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

TL;DR

This paper introduces an approximate inversion method for variable-exponent Abel integral operators, enabling better analysis and control of fractional Cauchy problems by adjusting initial conditions and exponents.

Contribution

It develops a novel approximate inversion technique for variable-exponent Abel operators and demonstrates how adjusting initial exponents improves well-posedness and solution regularity.

Findings

The inversion technique converts complex variable-exponent problems into feasible forms.

Adjusting initial exponents resolves sensitivity issues in fractional differential equations.

Variable exponents can eliminate solution singularities at initial time.

Abstract

We investigate the variable-exponent Abel integral equations and corresponding fractional Cauchy problems. The main contributions of the work are enumerated as follows: (i) We develop an approximate inversion technique for variable-exponent Abel integral operators, based on which we analyze the corresponding integral and differential equations; (ii) We prove that the sensitive dependence of the well-posedness of classical Riemann-Liouville fractional differential equations on the initial value could be resolved by adjusting the initial value of the variable exponent; (iii) We prove that the singularity of the solutions to the Riemann-Liouville fractional differential equations could also be eliminated by adjusting the variable exponent and its derivatives at the initial time, which, together with (ii), demonstrates the advantages of introducing the variable exponent. The proposed approximate inversion technique provides a potential means to convert the intricate variable-exponent integral and differential problems to feasible forms, and 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.