# NEW LAPLACE-TYPE INTEGRAL TRANSFORM FOR SOLVING STEADY HEAT-TRANSFER PROBLEM

Shehu Maitama*, Weidong Zhao  School of Mathematics, Shandong University,
Jinan, People’s Republic of China

###### Abstract

The fundamental purpose of this paper is to propose a new Laplace-type
integral transform (NL-TIT) for solving steady heat-transfer problems. The
proposed integral transform is a generalization of the Sumudu, and the Laplace
transforms and its visualization is more comfortable than the Sumudu
transform, the natural transform, and the Elzaki transform. The suggested
integral transform is used to solve the steady heat-transfer problems, and
results are compared with the results of the existing techniques.

###### keywords:

integral transforms, analytical solutions, heat transfer problems, Laplace-
type integral transform.

††journal: Thermal Science **footnotetext: *Corresponding author e-mail
address<EMAIL_ADDRESS>(S. Maitama<EMAIL_ADDRESS>(W. Zhao).

##  1 Introduction

For more than 150 years, the motivation behind integral transforms is easy to
understand. The integral transforms have a widely-applicable spirit of
converting differential operators into multiplication operators from its
original domain into another domain. Besides, the symbolic manipulating and
solving the equation in the new domain is easier than manipulation and
solution in the original domain of the problem [1]-[8]. The inverse integral
transforms are always used to mapped the manipulated solution back to the
original domain to obtain the required result.

In the mathematical literature, the famous classical integral transforms used
in differential equations, analysis, theory of functions and integral
transforms are the Laplace transform [9] which was first introduced by a
French mathematician Pierre-Simon Laplace (1747-1827), the Fourier integral
transform [10] devised by another French mathematician Joseph Fourier
(1768-1830), and the Mellin integral transform [11] which was introduced by a
Finnish mathematician Hjalmar Mellin (1854-1933). Besides, the Laplace
transform, the Fourier transform, and the Mellin integral transforms are
similar, except in different coordinates and have many applications in science
and engineering [12]. Moreover, in mathematics there are many Laplace-types
integral transforms such as the Laplace-Carson transform used in the railway
engineering [13], the z-transform applied in signal processing [14], the
Sumudu transform used in engineering and many real-life problems [15], the
Hankel’s and Weierstrass transform applied in heat and diffusion equations
[16, 17]. In addition, we have the natural transform [18] and Yang transform
[19, 20] used in many fields of physical science and engineering.

This paper aims to further introduce a suitable Laplace-type integral
transform for solving steady heat-transfer problems. We will prove some
important theorems and properties of the suggested integral transform and
illustrated their applications. In the next section, we begin with the
definition of the proposed Laplace-type integral transform and introduce some
useful theorems of the integral transform.

## Definition and Theorems

###### Definition 1.1

The new Laplace-type integral transform of the function $v(t)$ of exponential
order is defined over the set of functions,
$A=\left\\{v(t):\exists\,C,\,\xi_{1},\,\xi_{2}>0,\,\,\left|v(t)\right|<Ce^{\frac{\left|t\right|}{\xi_{i}}},\,\textnormal{if}\
t\in(-1)^{i}\times\left[0,\infty\right)\right\\}$,
by the following integral:

$\Theta\left[v(t)\right](s,u)=V(s,u)=u\int_{0}^{\infty}e^{-st}\,v(ut)\,dt=\lim_{\beta\rightarrow\infty}\int_{0}^{\beta}e^{\frac{-st}{u}}v(t)dt,\,\,s\,>0,\,\,u>0,$
(1)

where $\Theta$ is the NL-TIT operator. It converges if the limit of the
integral exists, and diverges if not.

The inverse NL-TI transform is given by:

${\Theta^{-1}}\left[V(s,u)\right]=v(t),\,\,for\,\,t\geq 0.$ (2)

Equivalently, the complex inversion formula of the NL-TI transform is given
by:

$v(t)={\Theta^{-1}}\left[V(s,u)\right]=\frac{1}{2\pi
i}\int_{\beta-i\infty}^{\beta+i\infty}\frac{1}{u}e^{\frac{st}{u}}V(s,u)\,ds,\,\,t>0,$
(3)

where $s$ and $u$ are the NL-TI transform variables, and $\beta$ is a real
constant and the integral in eq. (3) is computed along $s=\beta$ in the
complex plane $s=x+iy$.

###### Theorem 1.1

The sufficient condition for the existence of the new Laplace-type integral
transform. If the function $v(t)$ is piecewise continues on every finite
interval $[0,t_{0}]$ and satisfies

$|v(t)|\leq Ce^{\beta t},$ (4)

for all $t\in[t_{0},\infty)$, and a constant $\beta$, then
${\Theta}\left[v(t)\right](s,u)$ exists for all $\frac{s}{u}>\beta$.

Proof.

To prove the theorem, we must first show that the improper integral converges
for $\frac{s}{u}>\beta$. Without loss of generality, we first split the
improper integral into two parts namely:

$\int_{0}^{\infty}e^{-\frac{st}{u}}v(t)dt=\int_{0}^{t_{0}}e^{-\frac{st}{u}}v(t)dt+\int_{t_{0}}^{\infty}e^{-\frac{st}{u}}v(t)dt.$
(5)

The first integral on the right hand side of eq. (5) exists by the first
hypothesis, hence the existence of the Laplace-type integral transform
completely depends on the second integral. Then by the second hypothesis we
deduce:

$\left|e^{-\frac{st}{u}}v(t)\right|\leq Ce^{-\frac{st}{u}}e^{\beta
t}=Ce^{-\frac{(s-\beta u)t}{u}}.$ (6)

Thus

$\int_{t_{0}}^{\infty}Ce^{-\frac{(s-\beta u)t}{u}}dt=\frac{u}{s-\beta
u}Ce^{-\frac{(s-\beta u)t_{0}}{u}}.$ (7)

Hence, eq. (7) converges for $\beta<\frac{s}{u}$. This implies by the
comparison test for improper integrals theorem,
${\Theta}\left[v(t)\right](s,u)$ exists for $\beta<\frac{s}{u}$.This complete
the proof. $\Box$

In the next theorem, we prove the uniqueness of the NL-TI transform.

###### Theorem 1.2

Uniqueness of the new Laplace-type integral transform.

Let $v(t)$ and $w(t)$ be continuous functions defined for $t\geq 0$ and having
NL-TI transforms of $V(s,u)$ and $W(s,u)$ respectively. If $V(s,u)=W(s,u)$,
then $v(t)=w(t)$.

Proof.

From the inverse NL-TI transform eq. (3), we have

$v(t)=\frac{1}{2\pi
i}\int_{\beta-i\infty}^{\beta+i\infty}\frac{1}{u}e^{\frac{st}{u}}V(s,u)\,ds.$
(8)

Since $V(s,u)=W(s,u)$ by the second hypothesis, then replacing this in eq.
(8), we obtain

$v(t)=\frac{1}{2\pi
i}\int_{\beta-i\infty}^{\beta+i\infty}\frac{1}{u}e^{\frac{st}{u}}W(s,u)\,ds.$
(9)

This implies

$v(t)=\frac{1}{2\pi
i}\int_{\beta-i\infty}^{\beta+i\infty}\frac{1}{u}e^{\frac{st}{u}}V(s,u)\,ds=w(t).$
(10)

Hence, eq. (10) proves the uniqueness of the NL-TI transform. $\Box$

###### Theorem 1.3

Convolution theorem of the NL-TI transform. Let the functions $v(t)$ and
$w(t)$ be in set A. If $V(s,u)$ and $W(s,u)$ are the NL-TI transforms of the
functions $v(t)$ and $w(t)$ respectively, then

${\Theta}\left[(v*w)(t)\right]=V(s,u)W(s,u).$ (11)

Where $v*w$ is the convolution of two functions $v(t)$ and $w(t)$ which is
defined as:

$\int_{0}^{t}v(\tau)w(t-\tau)d\tau=\int_{0}^{t}v(t-\tau)w(\tau)d\tau.\\\ $
(12)

Proof.

