partial x}u(x,y)=0\, . $

This relation implies that the function u(x,y) is independent of x. Hence the general solution of this equation is

$ u(x,y) = f(y),\, $


$ \frac{du(x)}{dx}=0\, $

which has the solution

$ u(x) = c,\, $

where c is any constant value (independent of x). These two examples illustrate that general solutions of ordinary differential equations (ODEs) involve arbitrary constants, but solutions of PDEs involve arbitrary functions. A solution of a PDE is generally not unique; additional conditions must generally be specified on the boundary of the region where the solution is defined. For instance, in the simple example above, the function $ f(y) $ can be determined if $ u $ is specified on the line $ x=0 $.

Existence and uniqueness Edit

Although the issue of existence and uniqueness of solutions of ordinary differential equations has a very satisfactory answer with the Picard–Lindelöf theorem, that is far from the case for partial differential equations. There is a general theorem (the Cauchy–Kowalevski theorem) that states that the Cauchy problem for any partial differential equation that is analytic in the unknown function and its derivatives has a unique analytic solution. Although this result might appear to settle the existence and uniqueness of solutions, there are examples of linear partial differential equations whose coefficients have derivatives of all orders (which are nevertheless not analytic) but which have no solutions at all: see Lewy (1957). Even if the solution of a partial differential equation exists and is unique, it may nevertheless have undesirable properties. The mathematical study of these questions is usually in the more powerful context of weak solutions.



with boundary conditions

$ u(x,0) = 0, \, $
$ \frac{\partial u}{\partial y}(x,0) = \frac{\sin n x}{n},\, $


$ u(x,y) = \frac{(\sinh ny)(\sin nx)}{n^2}.\, $


Notation Edit


That is:

$ u_x = {\partial u \over \partial x} $
$ u_{xy} = {\partial^2 u \over \partial y\, \partial x} = {\partial \over \partial y } \left({\partial u \over \partial x}\right). $

Especially in (mathematical) physics, one often prefers the use of del (which in cartesian coordinates is written $ \nabla=(\partial_x,\partial_y,\partial_z)\, $) for spatial derivatives and a dot $ \dot u\, $ for time derivatives. For example, the wave equation (described below) can be written as

$ \ddot u=c^2\nabla^2u\, $ (physics notation),


$ \ddot u=c^2\Delta u\, $ (math notation), where Δ is the Laplace operator.

This often leads to misunderstandings in regards of the $ \Delta $-(delta)operator.

Examples Edit

Heat equation in one space dimension Edit

The equation for conduction of heat in one dimension for a homogeneous body has the form

$ u_t = \alpha u_{xx} \, $

where u(t,x) is temperature, and α is a positive constant that describes the rate of diffusion. The Cauchy problem for this equation consists in specifying $ u(0,x)= f(x) $, where f(x) is an arbitrary function.







and the resulting solution of the heat equation is


This is a Gaussian integral. It may be evaluated to obtain


This result corresponds to a normal probability density for x with mean 0 and variance 2αt. The heat equation and similar diffusion equations are useful tools to study random phenomena.

Wave equation in one spatial dimension Edit


$ u_{tt} = c^2 u_{xx}. \, $


$ u(0,x) = f(x), \, $
$ u_t(0,x) = g(x), \, $




$ x - ct = \hbox{constant,} \quad x + ct = \hbox{constant}, \, $


Spherical waves Edit



This is equivalent to

$ (ru)_{tt} = c^2 \left[(ru)_{rr} \right],\, $




Laplace equation in two dimensions Edit


$ \varphi_{xx} + \varphi_{yy} = 0. $

Solutions of Laplace's equation are called harmonic functions.

Connection with holomorphic functions Edit


$ u_x = v_y, \quad v_x = -u_y,\, $

and it follows that



A typical boundary value problem Edit




Euler–Tricomi equation Edit


$ u_{xx} \, =xu_{yy}. $

Advection equation Edit

The advection equation describes the transport of a conserved scalar $ \psi $ in a velocity field $ {\bold u}=(u,v,w) $. It is:

$ \psi_t+(u\psi)_x+(v\psi)_y+(w\psi)_z \, =0. $

If the velocity field is solenoidal (that is, $ \nabla\cdot{\bold u}=0 $), then the equation may be simplified to

$ \psi_t+u\psi_x+v\psi_y+w\psi_z \, =0. $

In the one-dimensional case where $ u $ is not constant and is equal to $ \psi $, the equation is referred to as Burgers' equation.

