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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.