Based on eq. (1) and eq. (12), we get:

$\displaystyle{\Theta}\left[\int_{0}^{t}v(\tau)w(t-\tau)\right]$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}e^{\frac{-st}{u}}\left(\int_{0}^{t}v(\tau)w(t-\tau)\right)d\tau.$

Changing the limit of integration yields:

$\displaystyle{\Theta}\left[\int_{0}^{t}v(\tau)w(t-\tau)\right]$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}\left(v(\tau)\int_{\tau}^{\infty}e^{\frac{-st}{u}}w(t-\tau)dt\right)d\tau.$

Substituting $\vartheta=t-\tau$ in the inner integral, we deduce:

$\displaystyle\int_{\tau}^{\infty}e^{\frac{-st}{u}}w(t-\tau)dt=\int_{0}^{\infty}e^{-\frac{(\vartheta+\tau)}{u}s}w(\vartheta)d\vartheta$
$\displaystyle=$ $\displaystyle e^{-\frac{\tau
s}{u}}\int_{0}^{\infty}e^{-\frac{\vartheta
s}{u}}w(\vartheta)d\vartheta=e^{-\frac{\tau s}{u}}W(s,u).$

Hence,

$\displaystyle{\Theta}\left[\int_{0}^{t}v(\tau)w(t-\tau)\right]=\int_{0}^{\infty}v(\tau)e^{\frac{-s\tau}{u}}W(s,u)d\tau$
(13) $\displaystyle=$ $\displaystyle
W(s,u)\int_{0}^{\infty}v(\tau)e^{\frac{-s\tau}{u}}d\tau=V(s,u)W(s,u).$

This complete the proof. $\Box$

###### Theorem 1.4

Derivative of the NL-TI transform. Suppose that
${\Theta}\left[v(t)\right](s,u)$ exists and that $v(t)$ is differentiable
n-times on the interval $(0,\infty)$ with $n^{th}$ derivative $v^{(n)}(t)$,
then

${\Theta}\left[v^{\prime}(t)\right](s,u)=\frac{s}{u}V(s,u)-v(0),$ (14)
${\Theta}\left[v^{\prime\prime}(t)\right](s,u)=\frac{s^{2}}{u^{2}}V(s,u)-\frac{s}{u}v(0)-v^{\prime}(0),$
(15)
${\Theta}\left[v^{\prime\prime\prime}(t)\right](s,u)=\frac{s^{3}}{u^{3}}V(s,u)-\frac{s^{2}}{u^{2}}v(0)-\frac{s}{u}v^{\prime}(0)-v^{\prime\prime}(0),$
(16) $\vdots$
${\Theta}\left[v^{(n)}(t)\right](s,u)=\frac{s^{n}}{u^{n}}V(s,u)-\sum_{k=0}^{n-1}\left(\frac{s}{u}\right)^{n-(k+1)}v^{(k)}(0).$
(17)

Proof.

Using Definition 1.1 of the NL-TI transform and integration by parts, we
deduce:

$\displaystyle{\Theta}\left[v^{\prime}(t)\right](s,u)$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}e^{\frac{-st}{u}}v^{\prime}(t)dt$ (18)
$\displaystyle=$
$\displaystyle-v(0)+\frac{s}{u}\int_{0}^{\infty}e^{\frac{-st}{u}}v(t)dt=-v(0)+\frac{s}{u}V(s,u).$

$\displaystyle{\Theta}\left[v^{\prime\prime}(t)\right](s,u)$ $\displaystyle=$
$\displaystyle\frac{s}{u}{\Theta}\left[v^{\prime}(t)\right](s,u)-v^{\prime}(0)=\frac{s}{u}\left[-v(0)+\frac{s}{u}V(s,u)\right]-v^{\prime}(0)$
(19) $\displaystyle=$
$\displaystyle-v^{\prime}(0)-\frac{s}{u}v(0)+\frac{s^{2}}{u^{2}}V(s,u).$

$\displaystyle{\Theta}\left[v^{\prime\prime\prime}(t)\right](s,u)$
$\displaystyle=$
$\displaystyle\frac{s}{u}{\Theta}\left[v^{\prime\prime}(t)\right](s,u)-v^{\prime\prime}(0)=\frac{s}{u}\left[-v^{\prime}(0)-\frac{s}{u}v(0)+\frac{s^{2}}{u^{2}}V(s,u)\right]-v^{\prime\prime}(0)$
(20) $\displaystyle=$
$\displaystyle-v^{\prime\prime}(0)-\frac{s}{u}v^{\prime}(0)-\frac{s^{2}}{u^{2}}v(0)+\frac{s^{3}}{u^{3}}V(s,u).$

Finally, eq. (17) follows using mathematical induction. $\Box$

In the next theorems, we prove the NL-TI transform of Riemann-Liouville
fractional derivative ${}^{RL}\hskip-2.5ptD_{t}^{\alpha}$ [6], and the Caputo
fractional derivative ${}^{C}\hskip-2.5ptD_{t}^{\alpha}$ [6].

###### Theorem 1.5

New Laplace-type integral transform of Riemann-Liouville fractional
derivative. If $\alpha>0,\,n=1+[\alpha]$ and $v(t)$, $I^{n-\alpha}v(t)$,
$\frac{d}{dt}I^{n-\alpha}v(t)$,
$\cdots,\,\frac{d^{n}}{dt^{n}}I^{n-\alpha}v(t)$,
${}^{RL}\hskip-2.5ptD^{\alpha}$ $v(t)\in A$, then

$\Theta\left[{}^{RL}\hskip-2.5ptD_{t}^{\alpha}v(t)\right]=\left(\frac{s}{u}\right)^{\alpha}{\Theta}\left[v(t)\right]-\sum_{k=0}^{n-1}\left(\frac{s}{u}\right)^{n-k-1}\frac{d^{k-1}}{dt^{k-1}}I^{n-\alpha}v(0+),$
(21)

where $I^{\alpha}$ is the Riemann-Liouville fractional integral.

Proof.

Since
${}^{RL}\hskip-2.5ptD_{t}^{\alpha}v(t)=\frac{d^{n}}{dt^{n}}I^{n-\alpha}v(t)$.
Let $g(t)=I^{n-\alpha}v(t)$, then
${}^{RL}\hskip-2.5ptD_{t}^{\alpha}v(t)=\frac{d^{n}}{dt^{n}}v(t)$. Applying the
hypothesis of theorem (1.4), we get

$\displaystyle\Theta\left[{}^{RL}\hskip-2.5ptD_{t}^{\alpha}v(t)\right]=\left(\frac{s}{u}\right)^{n}{\Theta}\left[v(t)\right]-\sum_{k=0}^{n-1}\left(\frac{s}{u}\right)^{n-k-1}v^{(k)}(0+)$
$\displaystyle=$
$\displaystyle\left(\frac{s}{u}\right)^{\alpha}{\Theta}\left[v(t)\right]-\sum_{k=0}^{n-1}\left(\frac{s}{u}\right)^{n-k-1}\frac{d^{k-1}}{dt^{k-1}}I^{n-\alpha}v(0+).$

The proof ends. $\Box$

###### Theorem 1.6

New Laplace-type integral transform of Caputo fractional derivative. Assume
$\alpha>0,\,n=1+[\alpha]$ and
$v(t)$, $\frac{d}{dt}v(t)$, $\frac{d^{2}}{dt^{2}}v(t)$, $\cdots$,
$\frac{d^{n}}{dt^{n}}v(t)$, ${}^{C}\hskip-2.5ptD^{\alpha}$ $v(t)\in A$, then

$\Theta\left[{}^{C}\hskip-2.5ptD_{t}^{\alpha}v(t)\right]=\left(\frac{s}{u}\right)^{\alpha}{\Theta}\left[v(t)\right]-\sum_{k=0}^{n-1}\left(\frac{s}{u}\right)^{\alpha-k-1}v^{(k)}(0+),$
(22)

where ${}^{C}\hskip-2.5ptD_{t}^{\alpha}$ is the Caputo fractional derivative.

