Elzaki transform homotopy perturbation method for partial differential equations

Abstract

Partial differential equations (PDEs) occur in many applications and play a big role in engineering and applied sciences. Since some PDEs are quite difficult to solve, many new methods are introduced to the academic community. Some of them are homotopy perturbation method, variational iteration method, adomian decomposition method, differential transformation method, ELzaki transform, ELzaki transform homotopy perturbation method (ETHPM) and etc. In this study two methods are considered which is homotopy perturbation method and ELzaki transform. The two methods were introduced and examples were presented to illustrate the efficiency of both methods. It is shown that both methods can be used to solve different types of partial differential equations. Although they can be used to solve PDEs, they have their own limitations. There are certain nonlinear forms of PDEs that are quite difficult to solve using ELzaki transform, and for homotopy perturbation method, the expansion itself sometimes can be quite difficult to solve. Then, the combination of both methods was introduced and the efficiency of the method was shown by solving some applications of partial differential equations. ETHPM was used to solve some gas dynamics and Klein-Gordon equations. The results are compared with previous study to determine the efficiency of the method. The graph of each solution is illustrated by using Mathematica software. From the result, it is shown that ETHPM method produces anticipated exact solutions and the calculations is not that complicated.