Analytical and numerical solutions of time-fractional advection-diffusion-reaction equation

Abstract

The analytical and numerical solutions of non-autonomous time-fractional partial advection-diffusion-reaction equations are the key topics of this article. Here, by using the Sumudu decomposition method and the maximum-minimum principle we establish the existence and uniqueness of the analytical solution to these problems. Further, we propose a computational scheme to obtain the numerical solution of the fractional diffusion equation. To obtain the numerical solution, first, we discretize the time domain by a graded mesh, and by using the L1−scheme we semi-discretize the fractional time derivative, and study the stability and convergence of the method. Later, by using the cubic spline method over a uniform mesh, we approximate the spatial derivative and obtain the fully discrete scheme. As a result of our convergence study, we have obtained second-order error estimates in the spatial variable and (2−β)-order with regard to the temporal variable. Also, we have applied the proposed method to solve semilinear problems after linearizing by the Newton linearization process. Numerical experiments are carried out to validate the proposed method, and the outcomes are compared with those obtained using earlier techniques described in the literature.