Fully discretized collocation methods for nonlinear singular Volterra integral equations

Authors:
Teresa Diogo;Jingtang Ma;Magda Rebelo
Affiliations:
CEMAT-Departamento de Matemática, Instituto Superior Técnico, UTL, Av. Rovisco Pais, 1049-001 Lisbon, Portugal;School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, 611130, China;Departamento de Matemática, Faculdade de Ciências e Tecnologia, UNL, Quinta da Torre, 2829-516 Caparica, Portugal
Venue:
Journal of Computational and Applied Mathematics
Year:
2013

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The piecewise polynomial collocation method for nonlinear weakly singular Volterra equations

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A hybrid collocation method for a nonlinear Volterra integral equation with weakly singular kernel

Journal of Computational and Applied Mathematics

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Abstract

We consider a nonlinear weakly singular Volterra integral equation arising from a problem studied by Lighthill (1950) [1]. A series expansion for the solution is obtained and shown to be convergent in a neighbourhood of the origin. Owing to the singularity of the solution at the origin, the global convergence order of product integration and collocation methods is not optimal. However, the optimal orders can be recovered if we use the fully discretized collocation methods based on graded meshes. A theoretical proof is given and we present some numerical results which illustrate the performance of the methods.