Proof.

Applying the Caputo fractional derivative [6] and theorem (1.3), we deduce:

${}^{C}\hskip-2.5ptD_{t}^{\alpha}v(t)$ $\displaystyle=$
$\displaystyle\frac{1}{\Gamma(n-\alpha)}\int_{0}^{t}(t-\tau)^{n-\alpha-1}v^{(n)}(\tau)d\tau$
$\displaystyle=$
$\displaystyle\frac{1}{(n-\alpha)}t^{n-\alpha-1}*v^{(n)}(\tau).$

Finally, using the hypothesis of theorem (1.4) yields:

$\displaystyle\Theta\left[{}^{C}\hskip-2.5ptD_{t}^{\alpha}v(t)\right]=\frac{1}{\Gamma(n-\alpha)}\Theta\left[t^{n-\alpha-1}\right]\Theta\left[v^{(n)}(\tau)\right]$
$\displaystyle=$
$\displaystyle\left(\frac{s}{u}\right)^{\alpha}{\Theta}\left[v(t)\right]-\sum_{k=0}^{n-1}\left(\frac{s}{u}\right)^{\alpha-k-1}v^{(k)}(0+).$

This complete the proof. $\Box$

## Some Properties of the NL-TI transform

###### Property 1

Linearity property of the NL-TI transform. Let
${\Theta}\left[v(t)\right](s,u)=V(s,u)$ and
${\Theta}\left[w(t)\right](s,u)=W(s,u)$, then

${\Theta}\left[\alpha v(t)+\beta
w(t)\right](s,u)=\alpha{\Theta}\left[v(t)\right](s,u)+\beta{\Theta}\left[w(t)\right](s,u),$
(23)

where $\alpha$ and $\beta$ are constants.

Proof.

Linearity property follows directly from Definition 1.1. $\Box$

###### Property 2

Exponential Shifting Property of the NL-TI transform. Let the function
$v(t)\in A$ and $\alpha$ is an arbitrary constant, then

${\Theta}\left[e^{\alpha t}v(t)\right](s,u)=V(s-\alpha u).\\\ $ (24)

Proof.

Using Definition 1.1 of the NL-TI transform, we get:

$\displaystyle{\Theta}\left[v(t)\right](s,u)$ $\displaystyle=$ $\displaystyle
u\int_{0}^{\infty}e^{-st}v(ut)dt.$ (25)

Then

$\displaystyle{\Theta}\left[e^{\alpha t}v(t)\right](s,u)$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}e^{(\alpha
t)}e^{\frac{-st}{u}}v(t)dt=\int_{0}^{\infty}e^{-\frac{(s-\alpha u)}{u}}v(t)dt$
(26) $\displaystyle=$ $\displaystyle u\int_{0}^{\infty}e^{-(s-\alpha
u)t}v(ut)dt={\Theta}\left[v(t)\right](s-\alpha u)=V(s-\alpha u).$

In particular,

${\Theta}\left[\sin(3t)e^{-4t}\right](s,u)={\Theta}\left[\sin(3t)\right](s-(-4u))={\Theta}\left[\sin(3t)\right](s+4u).$
(27)

Based on Definition 1.1, the NL-TI transform of $\sin(3t)$ is given by:

${\Theta}\left[\sin(3t)\right](s,u)=\int_{0}^{\infty}e^{\frac{-st}{u}}\sin(3t)dt=\frac{1}{2i}\left(\frac{u}{s-3iu}-\frac{u}{s+3iu}\right)=\frac{3u^{2}}{s^{2}+9u^{2}}.$
(28)

So, replacing the variable $s$ with $(s+4u)$ in eq. (28), we obtain:

${\Theta}\left[\sin(3t)\right](s+4u)=\frac{3u^{2}}{(s+4u)^{2}+9u^{2}}=\frac{3u^{2}}{s^{2}+8us+25u^{2}}.$
(29)

Alternatively,

$\int_{0}^{\infty}\sin(3t)e^{-4t}e^{\frac{-st}{u}}dt=\frac{3u^{2}}{s^{2}+8us+25u^{2}}.$
(30)

Moreover,

${\Theta}\left[te^{\alpha t}\right](s,u)=\frac{u^{2}}{(s-\alpha
u)^{2}}=\left\\{\begin{array}[]{ll}\frac{1}{(s-\alpha)^{2}},\,\,u=1,\,\,\,\textnormal{Laplace
transform}\ [6]&\\\ \\\ \frac{u^{2}}{(1-\alpha
u)^{2}},\,\,s=1,\,\,\,\textnormal{Yang transform}\ [19].&\\\ \\\
\end{array}\right.$ (31)

This complete the proof. $\Box$

###### Property 3

New Laplace-type transform of integral. Let ${\Theta}\left[v(t)\right]=V(s,u)$
and $v(t)\in A$, then

${\Theta}\left[\int_{0}^{t}v(\zeta)d\zeta\right]=\frac{u}{s}V(s,u).$ (32)

Proof.

Let $w(t)=\int_{0}^{t}v(\zeta)d\zeta$, then $w^{\prime}(t)=v(t)$ and $w(0)=0$.
Computing the NL-TI transform of both sides, we get:

$\displaystyle{\Theta}\left[w^{\prime}(t)\right](s,u)$ $\displaystyle=$
$\displaystyle{\Theta}\left[v(t)\right](s,u)=\frac{s}{u}{\Theta}\left[w(t)\right](s,u)-w(0)=V(s,u).$
(33)

This implies

${\Theta}\left[\int_{0}^{t}v(\zeta)d\zeta\right]=\frac{u}{s}V(s,u).$ (34)

This complete the proof. $\Box$

###### Property 4

Multiple Shift Property of the NL-TI transform. Let
${\Theta}\left[v(t)\right](s,u)=V(s,u)$ and $v(t)\in A$, then

${\Theta}\left[t^{n}v(t)\right](s,u)=(-u)^{n}\frac{d^{n}}{ds^{n}}V(s,u).$ (35)

Proof.

