Application of Homotopy Perturbation and Modified Adomian Decomposition Methods for Higher Order Boundary Value Problems

This work considers the numerical solution of higher order boundary value problems using Homotopy perturbation method (HPM) and modified Adomian decomposition method (MADM). HPM is applied without any transformation or calculation of Adomian polynomials. The differential equations are transformed into an infinite number of simple problems without necessarily using the perturbation techniques. Two numerical examples are solved to illustrate the method and the results are compared with the exact and MADM solutions. The accuracy and rapid convergence of HPM in handling the equations without calculating Adomian polynomials reveals its advantage over MAD

Comparison Homotopy Perturbation and Adomian Decomposition Techniques for Parabolic Equations

This paper compares homotopy perturbation and Adomian decomposition techniques for the solution of parabolic equations. Some examples are considered to illustrate the techniques. The results reveal that the two techniques gave closed form of solution and as such considered most suitable for solving heat flow problems

Elzaki transform homotopy perturbation method for partial differential equations

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

Exact traveling wave solutions to the Klein–Gordon equation using the novel (G′/G)-expansion method

AbstractThe novel (G′/G)-expansion method is one of the powerful methods that appeared in recent times for establishing exact traveling wave solutions of nonlinear partial differential equations. Exact traveling wave solutions in terms of hyperbolic, trigonometric and rational functions to the cubic nonlinear Klein–Gordon equation via this method are obtained in this article. The efficiency of this method for finding exact solutions and traveling wave solutions has been demonstrated. It is shown that the novel (G′/G)-expansion method is a simple and valuable mathematical tool for solving nonlinear evolution equations (NLEEs) in applied mathematics, mathematical physics and engineering

Numerical solution of singularly perturbed problems using Haar wavelet collocation method

Abstract: In this paper, a collocation method based on Haar wavelets is proposed for the numerical solutions of singularly perturbed boundary value problems. The properties of the Haar wavelet expansions together with operational matrix of integration are utilized to convert the problems into systems of algebraic equations with unknown coefficients. To demonstrate the effectiveness and efficiency of the method various benchmark problems are implemented and the comparisons are given with other methods existing in the recent literature. The demonstrated results confirm that the proposed method is considerably efficient, accurate, simple, and computationally attractive