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

Abstract

We present an exponentially convergent numerical method to approximate the solution of the Cauchy problem for the inhomogeneous fractional differential equation with an unbounded operator coefficient and Caputo fractional derivative in time. The numerical method is based on the newly obtained solution formula that consolidates the mild solution representations of sub-parabolic, parabolic and sub-hyperbolic equations with sectorial operator coefficient A and non-zero initial data. The involved integral operators are approximated using the sinc-quadrature formulas that are tailored to the spectral parameters of A, fractional order α and the smoothness of the first initial condition, as well as to the properties of the equation’s right-hand side f(t). The resulting method possesses exponential convergence for positive sectorial A, any finite t, including t=0 and the whole range α∈(0,2). It is suitable for a practically important case, when no knowledge of f(t) is available outside the considered interval t∈[0,T]. The algorithm of the method is capable of multi-level parallelism. We provide numerical examples that confirm the theoretical error estimates.

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

“…Under the same conditions, the norm of the remaining integrand from ( 6) is asymptotically equal to e z(t−s) |z| −α , which leads to a slower-than-linear decay with respect to z for t = s and α < 1. This fact is obviously detrimental to the practical applications of (6) relying on the quadrature-based numerical evaluation of the solution [35][36][37][38]. The same observation applies to the solution representations from [18].…”

“…To justify the new solution representation, we rely upon the results from the theory of abstract integral equations [25] instead of the usual toolkit from the holomorphic function calculus. The new representation is thoroughly validated in [38], where it is used as a base for the exponentially convergent numerical method.…”

“…The mentioned drawbacks of finite-difference methods have put an additional spotlight on the alternative numerical solution evaluation strategies [47][48][49] and, in particular, on the quadrature-based numerical methods [35][36][37][38]. Unlike finite differences, the methods from the latter group perform the direct numerical evaluation of ( 17) via the quadrature of the integral in Formula ( 13), defining the operator function S α,β (t).…”

“…For that reason, the numerical potential of ( 22) has been largely left unexplored, with the exception of [54]. The direct approximation of (24) might become a worthwhile complement to the numerical methods from [36][37][38], provided that the underlying integrals can be numerically evaluated in an accurate and efficient manner for small values of α ∈ (0, 0.1) or when αϕ s > π/2. The evidence supplied in [38] indicates that all the mentioned complications with (17) can be avoided if the solution formula involves only the propagators S α (t), S α,2 (t).…”

“…The direct approximation of (24) might become a worthwhile complement to the numerical methods from [36][37][38], provided that the underlying integrals can be numerically evaluated in an accurate and efficient manner for small values of α ∈ (0, 0.1) or when αϕ s > π/2. The evidence supplied in [38] indicates that all the mentioned complications with (17) can be avoided if the solution formula involves only the propagators S α (t), S α,2 (t). The next section is devoted to deriving the alternative representation of the solution to the fractional Cauchy problem (1), (7) that fulfills this property.…”