^{1}

^{2}

^{1}

^{2}

Double Laplace transform is applied to solve general linear telegraph and partial integrodifferential equations. The scheme is tested through some examples, and the results demonstrate reliability and efficiency of the proposed method.

The wave equation is known as one of three fundamental equations in mathematical physics and occurs in many branches of physics, applied mathematics, and engineering. It is also known that there are two types of these equation: the homogenous equation that has constant coefficient with many classical solutions such as separation of variables [

In this study, we use double Laplace transform to solve telegraph equation and partial integrodifferential equation. We follow the method that was proposed by Kılıçman and Eltayeb [

First of all, we recall the following definitions given by Kılıçman and Gadain [

Double Laplace transform for second partial derivative with respect to

If at the point

See [

Next, we study the uniqueness and existences of double Laplace transform. First of all, let

Let

If

A function

Of course, the main difficulty in using Theorem

Let

In this part, we consider some of the properties of the double Laplace Transform that will enable us to find further transform pairs

Was first verify (I) as

We calculate the integral inside bracket as

Second, the right hand side of (II) can be written in the form of

The last property, from definition of double Laplace transform

Owing to the convergence properties of the improper integral involved, we can interchange the operation of differentiation and integration and differentiate with respect to

The previous three properties are very useful at the proof of Theorem

Let us define the set of functions depending on parameters

We apply property (III) (we must evaluate the

Find double Laplace transform for a regular generalized function

Double Laplace transform of (

A linear continuous function over the space

Let us find double laplace transform of the function

Consider the general telegraph equation in the following form:

Here, we assume that the double inverse Laplace transform exists for each term in the right side of (

Consider the homogeneous telegraph equation given by

By taking double Laplace transform for (

By using double inverse Laplace transform for (

In the next example we apply double Laplace transform for nonhomogenous telegraphic equation as follows.

Consider the nonhomogenous telegraphic equation denoted by

By taking double Laplace transform for (

By applying double inverse Laplace transform for (

Consider the following partial integrodifferential equation:

By taking double Laplace transform for (

By applying double inverse Laplace transform for (

We provide the double inverse Laplace transform existing for each terms in the right side of (

Consider the partial integro-differential equation

By taking double Laplace transform for (

By using double inverse Laplace transform for (

The authors would like to express their sincere thanks and gratitude to the reviewers for their valuable comments and suggestions for the improvement of this paper. The first author gratefully acknowledges that this project was partially supported by the Research Center, College of Science, King Saud University.