A Kernel Compression Scheme for Fractional Differential Equations

Abstract

Abstract. The nonlocal nature of the fractional integral makes the numerical treatment of fractional differential equations expensive in terms of computational effort and memory requirements. In this paper we propose a method to reduce these costs while controlling the accuracy of the scheme. This is achieved by splitting the fractional integral of a function f into a local term and a history term. Observing that the history term is a convolution of the history of f and a regular kernel, we derive a multipole approximation to the Laplace transform of the kernel. This enables the history term to be replaced by a linear combination of auxiliary variables defined as solutions to standard ordinary differential equations. We derive a priori error estimates, uniform in f , and obtain estimates on the number of auxiliary variables required to satisfy an error tolerance. The resulting formulation is discretized to produce a time stepping method. The method is applied to some test cases to illustrate the performance of the scheme.

High-Order Accurate Adaptive Kernel Compression Time-Stepping Schemes for Fractional Differential Equations

“…A discontinuous Galerkin time-stepping method has also been proposed for Volterra equations [5,6]. In this paper we propose high-order adaptive methods that are based on an efficient kernel compression [7] approximation of the history term.…”

“…The kernel compression scheme [7] prescribes an approximation to (1.1) given by a linear combination of solutions ψ 1 , . .…”

“…The main result of [7] is the following a priori estimate on J: for T > 0, δ > 0, and an error tolerance ε > 0, the approximation operator I δ,r prescribed by the scheme, given by I δ,r f = S * f , with J = pm, where…”

“…See also [11] for approximation of some functions by sums of exponentials. As the kernel compression scheme is not the main focus of this work, we refer the reader to [7] for a qualitative discussion of the different methods. With relevance to the context of this paper, we mention the following.…”

“…In [12] a fast collocation method is proposed. In addition to the efficient account of the history term, the kernel compression scheme proposed in [7] has other advantages. One is that the theoretical estimates are uniform in α ∈ (0, 1), and f ∈ L 2 (0, T ).…”