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

Abstract

High-order adaptive methods for fractional differential equations are proposed. The methods rely on a kernel compression scheme for the approximation and localization of the history term. To avoid complications typical to multistep methods, we focus our study on 1-step methods and approximate the local part of the fractional integral by integral deferred correction to enable high order accuracy. We study the local truncation error of integral deferred correction schemes for Volterra equations and present numerical results obtained with both implicit and the explicit methods applied to different problems.

A Kernel Compression Scheme for Fractional Differential Equations

“…For applications, however, methods are required to possess some additional features such as adaptive step size and high-order convergence. We explore some ideas addressing this in [15]. Another topic deserving attention in future work is the study of fully discrete schemes of the type proposed.…”

“…The kernel compression scheme also has features making it convenient for use with adaptive step size methods. In [15] we propose high order adaptive methods for FDEs based on the kernel compression scheme developed in this paper.…”

A Gauss–Jacobi Kernel Compression Scheme for Fractional Differential Equations

“…A similar method employing the kernel compression scheme of [11] is tested therein. The second method is a high order and adaptive method, denoted LER-IDC in [15], which we test here with the kernel compression scheme (4.5). The method is obtained by applying an integral deferred correction scheme based on the left endpoint rule for the approximation of (2.14) and a 4th order, L-stable, diagonally implicit Runge-Kutta scheme for the approximation of (2.9).…”

“…It employs adaptive step size control and modifies the kernel compression approximation accordingly. For simplicity, we denote this scheme LER-IDR, similarly to [15].…”

“…To apply the scheme we write (6.5a) as a system by substituting y = D α x. Thus, we have Figure 5, is computed with the an adaptive TR-IDC scheme [15], with error tolerance ε = 10 −10 and the kernel compression scheme of [11]. Figure 6 shows, starting at the top right and continuing counterclockwise, the number P of auxiliary variables, the step size h, and the local error…”