By Definition 1.1 of the NL-TI transform and Leibniz s rule, we obtain:

$\displaystyle\frac{d}{ds}V(s,u)$ $\displaystyle=$
$\displaystyle\frac{d}{ds}\int_{0}^{\infty}e^{\frac{-st}{u}}v(t)dt=\int_{0}^{\infty}\frac{\partial}{\partial
s}\left(e^{\frac{-st}{u}}\right)v(t)dt$ (36) $\displaystyle=$
$\displaystyle-\frac{1}{u}\int_{0}^{\infty}e^{\frac{-st}{u}}tv(t)dt={\Theta}\left[tv(t)\right](s,u)=-u\frac{d}{ds}V(s,u).$

Thus, eq. (36) above proves the theorem for $n=1$. To generalized the theorem,
we apply the induction hypothesis. Let assume the theorem holds for $n=k$ that
is

$\displaystyle\int_{0}^{\infty}e^{\frac{-st}{u}}t^{k}v(t)dt=(-u)^{k}\frac{d^{k}}{ds^{k}}V(s,u).$
(37)

Then

$\displaystyle\frac{d}{ds}\int_{0}^{\infty}e^{\frac{-st}{u}}t^{k}v(t)dt=(-u)^{k}\frac{d^{k+1}}{ds^{k+1}}V(s,u).$
(38)

Alternatively, using Leibniz s rule, we deduce:

$\displaystyle\frac{d}{ds}\int_{0}^{\infty}e^{\frac{-st}{u}}t^{k}v(t)dt=\int_{0}^{\infty}\frac{\partial}{\partial
s}\left(e^{\frac{-st}{u}}\right)t^{k}v(t)dt$ (39) $\displaystyle=$
$\displaystyle-\frac{1}{u}\int_{0}^{\infty}e^{\frac{-st}{u}}t^{k+1}v(t)dt=(-u)^{k}\frac{d^{k+1}}{ds^{k+1}}V(s,u).$

This implies

$\displaystyle\int_{0}^{\infty}e^{\frac{-st}{u}}t^{k+1}v(t)dt=(-u)^{k+1}\frac{d^{k+1}}{ds^{k+1}}V(s,u).$
(40)

Since, eq. (40) holds for $n=k+1$, then by induction hypothesis the prove is
complete. $\Box$

## Applications

In this section, we illustrate the applicability of the proposed Laplace-type
integral transform on steady heat-transfer problems to proves its efficiency
and high accuracy.

###### Example 1

Consider the following steady heat-transfer problem:

$-hMv(t)=\rho\Lambda_{c_{p}}v^{\prime}(t),$ (41)

subject to the initial condition

$v(0)=\beta,$ (42)

where $h-$ is the convection heat transfer coefficient, $M-$ is the surface
area of the body, $\rho-$ is the density of the body, $\Lambda-$ is the
volume, $c_{p}-$ is the specific heat of the material, and $v(t)$ is the
temperature.

Applying the NL-TI transform on both sides of eq. (41), we get:

$-hMV(s,u)=\rho\Lambda_{c_{p}}\left(\frac{s}{u}V(s,u)-v(0)\right).$ (43)

Substituting the given initial condition and simplifying, we get:

$V(s,u)=\frac{\beta u}{s+\frac{hM}{\rho\Lambda_{c_{p}}}u}$ (44)

Taking the inverse NL-TI transform of eq. (44), we get:

$v(t)=\beta e^{-\frac{hM}{\rho\Lambda_{c_{p}}}t}.$ (45)

The exact solution is in excellent agreement with the result obtained in [5,
20].

###### Example 2

Consider the following steady heat-transfer problem:

$v_{t}(x,t)=2v_{xx}(x,t),\,\,\,\,0<x<5,\,\,\,\,t>0.$ (46)

Subject to the boundary and initial conditions

$v(0,t)=0,\,\,\,v(5,t)=0,\,\,\,\,v(x,0)=10\sin(4\pi x)-5\sin(6\pi x).$ (47)

Applying the NL-TI transform on both sides of eq. (46), we deduce:

$\frac{s}{u}V(x,s,u)-v(x,0)=\frac{2d^{2}V(x,s,u)}{dx^{2}}.$ (48)

Substituting the given initial condition and simplifying, we get:

$\frac{2d^{2}V(x,s,u)}{dx^{2}}-\frac{s}{u}V(x,s,u)=-10\sin(4\pi x)+5\sin(6\pi
x).$ (49)

The general solution of eq. (49) can be written as:

$V(x,s,u)=V_{h}(x,s,u)+V_{p}(x,s,u),$ (50)

where $V_{h}(x,s,u)$ is the solution of the homogeneous part which is given
by:

$V_{h}(x,s,u)=\alpha_{1}e^{\sqrt{\frac{s}{u}}x}+\alpha_{2}e^{-\sqrt{\frac{s}{u}}x},$
(51)

and $V_{p}(x,s,u)$ is the solution of the inhomogeneous part which is given
by:

$V_{p}(x,s,u)=\alpha\sin(4\pi x)+\beta\sin(6\pi x).$ (52)

Applying the boundary conditions on eq. (51), yields

$\alpha_{1}+\alpha_{2}=0\Rightarrow\alpha_{1}e^{\sqrt{\frac{s}{u}}}+\alpha_{2}e^{-\sqrt{\frac{s}{u}}}=0\Rightarrow
V_{h}(x,s,u)=0,$ (53)

since, $\alpha_{1}=\alpha_{2}=0.$
Using the method of undetermined coefficients on the inhomogeneous part, we
get:

$V_{p}(x,s,u)=10\sin(4\pi x)\frac{u}{s+32\pi^{2}u}-5\sin(6\pi
x)\frac{u}{s+72\pi^{2}u},$ (54)

since, $\alpha=10\frac{u}{s+32\pi^{2}u},$ and
$\beta=-5\frac{u}{s+72\pi^{2}u}$.

Then eq. (50) will become:

$V(x,s,u)=10\sin(4\pi x)\frac{u}{s+32\pi^{2}u}-5\sin(6\pi
x)\frac{u}{s+72\pi^{2}u}.$ (55)

Taking the inverse NL-TI transform of eq. (55), we obtain:

$v(x,t)=10e^{-32\pi^{2}t}\sin(4\pi x)-5e^{-72\pi^{2}t}\sin(6\pi x).$ (56)

The exact solution is the same with the result obtained in [9].

###### Example 3

Consider the following fractional porous medium equation:

$D_{t}^{\alpha}v(x,t)=D_{x}(v(x,t)D_{x}v(x,t)),\,\,0<\alpha\leq 1,$ (57)

subject to the initial condition

$v(x,0)=x.$ (58)

Applying theorem (1.6) on eq. (57) subject to the initial condition, we
obtain:

$\Theta\left[v(x,t)\right]=\frac{u}{s}x+\frac{u^{\alpha}}{s^{\alpha}}\Theta\left[D_{x}(v(x,t)D_{x}v(x,t))\right].$
(59)

Computing the inverse NL-TI transform on both sides of eq. (59), we deduce:

$v(x,t)=x+\Theta^{-1}\left[\frac{u^{\alpha}}{s^{\alpha}}\Theta\left[D_{x}(v(x,t)D_{x}v(x,t))\right]\right].$
(60)

Based on the basic idea of the homotopy analysis method (see [6] and
references therein), we have:

$v(x,t)=\sum_{n=0}^{\infty}p^{n}v_{n}(x,t).$ (61)

Then eq. (60) will become:

$\sum_{n=0}^{\infty}p^{n}v_{n}(x,t)=x+p\left(\Theta^{-1}\left[\frac{u^{\alpha}}{s^{\alpha}}\Theta\left[D_{x}\left(\sum_{n=0}^{\infty}p^{n}H_{n}\right)\right]\right]\right),$
(62)

where $H_{n},$ is the He’s polynomials [6] which represent the nonlinear terms
$v(x,t)D_{x}v(x,t)$.
Some few components of the nonlinear terms $H_{n}$ are computed below:

$H_{0}=v_{0}v_{0x},\,\,H_{1}=v_{0}v_{1x}+v_{1}v_{0x},\,\,H_{2}=v_{0x}v_{2}+v_{1x}v_{1}+v_{2x}v_{0},\,\,\cdots$

