Approximate inversion for Abel integral operators of variable exponent and applications to fractional Cauchy problems

Abstract

We investigate the variable-exponent Abel integral equations and corresponding fractional Cauchy problems. The main contributions are twofold: We provide a novel inverse technique to convert the first-kind Volterra integral equation of variable exponent to a second-kind one, which, to our best knowledge, is not available in the literature; Based on this transformation, we carry out rigorous analysis to prove several theoretical results and their dependence on the variable exponent. In general, we conclude that by setting an integer limit of variable exponent $\alpha (t)$ at the initial time $t=0$, the variable-exponent problems have similar properties as their integer-order analogues. Otherwise, they behave like their constant-exponent counterparts of order $\alpha \equiv \alpha (0)$. To be specific, we prove that the sensitive dependence of the well-posedness of classical Riemann-Liouville fractional differential equations on the initial value and the singularity of their solutions could be resolved by adjusting the variable exponent at the initial time, which demonstrates the advantages of introducing the variable exponent. The above findings suggest that the variable-exponent fractional problems may serve as a connection between integer-order and fractional models by adjusting the variable exponent at the initial time.