Semi implicit Galerkin Legendre Spectral Schemes for Nonlinear Time Space Fractional Diffusion Reaction Equations with Smooth and Nonsmooth Solutions
For the first time in literature, semi implicit spectral approximations for nonlinear Caputo time and Riesz space fractional diffusion equations with both smooth and non smooth solutions are proposed. More precisely, the governing partial differential equation generalizes the Hodgkin Huxley, the Allen Cahn and the Fisher Kolmogorov Petrovskii Piscounov equations. The schemes employ a Legendre based Galerkin spectral method for the Riesz space fractional derivative, and L 1 type approximations with both uniform and graded meshes for the Caputo time fractional derivative. More importantly, by using fractional Gronwall inequalities and their associated discrete forms, sharp error estimates are proved which show an enhancement in the convergence rate compared with the standard L 1 approximation on uniform meshes. This analysis encompasses both uniform meshes as well as meshes that are graded in time, and guarantees the unconditional stability. The numerical results that accompany our analysis confirm our theoretical error estimates, and give significant insights into the convergence behavior of our schemes for problems with smooth and non smooth solutions.
J. Sci. Comput.