On comparing the coefficients of same powers of $p$ in eq. (62), the we
determine the following approximations:

$\displaystyle p^{0}\colon v_{0}(x,t)$ $\displaystyle=$ $\displaystyle x,$
$\displaystyle p^{1}\colon v_{1}(x,t)$ $\displaystyle=$
$\displaystyle\Theta^{-1}\left[\frac{u^{\alpha}}{s^{\alpha}}\Theta\left[D_{x}\left(H_{0}\right)\right]\right]$
$\displaystyle=$ $\displaystyle\frac{t^{\alpha}}{\Gamma(1+\alpha)},$
$\displaystyle p^{2}\colon v_{2}(x,t)$ $\displaystyle=$
$\displaystyle\Theta^{-1}\left[\frac{u^{\alpha}}{s^{\alpha}}\Theta\left[D_{x}\left(H_{1}\right)\right]\right]$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle\vdots$ $\displaystyle
p^{n}\colon v_{n}(x,t)$ $\displaystyle=$
$\displaystyle\Theta^{-1}\left[\frac{u^{\alpha}}{s^{\alpha}}\Theta\left[D_{x}\left(H_{n-1}\right)\right]\right]$
$\displaystyle=$ $\displaystyle 0,\,\,\textnormal{for}\ \,\,n\geq 2.$

Then the solution of eq. (57)-(58) is given by:

$\displaystyle v(x,t)$ $\displaystyle=$ $\displaystyle
x+\frac{t^{\alpha}}{\Gamma(1+\alpha)}.$ (63)

The result obtained in eq. (63) is in excellent agreement with the result
obtained in [6]. The special case of eq. (63) when $\alpha=1$ is given by:

$v(x,t)=x+t.$ (64)

The result of eq. (64) is in closed agreement with the result obtained in [6,
7].

## Conclusion

In this paper, we introduced a powerful Laplace-type integral transform for
finding a solution of steady heat-transfer problems. The proposed Laplace-type
integral transform converges to both Yang transform, and the Laplace
transforms just by changing variables. Many interesting properties of the
suggested integral transform are discussed and successfully applied to steady
heat-transfer problems. Finally, based on the efficiency and simplicity of the
Laplace-type integral transform, we conclude that it is a powerful
mathematical tool for solving many problems in science and engineering.

## Acknowledgment

The authors would like to thank the anonymous reviewers, managing editor, and
editor in chief for their valuable help in improving the manuscript. This work
is supported by the Natural Science Foundation of China (Grand No. 11571206).
The first author acknowledges the financial support of China Scholarship
Council (CSC) in Shandong University (Grand No. 2017GXZ025381).

## Nomenclature

Greek symbols
---
$c_{p}-$ is the specific heat of the material, [Jk$g^{-1}K^{-1}$] |  $x,t-$space co-ordinates, [m]
$h-$ is the convection heat transfer coefficient, [W$m^{-2}K^{-1}$] |  $v(t)-$ temperature, [K]
|  $v(x,t)-$temperature, [K]

## References

  * [1] Lokenath D., Bhatta, D.,  Integral Transform and Their Applications,  CRC Press, Boca Raton, Fla., USA., 2014\.
  * [2] Srivastava, H.M., Luo, M., Raina, R.K.,  A New Integral Transform and Its Applications,  Acta Mathematica Scientia, 35 (2015), 6, pp. 1386–1400.
  * [3] Yang, X.J.,  New Integral Transforms for Solving a Steady Heat-Transfer Problem,  Thermal Science, 21 (2017), pp. S79–S87.
  * [4] Yang, X.J.,  A New Integral Transform with an Application in Heat-Transfer Problem,  Thermal Science, 20 (2016), pp. S677–S681.
  * [5] Goodwine, B.,  Engineering Differential Equations: Theory and Applications,  Springer, New York, USA, 2010\.
  * [6] Yan, L.M.,  Modified Homotopy perturbation Method Coupled with Laplace Transform for Fractional Heat Transfer and Porous Media Equations,  Thermal Science, 17 (2013), 5, pp. 1409–1414.
  * [7] Pamuk, S.,  Solution of the Porous Media Equation by Adomian s Decomposition Method,  Physics Letters A, 344 (2005), pp. 184–188.
  * [8] Elzaki, T.M.,  The New Integral Transform ”Elzaki transform”,  Glob. J. of Pur. and Appl. Math., 7 (2011), 1, pp. 57–64.
  * [9] Spiegel, M.R.,  Theory and Problems of Laplace Transforms,  Schaum’s Outline Series, McGraw–Hill, New York, USA., 1965\.
  * [10] Bracewell, R.N.,  The Fourier Transform and Its Applications,  McGraw-Hill, Boston, Mass, USA., 2000.
  * [11] Boyadjiev, L., Luchko, Y.,  Mellin Integral Transform Approach to Analyze the Multidimensional Diffusion-Wave Equations,  Chaos Solitons and Fractals,, 102 (2017), pp. 127-134.
  * [12] Dattoli, G., Martinelli, M.R., Ricci, P.E.,  On New Families of Integral Transforms for the Solution of Partial Differential Equations,  Integral Transforms and Special Functions, 8 (2005), pp. 661–667.
  * [13] Lévesque, M., et al.,  Numerical Inversion of the Laplace-Carson Transform Applied to Homogenization of Randomly Reinforced Linear Viscoelastic Media,  Comput Mech., 40 (2007), pp. 771–789.
  * [14] Cui, Y.L., et al.,  Application of the Z-Transform Technique to Modeling the Linear Lumped Networks in the HIE-FDTD Method,  Journal of Electromagnetic Waves and Applications, 27 (2013), 4, pp. 529–538.
  * [15] Watugala, G.K.,  Sumudu Transform-A New Integral Transform to Solve Differential Equations and Control Engineering Problems,  Math Eng in Indust., 6 (1998), 1, pp. 319–329.
  * [16] Shah, P.C., Thambynayagam, R.K.M.,  Application of the Finite Hankel Transform to a Diffusion Problem Without Azimuthal Symmetry,  Transport in Porous Media, 14 (1994), 3, pp. 247–264.
  * [17] Karunakaran, V., Venugopal, T.,  The Weierstrass Transform for a Class of Generalized Functions,  Journal of Mathematical Analysis and Applications, 220 (1998), 2, pp. 508-527.
  * [18] Belgacem, F.B.M., Silambarasan, R.,  Theory of Natural Transform.  Math. in Eng. Sci., and Aeros., 3 (2012), 1, pp. 99–124.
  * [19] Yang, X.J.,  A New Integral Transform Operator for Solving the Heat-Diffusion Problem,  Applied Mathematics Letters, 64 (2017), pp. 193–197.
  * [20] Yang, X.J.,  A New Integral Transform Method for Solving Steady Heat-Transfer Problem,  Thermal Science, 20 (2016), pp. S639–S642.


# ERATTA: Extreme RAG for Table To Answers with Large Language Models

Sohini Roychowdhury1, Marko Krema1, Anvar Mahammad1, Brian Moore1, Arijit
Mukherjee 4, Punit Prakashchandra 4 4CDAO, Accenture, India  1 Corporate Data
and Analytics Office (CDAO), Accenture LLP, USA, Email:
<EMAIL_ADDRESS>
###### Abstract

Large language models (LLMs) with retrieval augmented-generation (RAG) have
been the optimal choice for scalable generative AI solutions in the recent
past. However, the choice of use-cases that incorporate RAG with LLMs have
been either generic or extremely domain specific, thereby questioning the
scalability and generalizability of RAG-LLM approaches. In this work, we
propose a unique LLM-based system where multiple LLMs can be invoked to enable
data authentication, user query routing, data retrieval and custom prompting
for question answering capabilities from data tables that are highly varying
and large in size. Our system is tuned to extract information from Enterprise-
level data products and furnish real time responses under 10 seconds. One
prompt manages user-to-data authentication followed by three prompts to route,
fetch data and generate a customizable prompt natural language responses.
Additionally, we propose a five metric scoring module that detects and reports
hallucinations in the LLM responses. Our proposed system and scoring metrics
achieve $>90\%$ confidence scores across hundreds of user queries in the
sustainability, financial health and social media domains. Extensions to the
proposed extreme RAG architectures can enable heterogeneous source querying
using LLMs.

