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

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