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June  2004, 3(2): 217-235. doi: 10.3934/cpaa.2004.3.217

High order product integration methods for a Volterra integral equation with logarithmic singular kernel

T. Diogo 1, , N. B. Franco 2, and  P. Lima 3,

1. 

Centro de Matematica e Aplicącoes, Instituto Superior Tecnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
2. 

Laboratorio Interdisciplinar de Computacao Cientıfica, Faculdades COC, 14096-160 Ribeirao Preto - SP, Brazil
3. 

Centro de Matemática Aplicacoes, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal

Received  June 2003 Revised  January 2004 Published  March 2004

This work is concerned with the construction and analysis of high order product integration methods for a class of Volterra integral equations with logarithmic singular kernel. Sufficient conditions for the methods to be convergent are derived and it is shown that optimal convergence orders are attained if the exact solution is sufficiently smooth. The case of non-smooth solutions is dealt with by making suitable transformations so that the new equation possesses smooth solutions. Two particular methods are considered and their convergence proved. A sample of numerical examples is included.
Keywords: product integration methods, convergence., Volterra integral equations, weakly singular kernel.
Mathematics Subject Classification: 65R20.
Citation: T. Diogo, N. B. Franco, P. Lima. High order product integration methods for a Volterra integral equation with logarithmic singular kernel. Communications on Pure & Applied Analysis, 2004, 3 (2) : 217-235. doi: 10.3934/cpaa.2004.3.217
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