Mousumi Datta, Md. Shah Alam, Umme Hahiba, Nigar Sultana, Md. Babul Hossain
Department of Mathematics, Mawlana Bhashani Science and Technology University, Santosh, Tangail, Bangladesh
Correspondence to: Md. Shah Alam, Umme Hahiba, Nigar Sultana, Md. Babul Hossain, Department of Mathematics, Mawlana Bhashani Science and Technology University, Santosh, Tangail, Bangladesh.
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Abstract
This article explores a new approach to achieving an exact solution of a special type of both linear and nonlinear partial differential equation (LNLPDE) with mixture derivatives, namely Goursat problems by Double Elzaki Decomposition Method (DEDM). The process utilizes characteristics of Double Elzaki Transform and Adomian Polynomials to obtain an exact solution for Goursat problems. Moreover, Adomian polynomials are used to decompose the nonlinear terms. The findings indicate the proposed method is very easy to use and more effective than other techniques because of seed and computational time. It can easily be expanded to other mixedtype LNLPDEs and the approximation task size can be simplified.
Keywords:
Linear and Nonlinear PDEs, Double Elzaki Transform, Mixed Partial Derivatives, Adomian Polynomials, Goursat Problem
Cite this paper: Mousumi Datta, Md. Shah Alam, Umme Hahiba, Nigar Sultana, Md. Babul Hossain, Exact Solution of Goursat Problem with Linear and Nonlinear Partial Differential Equations by Double Elzaki Decomposition Method, Applied Mathematics, Vol. 11 No. 1, 2021, pp. 511. doi: 10.5923/j.am.20211101.02.
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 reactiondiffusion, 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 mixedtype derivatives and it is known as hyperbolic PDE of second order. The standard form of the Goursat problems with conditions [710] isSeveral computational techniques, such as Variation Iteration Method (VIM) [7], Laplace Substitution–Variational Iteration Method (LSVIM) [15], Adomian Decomposition Method (ADM) [11], TwoDimensional Differential Transform Method (DTM) [8], Modified Variational Iteration Methods [12], RungeKutta 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 [1617] to solve secondorder linear and nonlinear 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 [26]. The proposed technique is combined the Double Elzaki Transform with the Adomian polynomials for solving nonlinear 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 nonlinear 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 transformUsing first partial derivatives of Elzaki transform, the inner integral provides Therefore, Taking Elzaki transform with respect to the above integrals is converted toBy 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 establishedThe 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, nonlinear 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 thatNow, 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 nonlinear mixed type PDEs by DEDM is  (9) 
Then the nonlinear 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 areThe 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 nonlinear 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 transformwhich is verified by the results obtained by VIM and ADM.
4.2. Linear NonHomogeneous Goursat Problem
Consider a Goursat problem which is linear and nonhomogeneous Where is a function of first degree.Example 2: Considering linear nonhomogeneous Goursat problem  (16) 
 (17) 
Taking double Elzaki transform in equation (16)  (18) 
By single Elzaki transform equation (17) becomeswith the above conditions, the equation (18) is converted to Applying inverse double Elzaki transform in the above equationWhich is the required exact solution. This result is completely similar with the result obtained by VIM and DTM.
4.3. Nonlinear Goursat Problem
To illustrate the proposed DEDM, two nonlinear homogeneous and nonhomogeneous Goursat problem is considered.Example 3: Considering the non linear nonhomogeneous Goursat problem  (19) 
 (20) 
Directly use the double Elzaki transform in equation (19) implies that Applying single Elzaki transform in equation (20), conditions reduce toThe 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 nonlinear 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 toThe 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 relationConsequently, 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 nonlinear 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 givesOperating inverse double Elzaki transform, find that  (26) 
The infinite series solution in DEDM is presented by  (27) 
and the nonlinear 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.Likewise 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 nonlinear 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 nonlinear 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.
References
[1]  Elzaki, T. M. (2011). Application of new transform “Elzaki transform” to partial differential equations. Global Journal of pure and applied Mathematics, 7(1), 6570. 
[2]  Hassaballa, A. A., & Salih, Y. A. (2015). On Double Elzaki Transform and Double Laplace Transform. IOSR Journal of Mathematics (IOSRJM), 11(1), 3541. 
[3]  Elzaki, T. M., Hilal, E. M., Arabia, J. S., & Arabia, J. S. (2012). Solution of Telegraph Equation by Modified of Double Sumudu Transform" Elzaki Transform. Mathematical Theory and Modeling, 2(4), 95103. 
[4]  Idrees, M. I., Ahmed, Z., Awais, M., & Perveen, Z. (2018). On the convergence of double Elzaki transform. Int Journal of Advanced and Applied Sciences, 5(6), 1924. 
[5]  Hassan, M. A. (2020). Double Elzaki Transform Decomposition Method for Solving NonLinear Partial Differential Equations. Journal of Applied Mathematics and Physics, 8(8), 14631471. 
[6]  MAHGOB, M. (2015). Solution of partial integrodifferential equations by double Elzaki transform method. Mathematical Theory and Modeling, 5(5), 6165. 
[7]  Wazwaz, A. M. (2007). The variational iteration method for a reliable treatment of the linear and the nonlinear Goursat problem. Applied Mathematics and Computation, 193(2), 455462. 
[8]  Taghvafard, H., & Erjaee, G. H. (2010). Twodimensional differential transform method for solving linear and nonlinear Goursat problem. International Journal for Engineering and Mathematical Sciences, 6(2), 103106. 
[9]  Wazwaz, A. M. (1995). The decomposition method for approximate solution of the Goursat problem. Applied Mathematics and Computation, 69(23), 299311. 
[10]  Day, J. T. (1966). A RungeKutta method for the numerical solution of the Goursat problem in hyperbolic partial differential equations. The Computer Journal, 9(1), 8183. 
[11]  Ahmad J, Mushtaq M. (2015). Exact Solution of Linear and Nonlinear Day, J. T. (1966). A RungeKutta method for the numerical solution of the Goursat problem in hyperbolic partial differential equations. The Computer Journal, 9(1), 8183. Goursat Problems. Univers J Comput Math. 3 (1): 1417. 
[12]  Noor, M. A., & MohyudDin, S. T. (2008). Modified variational iteration method for Goursat and Laplace problems. World Applied Sciences Journal, 4(4), 487498. 
[13]  Pandey, P. K. (2014). A Finite Difference Method for Numerical Solution of Goursat Problem of Partial Differential Equation. Open Access Libr J, 1, 16. 
[14]  Usman M, Zubair T, Ali U, Mohyuddin ST. (2012). On Goursat Problems. Int J Mod Math Sci.3 (3): 6376. 
[15]  AlFayadh, A., & Faraj, D. S. (2019). Laplace Substitution–Variational Iteration Method for Solving Goursat Problems Involving Mixed Partial Derivatives. American Journal of Mathematical and Computer Modelling, 4(1), 1620. 
[16]  Hossain, M. B. & Datta, M. (2018). Solutions of Linear Partial Differential Equations with Mixed Partial Derivatives by Elzaki Substitution Method. American Journal of Computational and Applied Mathematics 8:5964. 
[17]  Datta, M., Habiba, U., & Hossain, M. B. (2020). Elzaki Substitution Method for Solving Nonlinear Partial Differential Equations with Mixed Partial Derivatives Using Adomain Polynomial. International Journal, 8(1), 612. 