Existence and Attractivity of Mild Solutions for Fractional Diffusion Equations Involving the Regularized ψ-Hilfer Fractional Derivatives - Luyao Wang; Yuhang Jin; Wenchang He; Jia Mu (Axioms, 2025)
The regularized <semantics>ψ</semantics>-Hilfer derivative within the sense of Caputo is an improved version of the <semantics>ψ</semantics>-Hilfer fractional derivative, primarily because it addresses the issue where the initial conditions of problems involving the <semantics>ψ</semantics>-Hilfer fractional derivative lack clear physical significance unless <semantics>p=1</semantics>. This article’s main contribution is the use of the <semantics>ψ</semantics>-Laplace transform, which is the first to provide an explicit expression for mild solutions to the fractional diffusion equations with the regularized <semantics>ψ</semantics>-Hilfer derivative. Additionally, we investigate the existence and attractivity of mild solutions for fractional diffusion equations involving the regularized <semantics>ψ</semantics>-Hilfer fractional derivatives. Finally, we provide two examples to illustrate our main results.