Advanced Numerical and Semi-Analytical Methods for Differential Equations

10Exp‐Function Method

10.1 Introduction

In Chapter 9, we have studied Akbari–Ganji's Method (AGM), which provides the algebraic solution of linear and nonlinear differential equations. In the present chapter we study another straightforward and simple method called the exp‐function method by which one can find the solution (analytical/semi‐analytical) of differential equations. This method was first proposed by He and Wu [1] and was successfully applied to obtain the solitary and periodic solutions of nonlinear partial differential equations. Further, this method was used by many researchers for handling various other equations like stochastic equations , system of partial differential equations , nonlinear evaluation equation of high dimension [4], difference‐differential equation , and nonlinear dispersive long‐wave equation .

10.2 Basics of Exp‐Function Method

This method will be illustrated for partial differential equations as ordinary differential equations are straightforward while solving partial differential equations. Let us consider a nonlinear partial differential equation

10.1

to understand briefly the exp‐function method [17–10].

Here, subscripts indicate partial differentiation with respect to indicated variables in the subscript.

Next, using traveling wave transformation u = u(η), η = kx + ωt (where k and ω are constants) in Eq. (10.1), the partial differential ...