Composite Quadrature Methods for Weakly Singular Convolution Integrals

The well-known Caputo fractional derivative and the corresponding Caputo fractional integral occur naturally in many equations that model physical phenomena under inhomogeneous media. The relationship between the two fractional terms can be readily obtained by applying the Laplace transform to a given equation. We seek to numerically approximate Caputo fractional integrals using a Taylor series expansion for convolution integrals. This naturally extends into being able to approximate convolution integrals for a wider class of convolution integral kernels $K(t-s)$. One of the main advantages under this approach is the ability to numerically approximate weakly singular kernels, which fail to converge under traditional quadrature methods. We provide stability and convergence analysis for these composite quadratures, which offer optimal convergence for approximating functions in $C^{\gamma}[0,T]$, where $\alpha \leq \gamma \leq 5$ and $0<\alpha < 1$. For the order $\gamma = 1,2,3,4,5$ scheme, the resulting approximation is $O(\tau^{\gamma})$ accurate, where $\tau$ is the size of the partition of the time domain. By instead utilizing a fractional Taylor series expansion, we are able to obtain for $\gamma \in (0,5)-\{1,2,3,4\}$ order scheme, which yields an approximation of $O(\tau^{\gamma})$ with a constant dependent on the kernel function which improves the order of convergence. This allows for a far wider class of functions to be approximated, and by strengthening the regularity assumption, we are able to obtain more accurate results. General convolution integrals exhibit these results up to $\gamma = 2$ without the assumption of $K$ being decreasing. Finally, some numerical examples are presented, which validate our findings.