SUSHI for a Non-linear Time Fractional Diffusion Equation with a Time Independent Delay

We establish a linear implicit finite volume scheme for a non-linear time fractional diffusion equation with a time independent delay in any space dimension. The fractional order derivative is given in the Caputo sense. The discretization in space is performed using the SUSHI ((Scheme Using stabilized Hybrid Interfaces) developed in [11], whereas the discretization in time is given by a constrained time step-size. The approximation of the fractional order derivative is given by L1-formula. We prove rigorously new convergence results in $$L^\infty (L^2)$$ and $$L^2(H^1_0)$$ –discrete norms. The order is proved to be optimal in space and it is $$k^{2-\alpha }$$ in time, with k is the constant time step and $$\alpha $$ is the fractional order of the Caputo derivative. This paper is a continuation of some of our previous works which dealt either with only the linear fractional PDEs (Partial Differential Equations) without delays, e.g. [6, 7, 9, 10], or with only time dependent PDEs (the time derivative is given in the usual sense) with delays, e.g. [2, 4, 8].