Improvement of the Modified Decomposition Method for Handling Third-Order Singular Nonlinear Partial Differential Equations with Applications in Physics

Nemat Dalir

doi:10.1155/2014/607259

Abstract

The modified decomposition method (MDM) is improved by introducing new inverse differential operators to adapt the MDM for handling third-order singular nonlinear partial differential equations (PDEs) arising in physics and mechanics. A few case-study singular nonlinear initial-value problems (IVPs) of third-order PDEs are presented and solved by the improved modified decomposition method (IMDM). The solutions are compared with the existing exact analytical solutions. The comparisons show that the IMDM is effectively capable of obtaining the exact solutions of the third-order singular nonlinear IVPs.

Highlights

Singular nonlinear partial differential equations (PDEs) appear in many cases in physics and mechanics
Singular nonlinear PDEs appear in many cases in physics and mechanics
ODEs and PDEs of various types have been solved by the modified decomposition method (MDM), such that singular and nonsingular nonlinear ODEs and nonsingular nonlinear PDEs have been solved by the MDM [6,7,8,9,10]

Summary

Introduction

Singular nonlinear PDEs appear in many cases in physics and mechanics. Examples of singular nonlinear PDEs in physics include cylindrical and spherical KdV equations, Ernst equation, Clairaut’s equation, Hartree equation, Yamabe problem, Zakharov-Schulman system, Cauchy momentum equation, and reaction-diffusion equations [1,2,3]. Examples of singular nonlinear PDEs in mechanics are equation of motion of a point mass in a central force field, generalized equation of steady transonic gas flow, cylindrical and spherical NavierStokes equations, and cylindrical and spherical fluid hydrodynamic instability equations [4, 5]. Despite such importance in various fields of science and engineering, singular nonlinear PDEs are difficult to solve. The results of the IMDM solutions agree with the existing exact solutions of IVPs

IMDM for Third-Order Singular Nonlinear PDEs

Case Studies of Third-Order Singular Nonlinear PDEs Solved by IMDM

Conclusions