Investigation of Numerical Simulation and New Traveling Wave Solutions of Convection-Diffusion Equation with Reaction

Throughout this paper we will discuss and analyze of the convection-diffusion equation which included a reaction term. First, we will apply an alternating-direction implicit (ADI) scheme akin to that proposed by Polezhaev. Use of this implicit operator-splitting scheme allows the application of a tridiagonal Thomas solver to obtain the solution of steady convection-diffusion equation with reaction. According to the unsteady case, we will use the improved (G' /G) -expansion method to construct the traveling wave solutions, where (G' /G) satisfies a second order linear ordinary differential equation. In this paper we will explore new applications of this method to some special nonlinear convection-diffusion equation with reaction. The new type of exact travelling wave solution obtained in this paper might have significant impact on future researches.