A Nyström method for second-kind Volterra integral equations on the square

Luisa Fermo; Domenico Mezzanotte; Donatella Occorsio

doi:10.33205/cma.1960169

A Nyström method for second-kind Volterra integral equations on the square

Abstract

In this paper, we propose a Nyström-type method for the numerical approximation of second-kind bivariate Volterra integral equations on the unit square. The method is based on bivariate Generalized Bernstein (GB) polynomials and on a cubature formula defined on equally spaced nodes, allowing the integral operator to be discretized directly without requiring any change of variables. The use of uniform grids makes the proposed approach particularly suitable for applications in which data are naturally sampled at equidistant points. The approximation framework relies on tensor-product GB polynomials whose degrees may differ with respect to each variable. This flexibility allows different approximation orders to be prescribed according to the regularity of the target function in each variable. As a result, the computational cost can be reduced while preserving high level of accuracy. Compared with classical Bernstein operators, the proposed GB operators achieve higher approximation orders depending on the smoothness of the approximated functions. The resulting Nyström scheme leads to a well-conditioned linear system and is proved to be stable and convergent. Error estimates are derived in Sobolev and Zygmund spaces. Several numerical experiments are presented to validate the theoretical results and to illustrate the effectiveness of the method for kernels and right-hand side functions with different smoothness properties.

Keywords

Supporting Institution

Gruppo Nazionale Calcolo Scientifico-Istituto Nazionale di Alta Matematica (GNCS-INdAM), Fondazione di Sardegna

Thanks

This research has been accomplished within “Research ITalian network on Approximation” (RITA). The authors are members of the Gruppo Nazionale Calcolo Scientifico-Istituto Nazionale di Alta Matematica (GNCS-INdAM), the TAA-UMI Research Group, and the SIMAI Activity Group “Numerical and Analytical Approximation of Data and Functions with Applications” (ANA&A).

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Details

Primary Language

English

Subjects

Numerical Analysis

Journal Section

Research Article

Authors

Luisa Fermo
0000-0002-4398-8990
Italy

Domenico Mezzanotte ^*
0000-0001-5154-6538
Italy

Donatella Occorsio
0000-0001-9446-4452
Italy

Early Pub Date

July 28, 2026

Publication Date

-

Submission Date

May 29, 2026

Acceptance Date

July 24, 2026

Published in Issue

Year 2026 Number: Advanced Online Publication

DOI

https://doi.org/10.33205/cma.1960169

IZ

https://izlik.org/JA59LB59TJ

Cite

RIS / Bibtex

APA

Fermo, L., Mezzanotte, D., & Occorsio, D. (2026). A Nyström method for second-kind Volterra integral equations on the square. Constructive Mathematical Analysis, Advanced Online Publication. https://doi.org/10.33205/cma.1960169

AMA

1.Fermo L, Mezzanotte D, Occorsio D. A Nyström method for second-kind Volterra integral equations on the square. CMA. 2026;(Advanced Online Publication). doi:10.33205/cma.1960169

Chicago

Fermo, Luisa, Domenico Mezzanotte, and Donatella Occorsio. 2026. “A Nyström Method for Second-Kind Volterra Integral Equations on the Square”. Constructive Mathematical Analysis, no. Advanced Online Publication. https://doi.org/10.33205/cma.1960169.

EndNote

Fermo L, Mezzanotte D, Occorsio D (July 1, 2026) A Nyström method for second-kind Volterra integral equations on the square. Constructive Mathematical Analysis Advanced Online Publication

IEEE

[1]L. Fermo, D. Mezzanotte, and D. Occorsio, “A Nyström method for second-kind Volterra integral equations on the square”, CMA, no. Advanced Online Publication, July 2026, doi: 10.33205/cma.1960169.

ISNAD

Fermo, Luisa - Mezzanotte, Domenico - Occorsio, Donatella. “A Nyström Method for Second-Kind Volterra Integral Equations on the Square”. Constructive Mathematical Analysis. Advanced Online Publication (July 1, 2026). https://doi.org/10.33205/cma.1960169.

JAMA

1.Fermo L, Mezzanotte D, Occorsio D. A Nyström method for second-kind Volterra integral equations on the square. CMA. 2026. doi:10.33205/cma.1960169.

MLA

Fermo, Luisa, et al. “A Nyström Method for Second-Kind Volterra Integral Equations on the Square”. Constructive Mathematical Analysis, no. Advanced Online Publication, July 2026, doi:10.33205/cma.1960169.

Vancouver

1.Luisa Fermo, Domenico Mezzanotte, Donatella Occorsio. A Nyström method for second-kind Volterra integral equations on the square. CMA. 2026 Jul. 1;(Advanced Online Publication). doi:10.33205/cma.1960169