Numerical modelling of qualitative behaviour of solutions to convolution integral equations

Authors:
Neville J. Ford;Teresa Diogo;Judith M. Ford;Pedro Lima
Affiliations:
Mathematics Department, University of Chester, Parkgate Road, Chester, CH1 4BJ, UK;Departamento de Matematica, Instituto Superior Tecnico, Av. Rovisco Pais, 1049-001, Lisbon, Portugal;Mathematics Department, University of Chester, Parkgate Road, Chester, CH1 4BJ, UK;Departamento de Matematica, Instituto Superior Tecnico, Av. Rovisco Pais, 1049-001, Lisbon, Portugal
Venue:
Journal of Computational and Applied Mathematics
Year:
2007

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Abstract

We consider the qualitative behaviour of solutions to linear integral equations of the form(1)y(t)=g(t)+@!"0^tk(t-s)y(s)ds,where the kernel k is assumed to be either integrable or of exponential type. After a brief review of the well-known Paley-Wiener theory we give conditions that guarantee that exact and approximate solutions of (1) are of a specific exponential type. As an example, we provide an analysis of the qualitative behaviour of both exact and approximate solutions of a singular Volterra equation with infinitely many solutions. We show that the approximations of neighbouring solutions exhibit the correct qualitative behaviour.