Modified collocation-based numerical methods for Volterra Integral and Integro-Differential Equations

The aim of this talk is to present modified collocation based numerical methods for solving Volterra Integral and Integro-Differential Equations (VIEs and VIDEs respectively), which turn out to be at the heart of many modern applications of Mathematics to natural phenomena and are used more and more for the description of complex systems, in particular evolutionary problems with memory. In the literature many authors (see [1] and references therein contained) have analyzed one-step collocation methods for VIEs and VIDEs. As it is well known, a collocation method is based on the idea of approximating the exact solution of a given integral equation with a suitable function belonging to a chosen finite dimensional space, usually a piecewise algebraic polynomial, which satisfies the integral equation exactly on a certain subset of the integration interval (called the set of collocation points). The collocation technique allows the derivation of methods having many desirable properties. In fact, collocation methods provide an approximation at each point of the integration interval to the solution of the equation, thus leading, in general, to a cheap variable stepsize implementation. Moreover, the collocation function can be expressed as a linear combination of functions ad hoc for the problem we are integrating, in order to better reproduce the qualitative behaviour of the solution. The purpose of this talk is to present recently introduced families of collocation and modified collocation methods for VIEs [3, 4, 5, 6, 7] and VIDEs [2], methods which have been widely developed in the context of ordinary and delay differential equations. In particular we aim to present the main results obtained in the context of multistep collocation and almost collocation methods, i.e. methods obtained by relaxing some collocation and/or interpolation conditions in order to obtain desirable stability properties. We describe the derivation of the methods and also derive diagonally-implicit methods for VIEs, which result from the choice of suitable conditions to be satisfied by the collocation polynomial and from the application of special quadrature formulae for the approximation of the increment term, in such a way that the coefficient matrix of the nonlinear system for the computation of the internal stages is lower triangular or diagonal. The structured shape of the coefficient matrix can be suitably exploited in order to get an efficient implementation, also in a parallel environment. The constructed methods have an high uniform order of convergence together with strong stability properties (e.g A-stability). We present the constructive technique, discuss the order of convergence, provide examples of A-stable methods and show numerical experiments that confirm the theoretical expectations. References [1] H. Brunner, Collocation Methods for Volterra Integral and Related Functional Equations, Cambridge University Press (2004). [2] A. Cardone, D. Conte, B. Paternoster, A family of Multistep Collocation Methods for Volterra Integro–Differential Equations, in Numerical Analysis and Applied Mathematics ed. by T. E. Simos, G. Psihoyios, Ch. Tsitouras; AIP Conf. Proc. 1168 (1), Springer (2009), 358–361. [3] Conte, D., D’Ambrosio, and Paternoster, B., Two-step diagonally–implicit collocation–based methods for Volterra Integral Equations, in preparation. [4] Conte, D., D’Ambrosio, R., Ferro, M., and Paternoster, B., Modified Collocation Techniques for Volterra Integral Equations, in press on Applied and Industrial Mathematics in Italy III. [5] Conte, D., Jackiewicz, Z., and Paternoster, B., Two-step almost collocation methods for Volterra integral equations, Appl. Math. Comput. 204 (2008), 839–853. [6] Conte, D. and Paternoster, B., A Family of Multistep Collocation Methods for Volterra Integral Equations, in Numerical Analysis and Applied Mathematics ed. by T. E. Simos, G. Psihoyios, Ch. Tsitouras; AIP Conf. Proc. 936, Springer (2007), 128–131. [7] Conte, D. and Paternoster, B., Multistep collocation methods for Volterra Integral Equations, Appl. Numer. Math. 59 (2009), 1721–1736.