2021/02/09 by Ustim Khristenko, Barbara Wohlmuth, Khristenko, Ustim +1 · 1 citation
Computer Science · Mathematics · #26A33 #34A08 #35R11 #41A20 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Fractional Differential Equations Solutions #Iterative Methods for Nonlinear Equations #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2102.05139
openalex publication_date 2021/02/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Fractional differential equations (FDEs) describe subdiffusion behavior of dynamical systems. Its non-local structure requires taking into account the whole evolution history during the time integration, which then possibly causes additional memory use to store the history, growing in time. An alternative to a quadrature for the history integral is to approximate the fractional kernel with the sum of exponentials, which is equivalent to considering the FDE solution as a sum of solutions to a system of ODEs. One possibility to construct this system is to approximate the Laplace spectrum of the fractional kernel with a rational function. In this paper, we use the adaptive Antoulas--Anderson (AAA) algorithm for the rational approximation of the kernel spectrum which yields only a small number of real valued poles. We propose a numerical scheme based on this idea and study its stability and convergence properties. In addition, we apply the algorithm to a time-fractional Cahn-Hilliard problem.