Abstract Fractional Cauchy Problem: Existence of Propagators and Inhomogeneous Solution Representation

TL;DR

This paper introduces a new, more natural solution representation for fractional Cauchy problems involving Caputo derivatives, improving theoretical understanding and numerical feasibility across various equation types.

Contribution

It proposes an alternative solution formula based solely on the homogeneous propagator, simplifying analysis and enabling practical numerical methods for fractional differential equations.

Findings

New solution representation consolidates different equation types

Weaker regularity assumptions for convergence

Enhanced numerical applicability for all fractional orders

Abstract

We consider a Cauchy problem for the inhomogeneous differential equation given in terms of an unbounded linear operator $A$ and the Caputo fractional derivative of order $\alpha \in (0, 2)$ in time. The previously known representation of the mild solution to such a problem does not have a conventional variation-of-constants like form, with the propagator derived from the associated homogeneous problem. Instead, it relies on the existence of two propagators with different analytical properties. This fact limits theoretical and especially numerical applicability of the existing solution representation. Here, we propose an alternative representation of the mild solution to the given problem, that consolidates the solution formulas for sub-parabolic, parabolic and sub-hyperbolic equations with a positive sectorial operator $A$ and non-zero initial data. The new representation is solely based on the propagator of the homogeneous problem and, therefore, can be considered as a more natural fractional extension of the solution to the classical parabolic Cauchy problem. By exploiting a trade-off between the regularity assumptions on the initial data in terms of the fractional powers of $A$ and the regularity assumptions on the right-hand side in time, we show that the proposed solution formula is strongly convergent for $t \geq 0$ under considerably weaker assumptions compared to the standard results from the literature. Crucially, the achieved relaxation of space regularity assumptions ensures that the new solution representation is practically feasible for any $\alpha \in (0, 2)$ and is amenable to the numerical evaluation using uniformly accurate quadrature-based algorithms.