In this article, we consider the Galerkin and iterated Galerkin methods for solving the second kind weakly singular Volterra integral equation with mixed-type kernels, using piecewise polynomial basis functions based on graded mesh. The results of the study prove that the iterated Galerkin method superiors over the Galerkin method and we show that the Galerkin method converges with the order <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2831_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">$ \mathcal {O}(n^{-m})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow> <mo>-</mo> <mi>m</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, while the iterated Galerkin method converges with the order <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2831_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="64" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {O}(n^{-2m})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow> <mo>-</mo> <mn>2</mn> <mi>m</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> in infinity-norm, where n represents the number of partition of intervals and m corresponds to the highest degree of polynomials used in the approximation. Moreover, numerical examples are provided to validate the theoretical results.

Galerkin and Iterated Galerkin Methods for Linear Second Kind Weakly Singular Volterra Integral Equation with Mixed-Type Kernels

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发表日期：

2025年3月25日

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Krishna Murari Malav,
Kapil Kant,
Joydip Dhar

摘要

In this article, we consider the Galerkin and iterated Galerkin methods for solving the second kind weakly singular Volterra integral equation with mixed-type kernels, using piecewise polynomial basis functions based on graded mesh. The results of the study prove that the iterated Galerkin method superiors over the Galerkin method and we show that the Galerkin method converges with the order $ \mathcal {O}(n^{-m})$ $O (n^{- m})$ , while the iterated Galerkin method converges with the order $\mathcal {O}(n^{-2m})$ $O (n^{- 2 m})$ in infinity-norm, where n represents the number of partition of intervals and m corresponds to the highest degree of polynomials used in the approximation. Moreover, numerical examples are provided to validate the theoretical results.