###### Index Terms:

Retrieval Augmented Generation, authentication, sustainability, hallucination,
Unicorn, TextBison

##  I Introduction

Generative AI solutions using Large Language Models (LLMs) have been extremely
popular since the launch of Open AI’s chatGPT-3 in late 2022 [1]. Most LLM
solutions today can typically be categorized as Gen-AI applications, such as a
chatbot-like experience with a virtual assistant for code, language and
content generation [2]; or Gen-AI agents that are capable of automated
orchestration to perform specific tasks such as booking tickets, writing
blogs, articles and generating software automatically [3]. Most such solutions
implement the retrieval augmented-generation (RAG) approach, wherein the most
relevant data sources are first identified followed by the knowledge-graph
method to isolate only relevant data entities that are collated with natural
language instructions and answering guidelines and sent to the LLM to receive
the required responses [4]. In this paper, we present a novel system
architecture that maximizes the RAG impact through multiple LLM calls for
tasks such as user authentication, query routing and natural language
question-answering while ensuring accuracy standards by flagging
hallucinations, security standards and low throughput.

The RAG approaches are scalable and efficient for knowledge retrieval with
lower maintenance and operating costs when compared to LLM fine tuning [5].
For use-cases that require question-answering from large volumes of tabular
data at Enterprise level, RAG approaches have been accurate and scalable as
presented in [2]. The three major capabilities of RAG with LLMs for the
Tables-to-answers use-cases are as follows: 1) user-access authentication for
tabular data based on minimal user profile data, 2) semantic-selection of the
most relevant data chunks from a knowledge-base, 3) code generation using
textual prompts (also knows as seq2sql [6]).

The process of generating code (SQL, Python, C/C++) using LLMs to query
tabular-data has several advantages. First, LLMs can accept data in tabular
format which allows better manageability for ‘relevant data’ search to each
user query when compared to semantic retrieval on textual data from vector
databases as presented in the work [2]. Structured data tables eliminate the
need to manage a complex retrieval engine and results ranking system. Second,
passing data in smaller sub-tables to LLMs enables scalability on new data
tables without the need for the an offline table to text generation and vector
embedding process in [2]. For tabular data sets that are have high refresh
rates such as stock market prices, inflation indices, carbon footprint etc.,
passing the relevant sub-tables ensures most accurate data-to-answers. Third,
tabular data is structured and concise which minimizes the scope for data
misrepresentation and hallucinations. It is noteworthy that LLMs have been
found to be more accurate and stable for text-to-code generation than for
text-to-text generation in terms of ‘fake responses’ or hallucinations. Thus,
our novel system shown in Fig. 1 incorporates text to SQL code generation
followed by passing shorter data tables to LLM prompts to enable a variety of
table-to-answers that have been tested on Sustainability domain datasets.

Figure 1: The proposed Extreme-RAG system architecture that invokes 3-4 LLM
prompts per user query to authenticate, route, extract tabular data and fetch
the required response from fast-varying and large tabular data sources.

This paper makes three key contributions. First, we present three approaches
to implement scalable RAG-based retrieval systems using LLMs: 1) User access
to table authentication, 2) text to SQL code generation to identify relevant
data sub-tables, 3) tabular data input to LLM prompts for hallucination
minimized responses. Second, we present hallucination detection mechanisms on
the RAG system to ensure accurate and reliable responses. Third, we present a
multi-prompt system that can be extended to predictive and prescriptive
categories of tables-to-answers. For each user question, the first LLM prompt
routes the query and identifies the necessary data tables, the second LLM
prompt generates several SQL queries to retrieve the relevant data, and the
third LLM prompt constructs a coherent response in natural language. By
focusing on smaller, sub-tables with labeled information, our system
capitalizes on the LLM’s strengths in handling concise and well-defined
datasets.

##  II Related Work

Generating responses from tabular data sources has been an area of focus for
the past few decades. Early works such as [7] relied on the structure of
tabular data in xml/html documents to parse and extract the required answers.
Prior to the advent of LLMs, some major efforts in this domain included the
development of algorithms known as ‘seq2sql’ models that can accept a sequence
of input words and convert it into a corresponding SQL query. Some notable
‘seq2sql’ algorithms and their capabilities are shown in Table I.

TABLE I: SEQ2SQL algorithms prior to LLMs for tables-to-answers use-cases and
their capabilities.

Model Name | Year | Core Capabilities
---|---|---
Bidirectional Attention | June, 2018 | \- Three sub-modules were designed with deep learning models for inference.
for SQL Generation |  | \- Bi-directional attention mechanisms and character embeddings were implemented with convolutional neural networks (CNNs).
[8] |  | -Experimental evaluations were presented for the Wiki SQL dataset (87,726 hand annotated text and SQL).
Seq2seq | 2018 | \- This deep learning model used rewards from in-the-loop query execution over databases.
[9] |  | \- The model leveraged the structure of SQL to prune the space of generated queries and learned an optimal generation policy.
|  | -Tested on 8,0654 hand-annotated examples of questions and SQL queries leading to 23-25% improvement in logical accuracy.
TypeSQL+TC | 2018 | -Utilized type information to better understand rare entities and numbers in natural language questions.
[10] |  | -Tested on Wiki SQL dataset and outperformed prior state-of-the-art by 5.5% in less time.
Tranx | 2018 | -A transition-based neural semantic parser that mapped natural language (NL) utterances into formal meaning representations.
[11] |  | Highly generalizable system that can be applied to new representations by just writing a new abstract syntax description.
|  | Experimental verification performed for 4 semantic parsing and code generation tasks.
STAMP+RL | 2018 | -For typical seq2SQL models, inaccuracies occur due to mismatch between question words and table contents.
[12] |  | -Quality of generated SQL query was improved through content replication by column name, cells or SQL keywords.
|  | \- Generation of‘WHERE’ clause was further improved by using column-cell relations.
PT-MAML | 2018 | -Adapted meta learning to solve the situation where the number of conditions in a text-to-SQL query vary significantly.
[13] |  | -This method decoded a text sequence guided by a fixed syntax pattern by tagging for schema and constant values.

In this work, we extend the state-of-the-art by breaking down the tasks of
data fetching from an input sequence of words into three stages, 1) query
routing, 2) multiple-SQL generation, 3) SQL and text combination to yield
accurate solutions to complex, multi-tabular queries.

##  III Data and Methods

In this work we propose an automated LLM-based user access authenticator and
question-answering chatbot on sustainability-specific datasets containing
information regarding carbon footprint, water/electricity consumption and
renewable energy usage. The dataset comprises of 7 tables of sizes 50MB to 1.2
GB each with over 1000 rows and 50 columns of data in each table. Based on
these tables, the user questions can be categorized into the following 7 major
categories and their combinations as shown in Table II.

