Study of numerical treatment of functional first-kind Volterra integral equations

Hassanein Falah; Parviz Darania; Saeed Pishbin; Hassanein Falah; Parviz Darania; Saeed Pishbin

doi:10.3934/math.2024846

AIMS Mathematics

2024, Volume 9, Issue 7: 17414-17429. doi: 10.3934/math.2024846

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Research article

Study of numerical treatment of functional first-kind Volterra integral equations

Department of Mathematics, Faculty of Sciences, Urmia University, P. O. Box 165, Urmia, Iran

Received: 26 March 2024 Revised: 27 April 2024 Accepted: 30 April 2024 Published: 21 May 2024
MSC : 45L05, 65R20

First-kind Volterra integral equations have ill-posed nature in comparison to the second-kind of these equations such that a measure of ill-posedness can be described by $ \nu $-smoothing of the integral operator. A comprehensive study of the convergence and super-convergence properties of the piecewise polynomial collocation method for the second-kind Volterra integral equations (VIEs) with constant delay has been given in ^[1]. However, convergence analysis of the collocation method for first-kind delay VIEs appears to be a research problem. Here, we investigated the convergence of the collocation solution as a research problem for a first-kind VIE with constant delay. Three test problems have been fairly well-studied for the sake of verifying theoretical achievements in practice.
- piecewise polynomial numerical method,
- delay integral equation,
- convergence analysis
Citation: Hassanein Falah, Parviz Darania, Saeed Pishbin. Study of numerical treatment of functional first-kind Volterra integral equations[J]. AIMS Mathematics, 2024, 9(7): 17414-17429. doi: 10.3934/math.2024846

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Abstract

First-kind Volterra integral equations have ill-posed nature in comparison to the second-kind of these equations such that a measure of ill-posedness can be described by $ \nu $-smoothing of the integral operator. A comprehensive study of the convergence and super-convergence properties of the piecewise polynomial collocation method for the second-kind Volterra integral equations (VIEs) with constant delay has been given in ^[1]. However, convergence analysis of the collocation method for first-kind delay VIEs appears to be a research problem. Here, we investigated the convergence of the collocation solution as a research problem for a first-kind VIE with constant delay. Three test problems have been fairly well-studied for the sake of verifying theoretical achievements in practice.

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