1. Introduction

LNLPDE describes many problems in different fields of applied mathematics and sciences. These equations arise in various scientific models such as the fluid dynamics, chemical reaction-diffusion, shallow water wave propagation, and Schrodinger wave equation models. LNLDEs naturally occurs in various fields of study such as in engineering, physics, and applied mathematics. Many mathematicians and scientists have drawn too much attention to the extensive effectiveness of these equations. In the analysis of physical phenomena and applied sciences, the Goursat partial differential equation occurs in LNLPDEs including mixed-type derivatives and it is known as hyperbolic PDE of second order. The standard form of the Goursat problems with conditions [7-10] is

Several computational techniques, such as Variation Iteration Method (VIM) [7], Laplace Substitution–Variational Iteration Method (LSVIM) [15],

Adomian Decomposition Method (ADM) [11], Two-Dimensional Differential Transform Method (DTM) [8], Modified Variational Iteration Methods [12], Runge-Kutta Method [10] and Finite Difference Method (FDM) [13] and Homotopy Analysis Method (HAM) [14] have been developed to resolve the linear and nonlinear form of Goursat Problems. Previously, the ESM was used [16-17] to solve second-order linear and non-linear PDEs containing mixed partial derivatives. Recently, inspired and encouraged by research work in these fields, we introduce a new method named DEDM, based on Elzaki transform [1] and Double Elzaki Transform [2-6]. The proposed technique is combined the Double Elzaki Transform with the Adomian polynomials for solving non-linear Goursat problems. Here in section (2), definitions and partial derivatives of the Double Elzaki transform will be described. In Section (3), explanation of main advantage of DEDM applied directly without linearization or interruption to Goursat problems are presented. To clarify the applicability and effectiveness of the proposed method for several linear and non-linear problems are presented in section (4). We finally conclude the result at last section.

2. Preliminaries

2.1. Elzaki Transform

The Elzaki transform for exponential order of function f, in the set A is defined by:

(1)

where M is a finite constant and are finite or infinite in set A.

The Elzaki transform symbolized by the operator E (.) and expressed by the integral equation-

(2)

where is used to factor the variable in the argument of function .

Table 1

2.2. Inverse Elzaki Transform

If is the Elzaki transforms of then is called the inverse Elzaki transform of and in mathematical terms, it can be expressed as , where is an operator and is called inverse Elzaki transform.

2.3. Double Elzaki Transform

The function which can be expressed as an infinite convergent series, as a consequence Double Elzaki Transform is:

(3)

where are complex.

Table 2

2.4. Partial Derivatives of Double Elzaki Transform

Using Integration by parts to produce double Elzaki transforms of partial derivatives of and

Proof:

By the definition of double Elzaki transform-

Using first partial derivatives of Elzaki transform, the inner integral provides

Therefore,

Taking Elzaki transform with respect to the above integrals is converted to

By second order partial derivatives of Elzaki transform, the inner integral becomes-

Therefore,

Taking Elzaki transform with respect to

IV. Similarly the following formula can be established

The inner integral reduces to

Therefore,

3. DEDM for Goursat Problem

In order to explain the solution procedure for DEDM, considering a general Goursat problem rolling mixed partial derivatives with conditions is

(4)

(5)

where and are linear term, non-linear differential term and source term respectively.

Consequently, equation (4) becomes-

By double Elzaki transform the above equation is converted to-

(6)

Again taking single Elzaki transform in equation (5), then the equation reduces to

(7)

Substituting these values in equation (6), implies that

Now, on both sides of the above equation using inverse double Elzaki transform, finally-

(8)

where initial conditions are prescribed and reflects the resulting source term.

Consider an infinite series solution to solve non-linear mixed type PDEs by DEDM is

(9)

Then the non-linear term is reduced to

(10)

where are Adomian polynomials with components are defined by

(11)

Substituting u(x, y) and Nu(x, y) in equation (8), the equation is converted to-

(12)

Comparing on both sides of the above equation, obtained relationships are

In general, the recursive relations are

The values of all are obtained from the above recursive relation and substitute in equation (9), then the series solution for the given equation is found.

4. Applications

The DEDM for LNLPDEs with mixed type derivatives will be explained by considering Goursat problems in a linear and non-linear form.

4.1. Linear Homogeneous Goursat Problem

Considering a Goursat problem which is linear and homogeneous

where, is a function of first degree.

Example 1: Considering a Goursat problem which is linear and homogeneous-

(13)

(14)

Equation (13) is converted to this new one by double Elzaki transform

(15)

Operating single Elzaki transform, equation (14) reduces to

Using the above initial values, the equation (15) is reduced to-

The exact solution is found by using inverse double Elzaki transform

which is verified by the results obtained by VIM and ADM.

4.2. Linear Non-Homogeneous Goursat Problem

Consider a Goursat problem which is linear and non-homogeneous-

Where is a function of first degree.

Example 2: Considering linear non-homogeneous Goursat problem-

(16)

(17)

Taking double Elzaki transform in equation (16)

(18)

By single Elzaki transform equation (17) becomes

with the above conditions, the equation (18) is converted to

Applying inverse double Elzaki transform in the above equation

Which is the required exact solution. This result is completely similar with the result obtained by VIM and DTM.

4.3. Non-linear Goursat Problem

To illustrate the proposed DEDM, two non-linear homogeneous and non-homogeneous Goursat problem is considered.

Example 3: Considering the non- linear non-homogeneous Goursat problem

(19)

(20)

Directly use the double Elzaki transform in equation (19) implies that

Applying single Elzaki transform in equation (20), conditions reduce to

The transformed equation reduces with these conditions as

Again apply inverse double Elzaki transform in the above equation on both sides,

(21)

In DEDM, solution is represented in infinite series form as below,

(22)

And the non-linear term can be written as follows,

(23)

where are Adomian polynomials of components which is defined by the formula (11).

Some Adomian polynomials of are found as

Then the equation (21) is transformed to

The following recursive relation is found by comparing the above equation on both sides,

The following few components of u_n (x, y), n>0 are determined from the above relation

Consequently,

Substituting all values of u_n (x, y), n≥0 in equation (22), finally the series solution of the given equation is

This results are in complete compliance with the result given by VIM and DTM.

Example 4: Consider the non-linear homogeneous Goursat problem

(24)

(25)

Applying double Elzaki transform the equation (24) becomes-

Using the single Elzaki transform the conditions in equation (25) reduces to

with these new conditions the above equation gives

Operating inverse double Elzaki transform, find that

(26)

The infinite series solution in DEDM is presented by

(27)

and the non-linear term can be written as-

(28)

where are Adomian polynomials of components which is defined by the formula (11). Adomian polynomials are

Then the equation (26) becomes

The recursive relationships are obtained by comparing both sides of the above equation

The following new components of have been found from the above relationship.

Like-wise

Putting all values of u_n (x, y), n≥0 in equation (27), finally the series solution of the given equation is

This finding is in complete compliance with the result obtained by VIM and DTM.

5. Conclusions

To find the approximate series solutions we successfully implemented the DEDM of linear and non-linear Goursat problems. The mentioned process eliminates the computational difficulties that occur in other existing methods and is directly implemented without the use of linearization, transformation, discretization or limiting assumptions. By applying DEDM, many examples of linear and non-linear Goursat problems were evaluated and the findings have shown noticeable and more satisfactory results. The improvement of this methodology would lead to the possibility of increasing the order and accuracy of the system further.