7th -Order Caudrey-Dodd-Gibbon Equation and Fisher-Type Equation by Homotopy Analysis Method

Ankita Sharma,Rajan Arora

doi:10.33889/ijmems.2020.5.2.022

Abstract

In this paper, we first describe the methodology of the Homotopy Analysis Method (HAM) which is an analytical technique and then employ it to some of the non-linear problems which are used in different fields of sciences like plasma physics, fluid dynamics, laser optics, biology, chemical kinetics, nucleation kinetics, physiology, etc. Approximate series solutions have been obtained and the results are compared with the closed form solutions of the equations, which show that this technique gives high accurate results. HAM is a reliable technique, easy to use and is widely applicable to a large class of non-linear differential equations. MATHEMATICA software package has been used for computations.

Highlights

In view of all above limitations one can conclude that it all arises due to the assumption of that small parameter which is known as the perturbation quantity
The advantage of these techniques is that they don’t require any parameter. Both the perturbation and non-perturbation techniques themselves do not provide the facility of controlling the rate of approximate series and the convergence region
In the present paper, the Homotopy Analysis Method (HAM) technique is employed on 7th-order Caudrey-Dodd-Gibbon equation and Fisher-type equation to obtain an approximate analytical and numerical solution

Summary

Introduction

The non-linear equations are not very easy to solve explicitly. Some techniques like perturbation techniques which involve some parameter (small or large), known as the perturbation quantity, play an important role in solving the non-linear differential equations to some extent. In order to overcome these limitations there were developed some non-perturbation techniques, e.g., Sinc-Collocation method (Zakeri and Navab, 2010), Sinc-Galerkin method (Rashidinia and Nabati, 2013), finite element method (Jin, 2014), finite difference method (Li and Ding, 2014), Eulerian Lagrangian method (Chertock et al, 2014) and so on The advantage of these techniques is that they don’t require any parameter. The other advantage of this method is that it can be applied to a high range of non-linear problems as it does not depend on any type of physical parameter It ensures the convergence of the solution series and becomes valid even for strong non-linear problems. The HAM method has been successfully employed on non-linear partial differential equations, namely, the seventh-order Caudrey-Dodd-Gibbon equation and a Fisher-type equation Their approximate analytical and numerical solutions have been found out. This method is very simple, effective and is being widely used to study the differential equations to a large scale

Basic Concepts of the Homotopy Analysis Method

The Caudrey-Dodd-Gibbon Equation

Evaluation of Convergence and Numerical Results

The Fisher’s Type Equation

Conclusion