Exponentially-fitted direct quadrature methods for Volterra integral equations with periodic solution

We consider Volterra integral equations with periodic solution of the type: y(x) = f(x) + int_0^x k(x - s)y(s)ds; x in [0; xend] y(x) = fi(x); -inf < x <= 0; where k in L_1(R+), f is continuous and periodic on [0; xend], fi is continuous and bounded on R+. These equations model periodic phenomena with memory, such as the spread of seasonal epidemics and the response of nonlinear circuits to a periodic input. General purpose methods usually require a high computational cost to follows the oscillations of the solution, thus numerical methods specially tuned on the problem should be applied. With this aim, we propose direct quadrature (DQ) methods based on exponential fitting theory [2], which exploit the knowledge on the qualitative behavior of the solution, to improve accuracy without increasing the computational cost. In [1] we introduced a DQ method based on an exponentially-fitted Simpson-like formula. Now we go one step further, introducing a Gaussian DQ method, to increase the order of convergence. Here we illustrate the construction of these methods, analyze convergence and stability and furnish some numerical experiments of comparison with other numerical methods. This is a joint work with Beatrice Paternoster from Università di Salerno [1] Cardone, A.; Ixaru, L. Gr.; Paternoster, B. 2010 Exponential fitting direct quadrature methods for Volterra integral equations. Numer. Algorithms, vol. 55, no. 4, pp. 467-480. [2] Ixaru, L.Gr.; Vanden Berghe, G. 2004 Exponential fitting. Kluwer Academic Publishers, Dordrecht.