Modeling Nonlinear Fractional Reaction-Advection Systems with Implicit Time-Stepping Methods

Abstract

   The paper presents a numerical exploration of nonlinear reaction-advection equations of the fractional reaction-advective equations using the implicit low order L1 time-stepping method. The proposed approach is effective in term of capturing both the diffusion, advection and nonlinear reaction, besides considering the effects of memory using the fractional derivative. Various fractional orders were experimented numerically (alpha = 0.6, 0.8, 0.95) to research their effect on the amplitude of the solution, smoothness and temporal development. The outcomes reveal that solutions of lower fractional orders have elevated magnitudes caused by the greater impact of memory whereas those of greater fractional orders have smaller and smooth solutions. The stability of the proposed method and its accuracy (first-order accuracy) are verified in a convergence study. The paper presents a detailed systemology of specifying the nonlinear reaction-advection problems of fractional order, where some quantitative results combined with graphical understanding are provided on this system behavior. These results point to the success and soundness of the implicit L1 scheme to the modeling of complex fractional systems.