Ginzburg–Landau equation Edit


$ iu_t+pu_{xx} +q|u|^2u \, =i\gamma u $

where $ p,q\in\mathbb{C} $ and $ \gamma\in\mathbb{R} $ are constants and $ i $ is the imaginary unit.

The Dym equation Edit


$ u_t \, = u^3u_{xxx}. $

Initial-boundary value problems Edit

Πρότυπο:Main Πρότυπο:See


Vibrating string Edit


$ u(t,0)=0, \quad u(t,L)=0, \, $

as well as the initial conditions


The method of separation of variables for the wave equation

$ u_{tt} = c^2 u_{xx}, \, $

leads to solutions of the form

$ u(t,x) = T(t) X(x),\, $




$ k= \frac{n\pi}{L}, \, $


$ X(0) =0, \quad X'(L) = 0.\, $



Vibrating membrane Edit


$ \frac{1}{c^2} u_{tt} = u_{xx} + u_{yy}, \, $

if t>0 and (x,y) is in D. The boundary condition is $ u(t,x,y) = 0 $ if $ (x,y) $ is on $ C $. The method of separation of variables leads to the form

$ u(t,x,y) = T(t) v(x,y),\, $

which in turn must satisfy

$ \frac{1}{c^2}T'' +k^2 T=0, \, $
$ v_{xx} + v_{yy} + k^2 v =0.\, $


Other examples Edit



Also see the list of non-linear partial differential equations.

Classification Edit

Some linear, second-order partial differential equations can be classified as parabolic, hyperbolic or elliptic. Others such as the Euler–Tricomi equation have different types in different regions. The classification provides a guide to appropriate initial and boundary conditions, and to smoothness of the solutions.

Equations of first order Edit


Equations of second order Edit

Assuming $ u_{xy}=u_{yx} $, the general second-order PDE in two independent variables has the form

$ Au_{xx} + 2Bu_{xy} + Cu_{yy} + \cdots = 0, $

where the coefficients A, B, C etc. may depend upon x and y. If $ A^2 +B^2 + C^2 > 0 $ over a region of the xy plane, the PDE is second-order in that region. This form is analogous to the equation for a conic section:

$ Ax^2 + 2Bxy + Cy^2 + \cdots = 0. $

More precisely, replacing $ \partial_x $ by $ X, $ and likewise for other variables (formally this is done by a Fourier transform), converts a constant-coefficient PDE into a polynomial of the same degree, with the top degree (a homogeneous polynomial, here a quadratic form) being most significant for the classification.

Just as one classifies conic sections and quadratic forms into parabolic, hyperbolic, and elliptic based on the discriminant $ B^2 - 4AC $, the same can be done for a second-order PDE at a given point. However, the discriminant in a PDE is given by $ B^2 - AC, $ due to the convention of the xy term being 2B rather than B; formally, the discriminant (of the associated quadratic form) is $ (2B)^2 - 4AC = 4(B^2-AC), $ with the factor of 4 dropped for simplicity.

  2. $ B^2 - AC = 0\, $ : equations that are parabolic at every point can be transformed into a form analogous to the heat equation by a change of independent variables. Solutions smooth out as the transformed time variable increases. The Euler–Tricomi equation has parabolic type on the line where x=0.
  3. $ B^2 - AC \, > 0 $ : hyperbolic equations retain any discontinuities of functions or derivatives in the initial data. An example is the wave equation. The motion of a fluid at supersonic speeds can be approximated with hyperbolic PDEs, and the Euler–Tricomi equation is hyperbolic where x>0.


$ L u =\sum_{i=1}^n\sum_{j=1}^n a_{i,j} \frac{\partial^2 u}{\partial x_i \partial x_j} \quad \hbox{ plus lower order terms} =0. \, $

The classification depends upon the signature of the eigenvalues of the coefficient matrix.


Systems of first-order equations and characteristic surfaces Edit

The classification of partial differential equations can be extended to systems of first-order equations, where the unknown u is now a vector with m components, and the coefficient matrices $ A_\nu $ are m by m matrices for $ \nu=1, \dots,n $. The partial differential equation takes the form



$ \varphi(x_1, x_2, \ldots, x_n)=0, \, $


$ Q\left(\frac{\partial\varphi}{\partial x_1}, \ldots,\frac{\partial\varphi}{\partial x_n}\right) =\det\left[\sum_{\nu=1}^nA_\nu \frac{\partial \varphi}{\partial x_\nu}\right]=0.\, $


$ Q(\lambda \xi + \eta) =0, \, $


Equations of mixed type Edit


$ u_{xx} \, = xu_{yy} $


Infinite-order PDEs in quantum mechanics Edit

Weyl quantization in phase space leads to quantum Hamilton's equations for trajectories of quantum particles. Those equations are infinite-order PDEs. However, in the semiclassical expansion one has a finite system of ODEs at any fixed order of $ \hbar $. The equation of evolution of the Wigner function is infinite-order PDE also. The quantum trajectories are quantum characteristics with the use of which one can calculate the evolution of the Wigner function.

