Composite Quadrature Methods for Weakly Singular Convolution Integrals

2020/01/22 by Davis, Wesley, Noren, Richard
#65L07 #65M12 #65N12 #65R10 #65R20 #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2001.08207

Abstract

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^γ[0,T], where α≤ γ≤ 5 and 0

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