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A Numerical Method for Proportional Delay Volterra Integral Equations

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Abstract

In this paper we propose a numerical method for nonlinear second kind Volterra integral equations (VIEs) with (vanishing) proportional delays qt (\(0<q<1\)). We shall present the existence and uniqueness of analytic solution for these type equations and then analyze the convergence and order of convergence of the proposed numerical method. The numerical method is based on the Romberg quadrature rule and will be shown that the order of the convergence is \(O(N^{-5})\), where N is number of the nodes in the time discretization. The theoretical results are then verified by numerical examples, also they are compared with some other numerical methods.

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  1. Faculty of Sciences, Yasouj University, Yasouj, Iran

    R. Katani

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  1. R. Katani
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Correspondence to R. Katani.

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Appendix A

Appendix A

Theorem

Let x(t) be a continuous nonnegative function such that

$$\begin{aligned}x (t) \le a + \int ^t_{t_0} [b + cx (s)] ds,\ \ \ \ t \ge t_0,\end{aligned}$$

where \(a\ge 0\), \(b\ge 0\), \(c> 0\). Then for \(t\ge t_0,\) x(t) satisfies

$$\begin{aligned}x (t)\le \frac{ b}{ c} (\exp (c (t -t_0)) - 1) + a \exp (c (t - t_0)).\end{aligned}$$

Proof

see [15] (p. 15). \(\square \)

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Katani, R. A Numerical Method for Proportional Delay Volterra Integral Equations. Int. J. Appl. Comput. Math 7, 170 (2021). https://doi.org/10.1007/s40819-021-01106-2

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