Numerical algorithm for a generalized form of Schnakenberg reaction-diffusion model with gene expression time delay

Abstract

In this paper, we discuss the analysis and the numerical solution of the time-space fractional Schnakenberg reaction-diffusion model with a fixed time delay. This model is a natural system of autocatalysis, which often occurs in a variety of biological systems. The numerical solutions are obtained by constructing an efficient numerical algorithm to approximate Riesz-space and Caputo-time fractional derivatives. More precisely, the L1 approximation is applied to discretize the temporal Caputo fractional derivative, while the Legendre-Galerkin spectral method is used to approximate the spatial fractional operator. The described method is shown to be unconditionally stable, with a 2−β convergent order in time and an exponential rate of convergence in space in case of the smoothness of the solution. The error estimates for the obtained solution are derived by applying a proper discrete fractional Grönwall inequality. Moreover, we offer numerical simulations that demonstrate a close match with the theoretical study to evaluate the efficacy of our methodology.