Exact Solutions for Non-Linear Partial Differential Equations Using Vector Sub Spaces

Abstract

The study of nonlinear differential equations has been handicapped due to the absence of well-defined analytic techniques to deal with them in general and deriving their exact solution. The main aim of this research is to investigate the recently developed analytic method namely Invariant Subspace Method towards deriving exact solutions of certain physically important nonlinear partial differential and differential-difference equations of both integer and fractional order. Differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology. Almost all physical phenomena obey mathematical laws that can be formulated by differential equations.

Exact solutions play a vital role in the proper understanding of qualitative features of the concerned phenomena and processes in various areas of science and engineering. Nonlinear differential equations play a crucial role in many branches of applied physical sciences such as condensed matter physics, biophysics, atomic chains, molecular crystals, and discretization in solid-state and quantum physics. They also play an important role in numerical simulation of soliton dynamics in high-energy physics because of their rich structures. Not only these we may encounter so many problems which may involve differential equations. So, in order to find exact solutions to those problems we have to find new methods effectively. One of those analytic technique in solving the non-linear differential equations.