A Gauss-Jacobi Kernel Compression Scheme for Fractional Differential Equations
This provides an efficient kernel compression method for solving fractional differential equations, but the work is incremental as it extends existing quadrature techniques to a specific kernel representation.
The paper proposes a Gauss-Jacobi quadrature-based scheme to approximate the kernel of fractional integrals by a sum of exponentials, achieving rapid convergence with a bounded number of terms for all fractional orders α in (0,1).
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 $α\in(0,1)$, and that $J$ is bounded for $α\in(0,1)$, $T>0$, and $δ\in(0,T)$.