Two Methods Addressing Variable-Exponent Fractional Initial and Boundary Value Problems and Abel Integral Equation (CSIAM Transactions on Applied Mathematics, 2025)
Variable-exponent fractional models attract increasing attentions in variousapplications, while rigorous mathematical and numerical analysis for typical modelsremains largely untreated. This work provides general tools to address these models.Specifically, we first develop a convolution method to study the well-posedness, regularity, an inverse problem and numerical approximation for the subdiffusion of variable exponent. For models such as the variable-exponent two-sided space-fractionalboundary value problem (including the variable-exponent fractional Laplacian equation as a special case) and the distributed variable-exponent model, for which the convolution method does not apply, we develop a perturbation method to prove theirwell-posedness. The relations between the convolution method and the perturbationmethod are discussed, and we further apply the latter to prove the well-posedness ofthe variable-exponent Abel integral equation and discuss the constraint on the dataunder different initial values of variable exponent.
Keywords: Variable exponent; fractional differential equation; integral equation; mathematical analysis; numerical analysis.