TABLE II: Categories of Questions routed by Prompt 1

Intention | Type | Example user-queries
---|---|---
0 | Percent | What % of our offices are at 100% renewable electricity?
1 | Change | What is the annual reduction of emissions globally?
2 | Rank | Which country has the highest Emissions type 1 emissions?
3 | Level | What is scope 1 emission levels for offices in Argentina?
4 | Rank | Which city had the highest water consumption for Dec 2022?
5 | Multi | Which countries reduced scope 3 emissions consistently in
|  | the last 2 years and increased renewable electricity?
6 | FAQ | What is included in business travel?

The descriptions for the proposed Extreme RAG system components are presented
below.

###  III-A Extreme RAG System Components

####  III-A1 Authentication RAG

This process is a scalable extension to an otherwise rule-based look-up that
enables/maps tables to users based on the user-access restrictions. As shown
in Fig. 1, upon each user login, there is a need to enable a database lookup
to retain access only to the tables that the user can retrieve from and to
withhold access to all remaining tables. For example, a user from ‘North
America’ can only have access to power consumption information from North
America. So only the tables with user-access are pre-loaded to the Postgres DB
or BigQuery for future question answering while all remaining geo-location
tables are skipped. On the other hand, a user specializing in renewable energy
may require access to specific data tables across continents. For
authenticating access for hundreds of thousands of users on thousands of data
tables with varying schema, a custom prompt that outputs the access
information in json/xml formats is the proposed scalable solution.

####  III-A2 Prompt 1: Routing a User Question

Once authenticated, each user question needs to be routed for its intention
and appropriate data tables. The query routing is an essential step since the
instructions to respond to the different categories of questions in Table I
vary significantly. Query extension and query rewriting is then performed by
converting the user query into its high-dimensional embedded format and
matching it to at most five prior query samples in the vector space. Thus, the
first query prompt maps the data sources that are pre-loaded to the user-
access constraints to the query context and scope. The output of this prompt 1
is a comprehensive list of data sources relevant to the user query.

The routing prompt is trained on a specific template to identify table
relevance. The result is a list of table names in Python list format, which
can be easily integrated with other data-processing tasks. Although this can
be envisioned as a standard classification task as in [2], LLMs are found to
be more generalizable and scalable across a wide range of use queries and
source tables with minimal training than standard supervised learning models.

Following the identification of these data sources, the proposed system
proceeds to retrieve a list of data source configurations for each identified
source. These data source configurations represent tabular metadata as in [13]
and are important to isolate the appropriate SQL queries in Prompt 2.

####  III-A3 Prompt 2: Data Retrieval

Once the appropriate data tables are mapped to a user query, the data
retrieval prompt converts standard language to SQL code. This prompt takes in
3 inputs: 1) the rewritten and extended sub-queries, 2) a list of data source
configurations (metadata) obtained by semantic matching of the queries with
prototype questions, and 3) a sample question and its answer. Utilizing these
inputs, Prompt 2 generates a complex, nested, SQL query. A query runner
process runs this SQL query against the data sources specified by the previous
step to fetch the required information in tabular form. Only the ‘relevant’
rows and fields from the pre-loaded tables are fetched and loaded into a
specific BigQuery Table.

The metadata corresponding to the mapped tables, can be expressed as “Table
Configurations” and the example questions can be specified as “Prototype
Question configurations” as shown in Fig. 2. Here, Table Configurations
represents an encapsulation (class) with properties for the table name, a list
of relevant table fields, and a list of sample field values. Prototype
Question Query similarly is another class that maps prior user questions to
queries and data source names. The primary advantage of using these data
structures in Prompt 2 is that it eliminates the need for code changes
whenever the data domains change. For each new data source, the Table
Configuration List is automatically updated, and with each new user query
category, the Prototype Question Query list is updated. Thus, these data
structures enable generalizability and scalability.

Figure 2: Examples of class definitions to extract tabular metadata and enable
scaling prompt 2 across source tables and user query types.

####  III-A4 Prompt 3: Answer Retrieval

Finally, the third prompt per user query utilizes the re-written and extended
questions and the tabular data loaded at the end of prompt 2 in BigQuery to
generate a customized prompt that is then sent to the LLM for a natural
language response. This step is an extension to the classical seq2sql models
as it collects the SQL query outputs combines them with standard instructional
guardrails and a sample question and answer to retrieve a natural language
response. The style and format are specified as part of the prompt. It is
noteworthy that per user-query the requirement is for all 3 prompts to
complete under 15 seconds of run time to ensure real-time qualities. An
example of the three prompts per user query is shown in Fig. 3.

Figure 3: Example of the steps in Prompts 1, 2 and 3 to retrieve an
appropriate response from the sustainability data tables.

###  III-B Quality Assurance

LLMs have been well known to be riddled with “fake-responses” or
“hallucinations” [2] even after setting model paremeters like temperature and
top-P to zero. We propose the following factual checks (flags)
$[s_{1,i},s_{2,i},s_{3,i},s_{4,i},s_{5,i}]$ for each user query ($i$) in
combination with the outcomes in prompts 2 and 3 to detect possible
hallucinations for the Extreme-RAG responses generated by the proposed system.
For these checks, “spacy”-based modules are used to extract and compare the
textual named entities in the user questions ($Q_{Ni}$), with those in the
response ($R_{Ni}$), in the SQL queries ($\Sigma_{Ni}$) and the data loaded in
BigQuery by Prompt 2 ($D_{Ni}$).

  1. 1.

Number check: This flag in prompt 3 ensures that all numerical values
mentioned in the answers are directly derived from the underlying data sources
without any discrepancies or fabrications by the end of Prompt 2. This check
is crucial for maintaining the factual accuracy of numerical data presented in
responses as shown in (3).

  2. 2.

Entity Check: This flag verifies that all entities mentioned in the user query
are accurately reflected in the response, ensuring the response’s relevance
and completeness. This step is critical for confirming that the answer
addresses all aspects of the query comprehensively, thereby enhancing user
satisfaction and trust in the system. For example, in a user query regarding
“water consumption rates in the USA”, this check ensures that the named entity
“USA” is included in the response ($R_{Ni}$) and query ($Q_{Ni}$) as shown in
(2).

  3. 3.

Query Check: This flag confirms that all essential keywords and conditions
stated in the user question are included in the SQL commands that are executed
in Prompt 2. This step verifies the presence of specific filter words used in
the user question ($Q_{Ni}$) within the SQL queries ($\Sigma_{Ni}$). This flag
is configured using column metadata for each table and each filter check is
added individually per data schema. This step ensures that the query
accurately reflects the criteria and constraints specified in the question and
can be envisioned by equation (2) wherein $R_{Ni}$ is changed to
$\Sigma_{Ni}$.

  4. 4.

Regurgitation check: This warning flag identifies if the response merely
replicates information from the prompt 3 without paraphrasing. This step
checks for word sequences where ten consecutive words from the prompt 3 are
repeated. This step detects responses that fail to expand upon the query,
which is essential for providing value-added insights to the user [2].

  5. 5.

Increase/Decrease Modifier check: This flag checks for consistency between the
directional changes mentioned in the user query and that described in the
prompt 3 response. This step ensures that the response accurately reflects the
dynamics or trends requested by the user, critical for analyses involving
changes over time or comparisons.

$\displaystyle s_{1,i}=\left\\{\begin{array}[]{ll}1&\mbox{if
}numbers(R_{i})\subset numbers(D_{i})\\\ 0&otherwise\end{array}\right\\},$ (3)
$\displaystyle s_{2,i}=\left\\{\begin{array}[]{ll}1&\mbox{if
}Q_{N_{i}}==R_{N_{i}}\\\ 0&otherwise\\\ \end{array}\right\\},$ (6)

##  IV Experiments and Results

For validation of the proposed system we have deployed the proposed Extreme-
RAG system to answer specific questions across domains and databases using the
Vertex AI platform in Google cloud. Prompt 1 and 3 are built using
TextBision002 while Prompt 2 is built using the Unicorn LLM. The proposed
system is scalable and can be generalized to other cloud-based and on-prem
LLMs as well.

