Finite difference/finite element method for a nonlinear time-fractional fourth-order reaction-diffusion problem.

In this article, a finite difference/finite element algorithm, which is based on a finite difference approximation in time direction and finite element method in spatial direction, is presented and discussed to cast about for the numerical solutions of a time-fractional fourth-order reaction–diffusion problem with a nonlinear reaction term. To avoid the use of higher-order elements, the original problem with spatial fourth-order derivative need to be changed into a second-order coupled system by introducing an intermediate variable σ=Δu. Then the fully discrete finite element scheme is formulated by using a finite difference approximation for time fractional and integer derivatives and finite element method in spatial direction. The unconditionally stable result in the norm, which just depends on initial value and source item, is derived. Some a priori estimates of L2-norm with optimal order of convergence O(Δt2−α+hm+1), where Δt and h are time step length and space mesh parameter, respectively, are obtained. To confirm the theoretical analysis, some numerical results are provided by our method.