A second-order scheme for a time-fractional diffusion equation (Q1726585)

The authors are concerned with a numerical second-order approximation scheme for solutions of the reaction-diffusion equation \[ D_t^\alpha u(x,t)+Lu(x,t)=f(x,t),\quad (x,t)\in (0,\ell)\times (0,T], \] subject to initial and boundary conditions. Here, $D_t^\alpha$ denotes the Caputo derivative whith $0<\alpha<1$ and $Lu(x,t)=-p\frac{\partial^2 u}{\partial x^2}(x,t)+c(x)u(x,t)$, where $p>0$ is a real number and $c\in C[0,\ell]$. It is known that the above problem admits a unique solution which typically exhibits a weak singularity at $t=0$. \par In the present work, the authors construct and analyze an integral discretization scheme on a graded mesh along with a decomposition of the exact solution of above problem. It is shown that the convergence order of the proposed scheme is $O(M^{-2}+N^{-2})$, where $M$ and $N$ are the spatial and temporal discretization parameters. Numerical experiments are provided to validate the theoretical findings.