Inverse source in two-parameter anomalous diffusion, numerical algorithms, and simulations over graded time meshes

We consider an inverse source two-parameter sub-diffusion model subject to a non-local initial condition. The problem models several physical processes, among them are the microwave heating and light propagation in photoelectric cells. A bi-orthogonal pair of bases is employed to construct a series representation of the solution and a Volterra integral equation for the source term. We develop a stable numerical algorithm, based on discontinuous collocation method, for approximating the unknown time-dependent source term. Due to the singularity of the solution near t= 0 , a graded mesh is used to maintain optimal convergence rates, both theoretically and numerically. Numerical experiments are provided to illustrate the expected analytical order of convergence.