Analytical methods to solve PDEs Edit

Separation of variablesEdit




Method of characteristicsEdit


More generally, one may find characteristic surfaces.

Integral transformEdit




Change of variablesEdit



is reducible to the heat equation


by the change of variables (for complete details see Solution of the Black Scholes Equation)

$ V(S,t) = K v(x,\tau)\, $
$ x = \ln(S/K)\, $
$ \tau = \frac{1}{2} \sigma^2 (T - t) $

Fundamental solutionEdit



Superposition principleEdit


Methods for non-linear equationsEdit

See also the list of nonlinear partial differential equations.



The method of characteristics (Similarity Transformation method) can be used in some very special cases to solve partial differential equations.


Lie Group MethodsEdit




Numerical methods to solve PDEs Edit


Finite Element Method Edit


Finite Difference Method Edit


Finite Volume Method Edit

Πρότυπο:Main Similar to the finite difference method or finite element method, values are calculated at discrete places on a meshed geometry. "Finite volume" refers to the small volume surrounding each node point on a mesh. In the finite volume method, volume integrals in a partial differential equation that contain a divergence term are converted to surface integrals, using the divergence theorem. These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods are conservative.

Almost-solution of PDEEdit

Almost-solution of PDE is a concept introduced by a Russian mathematician Vladimir Miklyukov in connection with research of solutions with nonremovable singularities.Πρότυπο:Citation needed

Εσωτερική ΑρθρογραφίαEdit


  • Berg, P.W. & Mc Gregor, J.L., Elementary Partial Differential Equations, Holden-Day Inc


  • Churchill, R.V., Fourier Series and Boundary Value Problems, 2nd Ed., McGraw-Hill Book Co


  • Copson, E.T., Partial Differential Equations, Cambridge University Press (1975).
  • Courant, R. & Hilbert D., Methods of Mathematical Physics, Vols I-II, J. Wiley & Sons (1989).
  • Δάσιος, Γ. & Κυριάκη, Κ., Μερικές Διαφορικές Εξισώσεις, Αθήνα (1994).
  • Dettman, J.W., Apllied Complex Variables, Dover (1984).
  • Duff, G.F.D. & Naylor, D., Differential Equations of Applied Mathematics, 3rd Ed., McGrawHill Inc (1966).
  • Evans, L.C, Partial Differential Equations, American Mathematical Society (1998).
  • Folland, G.B., Fourier Analysis and its Applications, Brooks/Cole Publ. Co. (1992).
  • Folland, G.B., Introduction to Partial Differential Equations, 2nd Ed., Princeton University

Press (1995).

  • Friedman, A. , Partial Differential Equations, 3rd Ed., Rinehart & Winston (1969).
  • Garabedian, P., Partial Differential Equations, J. Wiley (1964).
  • Guenther, B.R & Lee, J.W., Partial Differential Equations of Mathematical Physics and

Integral Equations, Dover (1996).

  • Hildebrand, F.B., Advanced Calculus for Applications, 2nd Ed., Prentice-Hall (1976).
  • John, F., Partial Differential Equations, 4td Ed., Springer-Verlag (1982).
  • Kevorkian, J., Partial Differential Equations, Analytical Solution Techniques, Brooks/Cole


  • Lieberstein, H.M., Theory of Partial Differential Equations, Academic Press (1972).
  • Morse, P.M. & Feshbach, H., Methods of Theoritical Physics, McGraw-Hill Inc (1953).
  • Petrovsky, I.G., Lectures of Partial Differential Equations, Willey (Intersciense) (1954).
  • Renardy, M. & Rogers, R.C, An Introduction to Partial Differential Equations, Springer-Verlag


  • Shaw, W.T., Complex Analysis with Mathematica, Cambridge University Press (2006).
  • Shilov, G.E., Elementary Real and Complex Analysis of Partial Differential Equations, Dover


  • Silverman, R.A., Complex Analysis with Applications, Dover (1984).
  • Sneddon, I., Elements of Partial Differential Equations, McGraw-Hill, Inc (1985).


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