Numerical Solution of Integral Algebraic Equations with Singular Points Using the Least Squares Method

Vo, Van Truong; Sidorov, Denis; Chistyakova, Elena; Chistyakov, Viktor; Dreglea, Aliona

doi:10.3390/math14040693

Open AccessArticle

Numerical Solution of Integral Algebraic Equations with Singular Points Using the Least Squares Method

by

Van Truong Vo

¹

,

Denis Sidorov

^1,2,3

,

Elena Chistyakova

^1,4

,

Viktor Chistyakov

⁴ and

Aliona Dreglea

^1,3,*

¹

Scientific Research Department, Irkutsk National Research Technical University, Irkutsk 664074, Russia

²

Sino-Russian Joint Research Center for Advanced Energy and Power Systems, Energy Systems Institute, Siberian Branch, Russian Academy of Sciences, Irkutsk 664033, Russia

³

School of Electrical Engineering and Automation, Harbin Institute of Technology, Harbin 150001, China

⁴

Institute of System Dynamics and Control Theory, Siberian Branch, Russian Academy of Sciences, Irkutsk 664033, Russia

^*

Author to whom correspondence should be addressed.

Mathematics 2026, 14(4), 693; https://doi.org/10.3390/math14040693

Submission received: 2 January 2026 / Revised: 3 February 2026 / Accepted: 13 February 2026 / Published: 16 February 2026

(This article belongs to the Section C2: Dynamical Systems)

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Abstract

We conduct a numerical study of integral algebraic equations (IAEs) with singular points, which pose significant challenges for standard computational methods. The presence of singular points often renders classical discretization schemes unstable and inaccurate. This work explores the reformulation of such problems using a least squares framework to restore numerical stability. By recasting the singular IAE as a minimization problem, the least squares method effectively handles the non-integrability and ill-conditioning inherent in direct approaches. We provide a numerical analysis of the proposed scheme and present results from several test cases, demonstrating its superior performance in terms of the convergence rate and solution quality compared to conventional methods. Our findings establish the least squares method as a viable and effective tool for solving singular IAEs.

Keywords: integral algebraic equations; index; least squares method; singular points; ill-posed problem

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MDPI and ACS Style

Vo, V.T.; Sidorov, D.; Chistyakova, E.; Chistyakov, V.; Dreglea, A. Numerical Solution of Integral Algebraic Equations with Singular Points Using the Least Squares Method. Mathematics 2026, 14, 693. https://doi.org/10.3390/math14040693

AMA Style

Vo VT, Sidorov D, Chistyakova E, Chistyakov V, Dreglea A. Numerical Solution of Integral Algebraic Equations with Singular Points Using the Least Squares Method. Mathematics. 2026; 14(4):693. https://doi.org/10.3390/math14040693

Chicago/Turabian Style

Vo, Van Truong, Denis Sidorov, Elena Chistyakova, Viktor Chistyakov, and Aliona Dreglea. 2026. "Numerical Solution of Integral Algebraic Equations with Singular Points Using the Least Squares Method" Mathematics 14, no. 4: 693. https://doi.org/10.3390/math14040693

APA Style

Vo, V. T., Sidorov, D., Chistyakova, E., Chistyakov, V., & Dreglea, A. (2026). Numerical Solution of Integral Algebraic Equations with Singular Points Using the Least Squares Method. Mathematics, 14(4), 693. https://doi.org/10.3390/math14040693

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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Numerical Solution of Integral Algebraic Equations with Singular Points Using the Least Squares Method

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