A novel Jacob spectral method for multi-dimensional weakly singular nonlinear Volterra integral equations with nonsmooth solutions

• Published in: Engineering with computers Vol. 37; no. 4; pp. 2623 - 2631
• Main Authors: Zaky, Mahmoud A., Ameen, Ibrahem G.
• Format: Journal Article
• Language: English
• Published: London Springer London 01-10-2021; Springer Nature B.V
• Subjects: Approximation; CAE) and Design; Calculus of Variations and Optimal Control; Optimization; Classical Mechanics; Collocation methods; Computer Science; Computer-Aided Engineering (CAD; Control; Integral equations; Math. Applications in Chemistry; Mathematical and Computational Engineering; Original Article; Singular integral equations; Singularities; Spectra; Spectral methods; Systems Theory; Volterra integral equations; Nonsmooth solution; Weakly singular kernels; Spectral methods; Volterra integral equations; Convergence analysis
• ISSN: 0177-0667; 1435-5663
• DOI: 10.1007/s00366-020-00953-9

The main purpose of this work is to develop a spectrally accurate collocation method for solving weakly singular integral equations of the second kind with nonsmooth solutions in high dimensions. The proposed spectral collocation method is based on a multivariate Jacobi approximation in the frequency space. The essential idea is to adopt a smoothing transformation for the spectral collocation method to circumvent the curse of singularity at the beginning of time. As such, the singularity of the numerical approximation can be tailored to that of the singular solutions. A rigorous convergence analysis is provided and confirmed by numerical tests with nonsmooth solutions in two dimensions. The results in this paper seem to be the first spectral approach with a theoretical justification for high-dimensional nonlinear weakly singular Volterra type equations with nonsmooth solutions.