This is an early access version, the complete PDF, HTML, and XML versions will be available soon.
Open AccessArticle
Numerical Solutions of Fractional Weakly Singular Two-Dimensional Partial Volterra Integral Equations Using Euler Wavelets
by
Seyed Sadegh Gholami
Seyed Sadegh Gholami 1,
Ali Ebadian
Ali Ebadian 2,
Amirahmad Khajehnasiri
Amirahmad Khajehnasiri 2 and
Kareem T. Elgindy
Kareem T. Elgindy
Dr. Kareem T. Elgindy is an Associate
Professor in the Department of Mathematics and Sciences at He [...]
Dr. Kareem T. Elgindy is an Associate
Professor in the Department of Mathematics and Sciences at Ajman University,
UAE. He holds a Ph.D. in Applied and Computational Mathematics from Monash
University, Australia, where his dissertation was nominated for the prestigious
Mollie Holman Doctoral Medal. Dr. Elgindy's academic journey includes a
master's degree in Scientific Computations from Assiut University, Egypt, and a
bachelor's degree in Mathematics with first-class honors from the same
institution, where he ranked as the top graduate in the Faculty of Science. His
research focuses on numerical analysis, numerical fractional calculus,
fractional optimal control theory, mathematical biology, and nonlinear
programming, with numerous publications in high-impact ISI journals. Dr.
Elgindy has held academic positions at King Fahd University of Petroleum &
Minerals (KFUPM), Xiamen University Malaysia (XMUM), and Assiut University, and
he was granted the title of Visiting Scholar at the California Institute of
Technology (Caltech) under the Fulbright Egyptian Visiting Scholar Award. A
former member of the Australian Mathematical Society (AustMS), Dr. Elgindy
remains actively engaged in the academic community as an editor for the
International Journal of Mathematics and Mathematical Sciences and through his
contributions to teaching, research mentorship, and professional service.
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
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.
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
Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details
here.
Article Metrics
Article Access Statistics
For more information on the journal statistics, click
here.
Multiple requests from the same IP address are counted as one view.