Computation and analysis of subdiffusion with variable exponent

We consider the subdiffusion of variable exponent modeling subdiffusion phenomena with varying memory properties. The main difficulty is that this model could not be analytically solved and the variable-exponent Abel kernel may not be positive definite or monotonic. This work develops a tool called the generalized identity function to convert this model to more feasible formulations for mathematical and numerical analysis, based on which we prove its well-posedness and regularity. In particular, we characterize the singularity of the solutions in terms of the initial value of the exponent. Then the semi-discrete and fully-discrete numerical methods are developed and their error estimates are proved, without any regularity assumption on solutions or requiring specific properties of the variable-exponent Abel kernel. The convergence order is also characterized by the initial value of the exponent. Finally, we investigate an inverse problem of determining the initial value of the exponent.