SciCADE 2013
International Conference on Scientific Computation and Differential Equations
September 16-20, 2013, Valladolid (Spain)

Invited Talk

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Sinc-collocation methods for Volterra integro-differential equations

T. Okayama

Abstract
In this talk, numerical schemes by means of Sinc methods for Volterra integro-differential equations $ u'(t)&=g(t)+\mu(t)u(t)+\mathcal{V}[u](t), a\leq t\leq b, u(a)&=u_a $ are considered. Here, $g$ and $\mu$ are known functions, $u$ is the solution to be determined, and $\mathcal{V}$ is the Volterra integral operator defined by $\mathcal{V}[f](t)=\int_a^tk(t,r)f(r)\mathrm{d}r$, where $k(t,r)$ is a known function. As a related study, Zarebnia [1] reduced the given equations to Volterra integral equations, and applied the Sinc-Nyström method developed by Muhammad et al. [2]. However, the computational cost of the method is relatively high, because the approximate solution includes double summation. Instead, a Sinc-collocation method is developed in this study, where only single summation is included. This study further improves the method by replacing the standard variable transformation with the trending one [3].

Bibliography
[1] M. Zarebnia, Sinc numerical solution for the Volterra integro-differential equation, Commun. Nonlinear Sci. Numer. Simul., 15 (2010), pp. 700-706.
[2] M. Muhammad, A. Nurmuhammad, M. Mori and M. Sugihara, Numerical solution of integral equations by means of the Sinc collocation method based on the double exponential transformation, J. Comput. Appl. Math., 177 (2005), pp. 269-286.
[3] M. Sugihara and T. Matsuo, Recent developments of the Sinc numerical methods, J. Comput. Appl. Math., 164/165 (2004), pp. 673-689.

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