The interest for numerical solution of physical and biological problems with oscillating and/or periodic behaviour requires the use of special-purpose methods. Examples include the electromagnetic scattering, the response of nonlinear circuits to a periodic input and the evolution of an age-structuredpopulation. These problems are characterized by either infinite integrals where the integrand function is oscillatory function, or by Volterra integral equations of type with periodic solution. By exploting the Exponential Fitting theory [1, 2, 3, 4, 5], a new class of quadrature rules, that are a generalization of the usual Gauss-Laguerre formulae, for problem (1) and a new direct quadrature (DQ) method for problem (2) are derived, respectively. Two extra problems appear in the context of building the exponentially-fitted (ef ) DQ method. The first one is the construction of a two-nodes ef quadrature rule of Gaussian type, that is a generalization of the usual two-nodes Gauss-Legendre formula, on which the DQ method is based. The second problem is the building of a suitable ef interpolation technique on four points which preserves the order of convergence of the overall method. These works are in collaboration with L. Gr. Ixaru (National Institute of Physics and Nuclear Engineering, Bucharest, Romania), B. Paternoster, A. Cardone and D. Conte (University of Salerno). References [1] Ixaru, L.Gr.; Vanden Berghe, G., Exponential fitting, Kluwer Academic Publishers, Dordrecht (2004). [2] A. Cardone, B. Paternoster, G. Santomauro, Exponential fitting quadrature rule for functional equations, AIP Conference Proceedings 1479 (2012) 1169-1172. [3] A. Cardone, L. Gr. Ixaru, B. Paternoster, G. Santomauro, Ef-Gaussian direct quadrature methods for Volterra integral equations with periodic solution, Mathematics and Computers in Simulation (submitted). [4] D. Conte, B. Paternoster, G. Santomauro, An exponentially fitted quadrature rule over unbounded intervals, AIP Conference Proceedings 1479 (2012) 1173-1176. [5] D. Conte, L. Gr. Ixaru, B. Paternoster, G. Santomauro, Gauss-Laguerre quadrature rule for oscillatory integrands, Computational and Applied Mathematics (submitted).
Exponentially-fitted quadrature methods for evolution problems with periodic solution
CARDONE, Angelamaria;CONTE, Dajana;PATERNOSTER, Beatrice;
2013-01-01
Abstract
The interest for numerical solution of physical and biological problems with oscillating and/or periodic behaviour requires the use of special-purpose methods. Examples include the electromagnetic scattering, the response of nonlinear circuits to a periodic input and the evolution of an age-structuredpopulation. These problems are characterized by either infinite integrals where the integrand function is oscillatory function, or by Volterra integral equations of type with periodic solution. By exploting the Exponential Fitting theory [1, 2, 3, 4, 5], a new class of quadrature rules, that are a generalization of the usual Gauss-Laguerre formulae, for problem (1) and a new direct quadrature (DQ) method for problem (2) are derived, respectively. Two extra problems appear in the context of building the exponentially-fitted (ef ) DQ method. The first one is the construction of a two-nodes ef quadrature rule of Gaussian type, that is a generalization of the usual two-nodes Gauss-Legendre formula, on which the DQ method is based. The second problem is the building of a suitable ef interpolation technique on four points which preserves the order of convergence of the overall method. These works are in collaboration with L. Gr. Ixaru (National Institute of Physics and Nuclear Engineering, Bucharest, Romania), B. Paternoster, A. Cardone and D. Conte (University of Salerno). References [1] Ixaru, L.Gr.; Vanden Berghe, G., Exponential fitting, Kluwer Academic Publishers, Dordrecht (2004). [2] A. Cardone, B. Paternoster, G. Santomauro, Exponential fitting quadrature rule for functional equations, AIP Conference Proceedings 1479 (2012) 1169-1172. [3] A. Cardone, L. Gr. Ixaru, B. Paternoster, G. Santomauro, Ef-Gaussian direct quadrature methods for Volterra integral equations with periodic solution, Mathematics and Computers in Simulation (submitted). [4] D. Conte, B. Paternoster, G. Santomauro, An exponentially fitted quadrature rule over unbounded intervals, AIP Conference Proceedings 1479 (2012) 1173-1176. [5] D. Conte, L. Gr. Ixaru, B. Paternoster, G. Santomauro, Gauss-Laguerre quadrature rule for oscillatory integrands, Computational and Applied Mathematics (submitted).I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
