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Article

Numerical Solutions of Fractional Weakly Singular Two-Dimensional Partial Volterra Integral Equations Using Euler Wavelets

by
Seyed Sadegh Gholami
1,
Ali Ebadian
2,
Amirahmad Khajehnasiri
2 and
Kareem T. Elgindy
3,4,*
1
Department of Mathematics Education, Farhangian University, Tehran 14665-889, Iran
2
Department of Mathematics, Faculty of Science, Urmia University, Urmia 57179-44514, Iran
3
Department of Mathematics and Sciences, College of Humanities and Sciences, Ajman University, Ajman P.O. Box 346, United Arab Emirates
4
Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman P.O. Box 346, United Arab Emirates
*
Author to whom correspondence should be addressed.
Mathematics 2025, 13(17), 2718; https://doi.org/10.3390/math13172718 (registering DOI)
Submission received: 14 June 2025 / Revised: 15 August 2025 / Accepted: 19 August 2025 / Published: 23 August 2025
(This article belongs to the Special Issue Fractional Calculus: Advances and Applications)

Abstract

This paper presents an innovative numerical method for solving two-dimensional weakly singular Volterra integral equations, including fractional Volterra integral equations with weak singularities. Solving these equations in higher dimensions and in the presence of fractional and weak singularities is highly challenging. The proposed approach uses Euler wavelets (EWs) within an operational matrix (OM) framework combined with advanced numerical techniques, initially transforming these equations into a linear algebraic system and then solving it efficiently. This method offers very high accuracy, strong computational efficiency, and simplicity of implementation, making it suitable for a wide range of such complex problems, especially those requiring high speed and precision in the presence of intricate features.
Keywords: fractional integral equation; two-dimensional Euler wavelets; fractional derivative; operational matrix fractional integral equation; two-dimensional Euler wavelets; fractional derivative; operational matrix

Share and Cite

MDPI and ACS Style

Gholami, S.S.; Ebadian, A.; Khajehnasiri, A.; Elgindy, K.T. Numerical Solutions of Fractional Weakly Singular Two-Dimensional Partial Volterra Integral Equations Using Euler Wavelets. Mathematics 2025, 13, 2718. https://doi.org/10.3390/math13172718

AMA Style

Gholami SS, Ebadian A, Khajehnasiri A, Elgindy KT. Numerical Solutions of Fractional Weakly Singular Two-Dimensional Partial Volterra Integral Equations Using Euler Wavelets. Mathematics. 2025; 13(17):2718. https://doi.org/10.3390/math13172718

Chicago/Turabian Style

Gholami, Seyed Sadegh, Ali Ebadian, Amirahmad Khajehnasiri, and Kareem T. Elgindy. 2025. "Numerical Solutions of Fractional Weakly Singular Two-Dimensional Partial Volterra Integral Equations Using Euler Wavelets" Mathematics 13, no. 17: 2718. https://doi.org/10.3390/math13172718

APA Style

Gholami, S. S., Ebadian, A., Khajehnasiri, A., & Elgindy, K. T. (2025). Numerical Solutions of Fractional Weakly Singular Two-Dimensional Partial Volterra Integral Equations Using Euler Wavelets. Mathematics, 13(17), 2718. https://doi.org/10.3390/math13172718

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