An efficient split-step method for distributed-order space-fractional reaction-diffusion equations with time-dependent boundary conditions.

A split-step predictor-corrector method, based on Adams-Moulton formula, is presented for the solution of nonlinear distributed-order space-fractional reaction-diffusion equations with time-dependent boundary conditions. The distributed-order space-fractional equation is first discretized to a multi-term space-fractional equation by applying a quadrature rule. Multi-term space-fractional equation is spatially discretized by matrix transfer technique and a linearly implicit split-step predictor-corrector method is applied for time-stepping to avoid solving nonlinear systems at each time step. The method is shown to be unconditionally stable and second order convergent. Numerical experiments are performed to confirm the stability and second order convergence of the method. The split-step predictor-corrector method is also compared with a second order IMEX predictor-corrector method. The IMEX predictor-corrector method is found to incur oscillatory behavior when implemented on problems with nonsmooth initial conditions or with mismatch between the initial and boundary conditions. Our method, which is an L -stable method, produces reliable and oscillation free results for such problems. [ABSTRACT FROM AUTHOR]