Computation and analysis of subdiffusion with variable exponent

Xiangcheng Zheng

Published 2024-04-15Version 1

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.