Document Type

Article

Publication Date

2021

DOI

10.1216/jie.2021.33.271

Publication Title

Journal of Integral Equations and Applications

Volume

33

Issue

3

Pages

271-288

Abstract

We obtain new numerical schemes for weakly singular integrals of convolution type called Caputo fractional order integrals using Taylor and fractional Taylor series expansions and grouping terms in a novel manner. A fractional Taylor series expansion argument is utilized to provide fractional-order approximations for functions with minimal regularity. The resulting schemes allow for the approximation of functions in Cγ [0, T], where 0 < γ <= 5. A mild invertibility criterion is provided for the implicit schemes. Consistency and stability are proven separately for the whole-number-order approximations and the fractional-order approximations. The rate of convergence in the time variable is shown to be O(𝜏γ), 0 < γ ≤ 5 for u ∈ Cγ [0, T], where 𝜏 is the size of the partition of the time mesh. Crucially, the assumption of the integral kernel K being decreasing is not required for the scheme to converge in second-order and below approximations. Optimal convergence results are then proven for both sets of approximations, where fractional-order approximations can obtain up to whole-number rate of convergence in certain scenarios. Finally, numerical examples are provided that illustrate our findings.

Comments

© 2021 Rocky Mountain Mathematics Consortium.

Included with the kind written permission of the copyright holders.

Original Publication Citation

Davis, W., & Noren, R. (2021). Stable and convergent difference schemes for weakly singular convolution integrals. Journal of Integral Equations Applications, 33(3), 271-288. https://doi.org/10.1216/jie.2021.33.271

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