# On Laplace transforms with respect to functions and their applications to fractional differential equations

## TL;DR

This paper develops an operational calculus approach for Laplace transforms with respect to functions, simplifying their analysis and application in solving fractional differential equations within the framework of $ ext{ extPsi}$-fractional calculus.

## Contribution

It introduces a new operational calculus method for Laplace transforms with respect to functions, enhancing the analysis and solution of fractional differential equations.

## Key findings

- Established properties and inversion formula for generalized Laplace transforms
- Applied the approach to solve fractional differential equations efficiently
- Enhanced understanding of fractional calculus with respect to functions

## Abstract

An important class of fractional differential and integral operators is given by the theory of fractional calculus with respect to functions, sometimes called $\Psi$-fractional calculus. The operational calculus approach has proved useful for understanding and extending this topic of study. Motivated by fractional differential equations, we present an operational calculus approach for Laplace transforms with respect to functions and their relationship with fractional operators with respect to functions. This approach makes the generalised Laplace transforms much easier to analyse and to apply in practice. We prove several important properties of these generalised Laplace transforms, including an inversion formula, and apply it to solve some fractional differential equations, using the operational calculus approach for efficient solving.