1. Introduction

We find in the literature, partial integro-differential equations are used in modelling various phenomena in science, engineering & social sciences. Jyoti Thorwe & Sachin Bhalekar [5] in 2012, used single Laplace transform method to solve linear partial integro-differential equations (PIDE) with a convolution kernel

(1.1)

(with prescribed conditions)

where are known functions.

are constants or the functions of x.

They converted the PIDE to an ordinary differential equation (ODE) using a Laplace transform. Solving this ODE & applying inverse Laplace transform they obtained the exact solution of the problem.

The double Laplace transform [4] of a function defined in the positive quadrant of the xt-plane is defined by the equation

(1.2)

whenever that integral exist. Here p, s are complex numbers.

Further the double Laplace transform [1] of first & second partial derivatives are given by

(1.3)

(1.4)

(1.5)

(1.6)

Double Laplace transform was used by Anwar et al. [2] to solve partial differential equations with Caputo fractional derivatives & fractional heat equation subject to certain initial & boundary conditions. Tarig M. Elazaki [8] in 2012, applied the modified variational iteration method & double Laplace transform to solve nonlinear convolution partial differential equations.

Recently in 2013, Eltayeb & Kilicman [3] have applied the double Laplace transform to solve general linear telegraph & partial integro-differential equations. Saeed Kazem [7] implemented single Laplace transform to obtain an exact solution of some linear fractional differential equations. Jafar Saberi-Nadjafi [6] proposed some new result in n-dimensional Laplace transforms involved with Fourier cosine transform; these results provide few algorithms for evaluating some n-dimensional Laplace transform pairs.

By using double Laplace transform method, we directly convert PIDE (1.1) into an algebraic equation instead of converting to ODE. Solving this algebraic equation & applying double inverse Laplace transform we obtain the exact solution. This method is illustrated by giving examples of various types already described in [5].

Convolution Theorem:

If single Laplace transform of with respect to t are given by

then

where, .

Therefore, by using double Laplace transform convolution theorem becomes

(1.7)

2. Solving PIDEs Using Double Laplace Transform Method

Consider PIDE (1.1),

(2.1)

(with prescribed conditions)

where are known functions & are constants.

Taking double Laplace transform of PIDE (2.1), we get

Using double Laplace transform of partial derivative & convolution theorem, we get

(2.2)

where

Equation (2.2) is an algebraic equation in . Solving this algebraic equation & taking inverse double Laplace transform of , we get an exact solution of (2.1).

3. Illustrative Examples

Example 3.1: Consider the PIDE

(3.1)

with initial condition

(3.2)

& boundary condition

(3.3)

Taking double Laplace transform of equation (3.1),

(3.4)

And single Laplace transforms of initial conditions (3.2) & boundary condition (3.3) are given by

Then equation (3.4) becomes,

(3.5)

Applying inverse double Laplace transform of equation (3.5), we get

(3.6)

This is the required exact solution of equation (3.1).

Example 3.2: Consider

(3.7)

(3.8)

(3.9)

Taking double Laplace transform of equation (3.7)

(3.10)

And single Laplace transforms of equation (3.8) & (3.9), we get

Then equation (3.10) becomes

(3.11)

Applying inverse double Laplace transform of equation (3.11), we get

(3.12)

This is the required exact solution of equation (3.7).

Example 3.3: Consider

(3.13)

(3.14)

(3.15)

Taking double Laplace transform of equation (3.13)

(3.16)

And single Laplace transforms of equation (3.14) & (3.15), we get

Then equation (3.16) becomes

(3.17)

Applying inverse double Laplace transform of equation (3.17), we get

(3.18)

This is the required exact solution of equation (3.13).

Example 3.4: Consider the PIDE

(3.19)

with initial condition

(3.20)

& boundary condition

(3.21)

Taking double Laplace transform of equation (3.19)

(3.22)

And single Laplace transforms of equation (3.20), we get

Then equation (3.22) becomes

(3.23)

This is first order linear differential equation in .

Therefore solution of (3.23) is

where C is a constant to be determined.

(3.24)

From the boundary condition (3.21),

(3.25)

Using (3.24) & (3.25), we get

(3.26)

Applying inverse double Laplace transform of equation (3.26), we get

(3.27)

This is the required exact solution of equation (3.19).