###  IV-A Qualitative Response Analysis

We assess our Extreme RAG system on 100 random queries from multiple domains
such as financial reporting, healthcare and social media and the user queries
along with the SQL generated and the quality scores are shown in Table III.
Here, we observe that although the prompt 3 contains instructions based on the
sustainability dataset, using a new-domain for validation dataset results in
similar response performances. For financial domain data tables and queries,
the proposed system achieves highest confidence scores (all scores are 1). For
healthcare related queries, we observe some numerical hallucinations in the
patient ID numbers that leads to inaccurate numerical predictions (80%
accuracy for $s_{1}$ metric). For the social media use case, the proposed
system hallucinates numbers (that can be flagged by our scoring module) and
the modifier flag returns a not applicable value of -1.

TABLE III: Sample Queries, generated SQL and Response Scores

Question | SQL command | $[s_{1},s_{2},s_{3},s_{4},s_{5}]$
---|---|---
What was the revenue growth for the Northeast region in | SELECT region, SUM(revenue) AS revenue FROM financials WHERE region $=$ ‘Northeast’ | [1,1,1,1,1]
last quarter compared to the previous quarter? | AND (quarter $=$ ‘Last’ OR quarter $=$ ‘Previous’) GROUP BY quarter |
How many new patients were admitted to the oncology department | SELECT department, COUNT (patient-id) AS new-patients FROM hospital-admissions | [0.8,1,1,1,1]
this month and how does this compare to last month? | WHERE department $=$ ‘Oncology’ AND (month $=$ ‘This’ OR month $=$ ‘Last’) GROUP BY month |
What was the click-through rate (CTR) for the digital marketing campaign | SELECT platform, AVG (click-through-rate) AS CTR FROM marketing-data | [0,1,1,1,-1]
on social media platforms in January? | WHERE campaign $=$ ‘Digital Marketing’ AND platform $=$ ‘Social Media’ AND month$=$ ‘January’ |

###  IV-B Quantitative Response Analysis

The quantitative responses on 500 sample variants of 60 queries generated on
the sustainability datasets are shown in Fig. 4. We observe that all responses
have average scores around 90%. The number check score is lowest at 89.09%
followed by entity check and modifier checks at 94.55% each. Averaged query
check and regurgitation check are around 92.73%. This shows the impact of the
proposed Extreme-RAG system towards hallucination control and scalability.

Figure 4: Averaged response quality scores on sustainability dataset [s1 to
s5]

###  IV-C Extreme RAG Extensions

The proposed RAG architecture can further be extended to scenario planning and
predictive capabilities using LLMs. Fig. 5 shows a sample flow where a real-
time optimization/predictive algorithm is invoked based on the named entities
extracted from prompt 1, and the result returned from the algorithm can then
be saved into BigQuery followed by a prompt 3 that furnishes the response in
natural language. Such extensions/modifications to the code generation LLM
prompt (prompt 2) enhance the scalability of the proposed system.

Figure 5: Example of an extension of the proposed Extreme RAG system to solve
for predictive and prescriptive trend questions using back-end optimization
algorithms that can be invoked by LLM prompts (prompt 2).

##  V Conclusion

In this work, we propose a novel multi-LLM system that enables scalable
tables-to-answers with an averaged accuracy of $>90\%$. The system flags
hallucinations with a series of checks that constantly monitor the multi-LLM
outputs. The proposed Extreme-RAG system enables several levels of checks to
ensure the appropriate data subset is fetched from large Enterprise-level data
tables while returning a natural language response in under 10 seconds per
user query. Future works can be directed towards extending RAG approaches to
heterogeneous data sources for automation and insight generation tasks.

## Acknowledgement

The authors would like to thank Dike Effidua and Bharat Jethwani for all
experimentation efforts and Priya Raman for all her leadership.

## References

  * [1] J. Achiam, S. Adler, S. Agarwal, L. Ahmad, I. Akkaya, F. L. Aleman, D. Almeida, J. Altenschmidt, S. Altman, S. Anadkat _et al._ , “Gpt-4 technical report,” _arXiv preprint arXiv:2303.08774_ , 2023.
  * [2] S. Roychowdhury, A. Alvarez, B. Moore, M. Krema, M. P. Gelpi, P. Agrawal, F. M. Rodríguez, Á. Rodríguez, J. R. Cabrejas, P. M. Serrano _et al._ , “Hallucination-minimized data-to-answer framework for financial decision-makers,” in _2023 IEEE International Conference on Big Data (BigData)_. IEEE, 2023, pp. 4693–4702.
  * [3] J. S. Park, J. O’Brien, C. J. Cai, M. R. Morris, P. Liang, and M. S. Bernstein, “Generative agents: Interactive simulacra of human behavior,” in _Proceedings of the 36th Annual ACM Symposium on User Interface Software and Technology_ , 2023, pp. 1–22.
  * [4] T. Bui, O. Tran, P. Nguyen, B. Ho, L. Nguyen, T. Bui, and T. Quan, “Cross-data knowledge graph construction for llm-enabled educational question-answering system: A~ case~ study~ at~ hcmut,” _arXiv preprint arXiv:2404.09296_ , 2024.
  * [5] O. Ovadia, M. Brief, M. Mishaeli, and O. Elisha, “Fine-tuning or retrieval? comparing knowledge injection in llms,” _arXiv preprint arXiv:2312.05934_ , 2023.
  * [6] X. Zhang, F. Yin, G. Ma, B. Ge, and W. Xiao, “M-sql: Multi-task representation learning for single-table text2sql generation,” _IEEE Access_ , vol. 8, pp. 43 156–43 167, 2020.
  * [7] X. Wei, B. Croft, and A. McCallum, “Table extraction for answer retrieval,” _Information retrieval_ , vol. 9, pp. 589–611, 2006.
  * [8] G. Huilin, G. Tong, W. Fan, and M. Chao, “Bidirectional attention for sql generation,” in _2019 IEEE 4th International Conference on Cloud Computing and Big Data Analysis (ICCCBDA)_. IEEE, 2019, pp. 676–682.
  * [9] Z. Li, J. Cai, S. He, and H. Zhao, “Seq2seq dependency parsing,” in _Proceedings of the 27th International Conference on Computational Linguistics_ , 2018, pp. 3203–3214.
  * [10] T. Yu, Z. Li, Z. Zhang, R. Zhang, and D. Radev, “Typesql: Knowledge-based type-aware neural text-to-sql generation,” _arXiv preprint arXiv:1804.09769_ , 2018.
  * [11] P. Yin and G. Neubig, “Tranx: A transition-based neural abstract syntax parser for semantic parsing and code generation,” in _Proceedings of the Conference on Empirical Methods in Natural Language Processing (Demo Track)_ , 2018.
  * [12] Y. Sun, D. Tang, N. Duan, J. Ji, G. Cao, X. Feng, B. Qin, T. Liu, and M. Zhou, “Semantic parsing with syntax-and table-aware sql generation,” in _Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)_ , 2018, pp. 361–372.
  * [13] P.-S. Huang, C. Wang, R. Singh, W.-t. Yih, and X. He, “Natural language to structured query generation via meta-learning,” in _Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 2 (Short Papers)_. Association for Computational Linguistics, 2018.