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<p>We propose a new spline collocation-based approach for solving Volterra integral equations of the first and second kinds. The solution is represented as a linear combination of quadratic minimal splines of maximal smoothness, with the coefficients determined through specialized local approximation schemes (quasi-interpolation). Numerical experiments demonstrate that the use of non-polynomial splines leads to higher accuracy of the approximate solution compared to previously proposed approaches based on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\varvec{B}\)</EquationSource> </InlineEquation>-splines, including approaches defined on nonuniform grids.</p>

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Spline collocation for Volterra integral equations with improved accuracy

  • 期刊论文

  • 发表日期：

    2026年5月15日

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  • Anton Makarov,
  • Egor Kulikov

摘要

We propose a new spline collocation-based approach for solving Volterra integral equations of the first and second kinds. The solution is represented as a linear combination of quadratic minimal splines of maximal smoothness, with the coefficients determined through specialized local approximation schemes (quasi-interpolation). Numerical experiments demonstrate that the use of non-polynomial splines leads to higher accuracy of the approximate solution compared to previously proposed approaches based on \(\varvec{B}\) -splines, including approaches defined on nonuniform grids.