1. Introduction

Linear partial differential equations involving mixed partial derivatives arise in various fields of physical science, astronomy and engineering. Sometimes it’s very difficult to solve them, either numerically or theoretically. There are various methods such as Method of Separation of Variables, Variation Iteration Method, Laplace Transform, Laplace Substitution Method [7], Sumudu Transform, to solve these kinds of equations. In recent, Tariq Elzaki introduced a new integral transform known as Elzaki transform [1-6] which is modified transform of Sumudu and Laplace transforms. Elzaki transform can be used to solve ordinary differential equations [1], partial differential equations [3, 8], partial integro-differential equations [9, 12], system of partial differential equations [10, 13] and wave equations [8, 15]. The main advantage of Elzaki transform is that it eliminates the need of linearization, perturbation or any other transformation. Elzaki transforms are widely used for solving ordinary and partial differential equations. The existing methods for solving partial differential equations involving mixed partial derivatives are time consuming with large computation. Our proposed method is more powerful and efficient to solve partial differential equations involving mixed partial derivatives with less computation.

2. Preliminaries

2.1. Elzaki Transform

A new transform called the Elzaki transform defined for function of exponential order we consider functions in the set defined by:

(2.1)

In a set is constant must be finite, may be finite or infinite.

The Elzaki transform is defined by

(2.2)

In this transform the variable is used to factor the variable in the argument of function . This transform has deeper connection with the Laplace transform. The aim of this study is to show the applicability of this interesting new transform and its ability in solving the mixed order linear partial differential equations.

2.2. Fundamental Properties of Elzaki Transform

Table 1

2.3. Elzaki Transformations of Partial Derivatives

Let be a function of two independent variables and , then

3. Elzaki Substitution Method

The aim of this section is to discuss the Elzaki substitution method. We consider the general form of non-homogeneous partial differential equation with initial conditions is given below

(3.1)

(3.2)

Where is the remaining linear terms in which contains only first order partial derivatives of with respect to either or and is the source term.

We can write equation (3.1) in the following form

(3.3)

Substituting in equation (3.3), we obtain

(3.4)

Taking Elzaki transform of equation (3.4) with respect to x, we get

(3.5)

Taking inverse Elzaki transform of equation (3.5) with respect to x, we get

(3.6)

Re-substitute the value of in equation (3.6), we get

(3.7)

This is a first order partial differential equation in the variables x and y.

Taking Elzaki transform of equation (3.7) with respect to y, we get

(3.8)

Taking the inverse Elzaki transform of equation (3.8) with respect to y, we get

(3.9)

The last equation (3.9) gives the exact solution of initial value problem (3.1)

4. Illustrative Examples

To illustrate this method for partial differential equations with mixed partial derivatives we take four examples in this section

Example 1: Consider the following linear partial differential equation

(4.1)

with initial conditions

(4.2)

In the above initial value problem and general linear term is zero.

We can re-write equation (4.1) in the following form

(4.3)

Substituting in equation (4.3), we get

(4.4)

which is non-homogeneous partial differential equation of first order.

Taking Ezaki transform on both sides of equation (4.4) with respect to x, we get

(4.5)

Taking inverse Elzaki transform of equation (4.5) with respect to x, we get

(4.6)

which is the partial differential equation of first order in the variables x and y.

Taking Elzaki transform of equation (4.6) with respect to y, we get

(4.7)

Taking inverse Elzaki transform of equation (4.7) with respect to y, we get

(4.8)

This is the required exact solution of equation (4.1). This can be verifying though the substitution.

Example 2: Consider the linear partial differential equation

(4.9)

With the initial conditions

(4.10)

Equation (4.10) we can write in the following form

(4.11)

Putting in equation (4.11), we get the non-homogeneous partial differential equation of first order is

(4.12)

Taking Elzaki transform on both sides of equation (4.12) with respect to x, we obtain

(4.13)

Taking inverse Elzaki transform on both sides of equation (4.13) with respect to x, we get

(4.14)

Taking Elzaki transform on both sides of equation (4.14) with respect to y, we get

(4.15)

Taking inverse Elzaki transform on both sides of equation (4.15) with respect to y, we get

This is the required exact solution of equation (4.9). This can be verifying though the substitution.

Example 3: Consider the linear partial differential equation

(4.16)

with the initial conditions

(4.17)

We can write the equation (4.16) in the following form as

(4.18)

Putting in equation (4.18), we get

(4.19)

This is the non-homogeneous partial equation of first order.

Taking Elzaki transform on both sides of equation (4.19) with respect to x, we get

(4.20)

Taking inverse Elzaki transform on both sides of equation (4.20) with respect to x, we get

(4.21)

Taking Elzaki transform on both sides of equation (4.21) with respect to y, we get

(4.22)

Taking inverse Elzaki transform on both sides of equation (4.22) with respect to y, we get

This is the required exact solution of equation (4.16). This can be verifying though the substitution.

Example 4: Consider the linear partial differential equation with

(4.23)

With the initial conditions

(4.24)

Equation (4.23) can be written in the following form as

(4.25)

Putting in equation (4.25), we get

(4.26)

Taking Elzaki transform on both sides of equation (4.26) with respect to x, we get

(4.27)

Taking inverse Elzaki transform on both sides of equation (4.27) with respect to x, we get

That is,

(4.28)

Taking Elzaki transform on both sides of equation (4.28) with respect to y, we get

(4.29)

Taking inverse Elzaki transform on both sides of equation (4.29) with respect to y, we get

We cannot solve the equation (4.30) because our goal is appeared in both sides of equation (4.30). Thus the equation (4.23) we cannot solve by using Elzaki substitution method because of

5. Conclusions

The main goal of this paper is to find exact solutions of partial differential equations involve mixed partial derivatives with general linear term by proposed Elzaki Substitution Method. This result has been extracted that Elzaki Substitution Method plays a key role in finding the solution of higher order initial value problem which involves the mixed partial derivatives with general linear term . The operation of this method is simple in use and time saving. But the result of example number four tell us that Elzaki Substitution Method is not applicable for those partial differential equations in which. Consequently the Elzaki Substitution Method can be applied for other equations that performing in various scientific fields.