TY - JOUR

T1 - Abstract Fractional Cauchy Problem

T2 - Existence of Propagators and Inhomogeneous Solution Representation

AU - Sytnyk, Dmytro

AU - Wohlmuth, Barbara

N1 - Publisher Copyright:
© 2023 by the authors.

PY - 2023/10

Y1 - 2023/10

N2 - 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 (Formula presented.) 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 (Formula presented.) 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 (Formula presented.) and is amenable to the numerical evaluation using uniformly accurate quadrature-based algorithms.

AB - 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 (Formula presented.) 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 (Formula presented.) 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 (Formula presented.) and is amenable to the numerical evaluation using uniformly accurate quadrature-based algorithms.

KW - Caputo fractional derivative

KW - contour representation

KW - inhomogeneous Cauchy problem

KW - mild solution

KW - propagator

KW - sub-hyperbolic problem

KW - sub-parabolic problem

UR - http://www.scopus.com/inward/record.url?scp=85175169408&partnerID=8YFLogxK

U2 - 10.3390/fractalfract7100698

DO - 10.3390/fractalfract7100698

M3 - Article

AN - SCOPUS:85175169408

SN - 2504-3110

VL - 7

JO - Fractal and Fractional

JF - Fractal and Fractional

IS - 10

M1 - 698

ER -