A structure-preserving Bhattacharya method for nonlinear parabolic equations with fractional diffusion and advection

Abstract

In this work, we investigate a parabolic equation with nonlinear reaction, and fractional diffusion and advection terms of the Riesz type. The model under investigation is a fractional generalization of the well-known Burgers–Fisher and Burgers–Huxley models from population and fluid dynamics, which are equations that admit positive, bounded and monotone solutions, some of them being traveling waves. A variable-step Bhattacharya-type finite-difference scheme based on fractional centered differences is proposed to approximate the solutions of the parabolic partial differential equation. The method is an explicit technique which, under suitable parameter conditions, is capable of preserving the positivity, the boundedness and the monotonicity of the approximations. Moreover, the method preserves the constant solutions of the fractional partial differential equation under investigation. The properties of consistency, stability and convergence of the technique are established thoroughly in this manuscript along with some a priori bounds for the numerical solutions. Some illustrative simulations and numerical comparisons against other numerical techniques are carried out in order to show that the method preserves these features of the approximations.