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.

翻译：本文考虑具有可变记忆特性的变指数次扩散现象建模问题。主要难点在于该模型无法解析求解，且变指数阿贝尔核可能不具备正定性或单调性。本研究建立了名为广义恒等函数的工具，将模型转化为更适用于数学分析与数值计算的表达形式，并在此基础上证明了模型的适定性与正则性。特别地，我们以指数初值表征了解函数的奇异性。随后发展了半离散与全离散数值方法，在无需对解的正则性假设或要求变指数阿贝尔核具备特定性质的前提下，证明了其误差估计。收敛阶同样通过指数初值进行刻画。最后，我们探讨了确定指数初值的反问题。