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The numerical analysis of functional integral and integro-differential equations of Volterra type. (English) Zbl 1122.65410

Summary: The qualitative and quantitative analysis of numerical methods for delay differential equations (DDEs) is now quite well understood, as reflected in the recent monograph by A. Bellen and M. Zennaro [Numerical methods for delay differential equations (2003; Zbl 1038.65058)]. This is in remarkable contrast to the situation in the numerical analysis of functional equations, in which delays occur in connection with memory terms described by Volterra integral operators. The complexity of the convergence and asymptotic stability analyses has its roots in new dimensions not present in DDEs: the problems have distributed delays; kernels in the Volterra operators may be weakly singular; a second discretization step (approximation of the memory term by feasible quadrature processes) will in general be necessary before solution approximations can be computed.

The purpose of this review is to introduce the reader to functional integral and integro-differential equations of Volterra type and their discretization, focusing on collocation techniques; to describe the state of the art in the numerical analysis of such problems; and to show that especially for many classical equations whose analysis dates back more than 100 years we still have a long way to go before we reach a level of insight into their discretized versions to compare with that achieved for DDEs.


MSC:
65R20Integral equations (numerical methods)
45G10Nonsingular nonlinear integral equations
45J05Integro-ordinary differential equations