Traveling wave solutions to a reaction-diffusion equation

Abstract

In this paper, we restrict our attention to traveling wave solutions of a reactiondiffusion equation. Firstly we apply the Divisor Theorem for two variables in the complex domain, which is based on the ring theory of commutative algebra, to find a quasi-polynomial first integral of an explicit form to an equivalent autonomous system. Then through this first integral, we reduce the reaction-diffusion equation to a first-order integrable ordinary differential equation, and a class of traveling wave solutions is obtained accordingly. Comparisons with the existing results in the literature are also provided, which indicates that some analytical results in the literature contain errors. We clarify the errors and instead give a refined result in a simple and straightforward manner.

The first integral method for constructing exact and explicit solutions to nonlinear evolution equations

“…The approach is currently known as the first integral method. The method has found several applications in the applied sciences (see, for instance, [21][22][23][24][25][26][27][28] and the references therein). The main idea behind the method is to construct a polynomial first integral (with polynomial coefficients) to an autonomous planar system which is equivalent to the equation to be solved.…”

Travelling Wave Solution of the Fisher-Kolmogorov Equation with Non-Linear Diffusion

“…Jiaqi et al [30] found a travelling wave solution of non-linear reaction diffusion equation by using the homotopic method and theory of travelling wave transform. Feng et al [31] found the travelling wave solution of reaction diffusion equation by two methods. Firstly they applied Divisor theorem for two variables in complex domain to find a quasi-polynomial first integral of an explicit form to an equivalent autonomous system.…”

A Method for Constructing Traveling Wave Solutions to Nonlinear Evolution Equations

“…The last decades have seen that many powerful and effective methods have been proposed to obtain analytical solutions to NLEEs, such as the inverse scattering transform [1,3], the tanh-function method [4,5], the homogeneous balance method [6,7], the Jacobi elliptical function expansion method [8], the extended mapping method [9], the first integral method [10][11][12], the Lie symmetry reduction method [13][14][15], the proper ansatzs method [16,17], and some numerical approaches [18,19] [20]. Khater et al explored traveling wave solutions by using an improved sine-cosine method and Wu's elimination method [22].…”