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Re-iterated approximation methods for nonlinear Volterra integral equations.

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Title: Re-iterated approximation methods for nonlinear Volterra integral equations.
Authors: Chakraborty, Samiran1 samiran2day@gmail.com, Kant, Kapil2, Nigam, Ritu3, Nelakanti, Gnaneshwar3
Source: Mathematical Modelling & Analysis. 2026, Vol. 31 Issue 2, p214-227. 14p.
Subjects: Volterra equations, Galerkin methods, Newton-Raphson method, Smoothness of functions, Numerical analysis, Integral equations
Abstract: In this article, the Newton-iteration scheme based upon iterated Galerkin operator is applied for solving non-linear Volterra Urysohn integral equations of the second kind for smooth and weakly singular kernels. A one step of improvement by iteration to the Galerkin method, named as iterated Galerkin method is a well discussed method and it gives improved convergence rates than Galerkin method. But if we iterate them one more time, then there is no guarantee that we get any improved convergence rates. The proposed Newton-iteration scheme based upon iterated Galerkin operator ensures improved convergence rates at every step of iteration. Specifically, we establish that the convergence rate in iterated Galerkin method increases by O(hr) for smooth kernel, and O(h1-α) for weakly singular kernel, in each step of reiteration, where h is the norm of the partition. Numerical examples are provided to justify the reliability and efficiency of the proposed technique. [ABSTRACT FROM AUTHOR]
Copyright of Mathematical Modelling & Analysis is the property of Vilnius Gediminas Technical University and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: <searchLink fieldCode="JN" term="%22Mathematical+Modelling+%26+Analysis%22">Mathematical Modelling & Analysis</searchLink>. 2026, Vol. 31 Issue 2, p214-227. 14p.
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  Data: <searchLink fieldCode="DE" term="%22Volterra+equations%22">Volterra equations</searchLink><br /><searchLink fieldCode="DE" term="%22Galerkin+methods%22">Galerkin methods</searchLink><br /><searchLink fieldCode="DE" term="%22Newton-Raphson+method%22">Newton-Raphson method</searchLink><br /><searchLink fieldCode="DE" term="%22Smoothness+of+functions%22">Smoothness of functions</searchLink><br /><searchLink fieldCode="DE" term="%22Numerical+analysis%22">Numerical analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Integral+equations%22">Integral equations</searchLink>
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  Data: In this article, the Newton-iteration scheme based upon iterated Galerkin operator is applied for solving non-linear Volterra Urysohn integral equations of the second kind for smooth and weakly singular kernels. A one step of improvement by iteration to the Galerkin method, named as iterated Galerkin method is a well discussed method and it gives improved convergence rates than Galerkin method. But if we iterate them one more time, then there is no guarantee that we get any improved convergence rates. The proposed Newton-iteration scheme based upon iterated Galerkin operator ensures improved convergence rates at every step of iteration. Specifically, we establish that the convergence rate in iterated Galerkin method increases by O(hr) for smooth kernel, and O(h1-α) for weakly singular kernel, in each step of reiteration, where h is the norm of the partition. Numerical examples are provided to justify the reliability and efficiency of the proposed technique. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of Mathematical Modelling & Analysis is the property of Vilnius Gediminas Technical University and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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      – Type: doi
        Value: 10.3846/mma.2026.22594
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      – Code: eng
        Text: English
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        PageCount: 14
        StartPage: 214
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      – SubjectFull: Volterra equations
        Type: general
      – SubjectFull: Galerkin methods
        Type: general
      – SubjectFull: Newton-Raphson method
        Type: general
      – SubjectFull: Smoothness of functions
        Type: general
      – SubjectFull: Numerical analysis
        Type: general
      – SubjectFull: Integral equations
        Type: general
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      – TitleFull: Re-iterated approximation methods for nonlinear Volterra integral equations.
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            NameFull: Kant, Kapil
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            NameFull: Nigam, Ritu
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            NameFull: Nelakanti, Gnaneshwar
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            – D: 01
              M: 04
              Text: 2026
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              Y: 2026
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            – TitleFull: Mathematical Modelling & Analysis
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