# On the fractional operators with respect to another function

## TL;DR

This paper studies fractional derivatives with respect to another function, establishing existence, uniqueness, and numerical methods, applicable to various fractional operators and integral equations, with demonstrated efficiency through examples and tests.

## Contribution

It introduces a unified approach for fractional derivatives with respect to another function, covering multiple operators and providing numerical schemes with optimal convergence.

## Key findings

- Results valid for Riemann-Liouville, Caputo, Hadamard, Erdélyi-Kober operators
- Numerical schemes achieve optimal convergence rates on graded meshes
- Applications include solving Volterra integral equations efficiently

## Abstract

This paper is in concern with Cauchy problems involving the fractional derivatives with respect to another function. Results of existence, uniqueness, and Taylor series among others are established in appropriate functional spaces. We prove that these results are valid at once for several standard fractional operators such as the Riemann-Liouville and Caputo operators, the Hadamard operators, the Erd\'elyi-Kober operators, etc., depending on the choice of the scaling function. We also show that our technique can be useful to solve a wide range of Volterra integral equations. The numerical approximation of solutions of systems involving the fractional derivatives with respect to another function is also investigated and the optimal convergence rate of the schemes is reached in graded meshes, even in the case of singular solutions. Various examples and numerical tests, with an application to the Erd\'elyi-Kober operators, are performed at the end to illustrate the efficiency of the proposed approach.