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

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 α∈(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≥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 α∈(0,2) and is amenable to the numerical evaluation using uniformly accurate quadrature-based algorithms.

Exponentially Convergent Numerical Method for Abstract Cauchy Problem with Fractional Derivative of Caputo Type

“…The illustration provided by Figure 1 shows that both scalar and operator parts of the parametrized integrand F α (t, ξ), t ∈ [0, T] remain analytic when ξ is extended into a certain complex neighborhood D of R. According to the general theory of numerical integration [49], an accuracy of quadrature formula is characterized by a norm of the error-term in the Hardy space H p (D) of functions, defined on the domain D ⊂ C. The shape of D depends on the chosen type of quadrature. For the reasons that are soon to be understood, we approximate integral (10) by the sinc-quadrature formula [31,50]:…”

“…Assume Γ is a contour fulfilling the conditions of Theorem 1. Due to the estimate [10], the integral representation of operator function S α,2 (t) is uniformly convergent for any bounded non-negative t. Formula ( 19) is obtained as a result of the parametrization of (7) on the contour Γ I defined by (8). In order to prove (18), we apply the identity 1 2πi Γ e zt /z dz = Res z=0 e zt /z = 1 to rewrite (7) with β = 1 in the following manner:…”

“…These cases also include the fractional powers of elliptic operators are encompassed by the class of strongly positive operators [13] considered in Theorem 1 and below. In this regard, the shape of Sp(A) justifies the choice of the range (0, 2) for α, as a maximally possible for the considered class of A (see [10] for a more detailed discussion).…”

“…The theory of fractional Cauchy problems for differential operators was developed in the works [3][4][5]. The abstract setting, considered here, has been theoretically studied in [6][7][8] for α ∈ (0, 1) then in [9] for α ∈ [1, 2) and, most recently, in [10]. In the current work, we focus on the numerical evaluation of the mild solution to problem (1), ( 4) that is given by the following result.…”

“…In addition to that, the rigorous analysis from [28,30] addresses a version of the proxy problem where ∂ α is a Riemann-Liouville (RL) fractional derivative. Cauchy problems with RL derivative are simpler in the sense of propagator representation [10], but they are compatible with (1), (4) only under some additional assumptions.…”