Abstract
This paper demonstrates a new technique based on a combined form of the new transform method with homotopy perturbation method to find the suitable accurate solution of autonomous Equations with initial condition. This technique is called the transform homotopy perturbation method (THPM). It can be used to solve the problems without resorting to the frequency domain.The implementation of the suggested method demonstrates the usefulness in finding exact solution for linear and nonlinear problems. The practical results show the efficiency and reliability of technique and easier implemented than HPM in finding exact solutions.Finally, all algorithms in this paper implemented in MATLAB version 7.12.
Highlights
The most of the phenomena that arise in mathematical physics and engineering fields can be described by partial differential Equations (PDEs), for example, in chemical diffusion, the heat flow, thermo elasticity and the wave propagation phenomena are well described by PDEs [1,2,3,4]
In this research we suggest new technique to solve one of the most important of amplitude Equations is the autonomous Equation which describes the appearance of the stripe pattern in two dimensional systems
Partial differential Equations are a type of differential Equation, i.e, a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables
Summary
The most of the phenomena that arise in mathematical physics and engineering fields can be described by partial differential Equations (PDEs), for example, in chemical diffusion, the heat flow, thermo elasticity and the wave propagation phenomena are well described by PDEs [1,2,3,4]. It's important tool for describing natural phenomena of science and engineering models Most of these problems are difficult to solve them analytically. In this research we suggest new technique to solve one of the most important of amplitude Equations is the autonomous Equation which describes the appearance of the stripe pattern in two dimensional systems. This Equation was applied to a number of problems in variety systems, e.g., Rayleigh-Benard convection, Faraday instability, nonlinear optics, chemical reactions and biological systems. In this paper a reliable transform homotopy perturbation method is proposed and applied for solving autonomous Equation. Some examples are illustrative for demonstrating the advantage of the technique
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