A Gauss–Jacobi Kernel Compression Scheme for Fractional Differential Equations

Abstract

A scheme for approximating the kernel w of the fractional α-integral by a linear combination of exponentials is proposed and studied. The scheme is based on the application of a composite Gauss-Jacobi quadrature rule to an integral representation of w. This results in an approximation of w in an interval [δ, T ], with 0 < δ, which converges rapidly in the number J of quadrature nodes associated with each interval of the composite rule. Using error analysis for Gauss-Jacobi quadratures for analytic functions, an estimate of the relative pointwise error is obtained. The estimate shows that the number of terms required for the approximation to satisfy a prescribed error tolerance is bounded for all α ∈ (0, 1), and that J is bounded for α ∈ (0, 1), T > 0, and δ ∈ (0, T ).

Trends, directions for further research, and some open problems of fractional calculus

“…In view of the observations recalled in the introduction to Sect. 3, a number of novel numerical methods for solving fractional differential equations have been developed; see, e.g., [65][66][67][68][69][70][71][72][73][74][75][76][77]. Although there are various differences in the details, all these methods share the common feature that they are based on some nonclassical representation of the Caputo-type fractional differential operator C D α a+ of order α > 0 that appears in the differential equations under consideration, i.e., instead of one of the traditional forms…”

Good (and Not So Good) Practices in Computational Methods for Fractional Calculus

“…Although the terminology "kernel compression scheme" has been introduced only recently for a few specific works [40][41][42], we use it here to describe a collection of methods that were proposed at various times by various authors and are all based on essentially the same principle: approximation of the solution of a non-local FDE by means of (possibly several) local ODEs. We provide here just the main ideas underlying this approach and we will refer the reader to the literature for a more comprehensive coverage of the subject.…”

Efficient high order algorithms for fractional integrals and fractional differential equations

“…To compute up to time T = N h using formula (2) requires O(N ) memory and O(N 2 ) arithmetic operations. Algorithms based on FFT can reduce the computational complexity to O(N log N ) [16] or O(N log 2 N ) [9], but not the memory requirements; for an overview of FFT algorithms see [7].…”

“…Both these decisions allow us to substantially reduce constants in the above asymptotic estimates of memory and computational costs. Recent references [27,12,2] also consider fast computation of (1), but do not address the approximation of the convolution quadrature approximation exploiting (3). Our main contribution here is the development of an efficient quadrature to approximate (3) and its use in a fast and memory efficient scheme for computing the discrete convolution (2).…”

“…To our knowledge, underlying high order solvers for ODEs have been considered for the approximation of (1) only at experimental level in [2,3] and in [23], where a fast and oblivious implementation of RK based CQ is considered for more general applications than (1). The fast and oblivious quadratures of [19] and [23] have the same asymptotic complexity as our algorithm, but have a more complicated memory management structure and require the optimization of the shape of the integration contour.…”