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<p>The paper presents a Nyström-type method to approximate the solution of second-kind Volterra–Fredholm integral equations. Two forms are considered, that is the disjoint form, in which the Volterra and Fredholm operators are additive integrals; and the mixed one, in which the two integrals appear in a single term through composition. In both situations, the right-hand side and the kernel functions may have algebraic singularities at <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10543_2025_1061_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">\(\pm 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>±</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and hence equations are treated in suitable weighted spaces equipped with the uniform norm. The proposed methods, based on product and Gauss rules, are stable and convergent. The error is of the order of the best polynomial approximation of the given functions. Numerical examples are presented to illustrate the accuracy of the method.</p>

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A global method for solving second-kind Volterra–Fredholm integral equations

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年4月4日

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  • Luisa Fermo,
  • Domenico Mezzanotte,
  • Donatella Occorsio

摘要

The paper presents a Nyström-type method to approximate the solution of second-kind Volterra–Fredholm integral equations. Two forms are considered, that is the disjoint form, in which the Volterra and Fredholm operators are additive integrals; and the mixed one, in which the two integrals appear in a single term through composition. In both situations, the right-hand side and the kernel functions may have algebraic singularities at \(\pm 1\) ± 1 and hence equations are treated in suitable weighted spaces equipped with the uniform norm. The proposed methods, based on product and Gauss rules, are stable and convergent. The error is of the order of the best polynomial approximation of the given functions. Numerical examples are presented to illustrate the accuracy of the method.