10.57647/ijm2c.2026.1602.08

Muntz Function Collocation Method for 2D Fredholm Integral Equations of the Second Kind

PDF
  • Jabbar Mahdy Hadaad1,
  • Masoud Allame*1,
  • Habeeb Abed Kadhim Aal-Rkhais2,
  • Majid Tavassoli Kajani1,
  1. Department of Mathematics, Isf. C., Islamic Azad University, Isfahan, Iran
  2. Department of Mathematics, College of Computer and Mathematics, Uinversity of Thi-Qar, Iraq

Received: 01-10-2025

Revised: 25-10-2025

Accepted: 01-11-2025

Published in Issue 30-06-2026

Published Online: 12-11-2025

Copyright (c) 2025 Jabbar Mahdy Hadaad, Masoud Allame, Habeeb Abed Kadhim Aal-Rkhais, Majid Tavassili Kajani (Author)

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite

Mahdy Hadaad, J., Allame, M., Abed Kadhim Aal-Rkhais, H., & Tavassoli Kajani, M. (2026). Muntz Function Collocation Method for 2D Fredholm Integral Equations of the Second Kind. International Journal of Mathematical Modelling & Computations, 16(2). https://doi.org/10.57647/ijm2c.2026.1602.08
Crossref
Scopus
Google Scholar
Europe PMC

Abstract

This paper presents a novel numerical approach for solving two-dimensional Fredholm integral equations of the second kind (2DFIEs) using M¨untz functions and a collocation method. In the proposed approach, both the unknown function and the integral kernel are approximated by M¨untztype orthogonal functions, and the integral equation is transformed into a system of linear equations using two-dimensional collocation points; the approximate solution is then obtained by solving this linear system. The approximation of bivariate functions is analyzed in the Sobolev space. Several numerical examples are provided to demonstrate the accuracy and efficiency of the method. The numerical results confirm that the two-dimensional M¨untz collocation method offers a flexible and effective tool for solving complex 2DFIEs, with potential applications in scientific and engineering problems involving integral equations. 

Keywords

  • 2D Fredholm integral equation,
  • Collocation Method,
  • Muntz Function,
  • Approximation
PDF

References

  1. R. Cavoretto, A. De Rossi, A.L. Laguardia, D. Mezzanotte, D. Occorsio, M.G. Russo, An RBF-based Nystrom method for Second-Kind Fredholm Integral Equations, J. Comput. Appl. Math. 474 (2026) 116968.
  2. K.E. Atkinson, D.D. Chien, J. Seol, Numerical analysis of the radiosity equation using the collocation method, Electron. Trans. Numer. Anal. 11 (2000) 94-120.
  3. S. Hatamzadeh-Varmazyar, Z. Masouri, A fast numerical method for analysis of one and twodimensional electromagnetic scattering using a set of cardinal functions, Eng. Anal. Bound. Elem. 36 (11) (2012) 1631-1639.
  4. K.E. Atkinson, The numerical solution of integral equations of the second kind, Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, 1997.
  5. A.M. Wazwaz, Linear and Nonlinear Integral Equations: Methods and Applications, first ed., Springer Publishing Company, Incorporated, 2011.
  6. F. Liang, F. Lin, A fast numerical solution method for two dimensional Fredholm integral equations of the second kind based on piecewise polynomial interpolation, Appl. Math. Comput. 216 (10) (2010) 3073-3088.
  7. D. Occorsio, M.G. Russo, Numerical methods for Fredholm integral equations on the square, Appl. Math. Comput. 218 (5) (2011) 2318-2333.
  8. D. Occorsio, M.G. Russo, Bivariate generalized Bernstein operators and their application to Fredholm integral equations, Publ. Inst. Math. 100 (114) (2016) 141-162.
  9. A.L. Laguardia, M.G. Russo, Numerical methods for Fredholm integral equations based on Padua points, Dolomites Res. Notes Approx. 15 (5) (2022) 39-50.
  10. K. Maleknejad, M.T. Kajani, Solving second kind integral equations by Galerkin methods with hybrid Legendre and Block-Pulse functions, Appl. Math. Comput. 145 (2-3) (2003) 623-629.
  11. M.T. Kajani, A.H. Vencheh, Solving second kind integral equations with Hybrid Chebyshev and Block-Pulse functions, Appl. Math. Comput. 163 (1) (2005) 71-77.
  12. B. Asady, M.T. Kajani, A.H. Vencheh, A. Heydari, Solving second kind integral equations with hybrid Fourier and block-pulse functions, Appl. Math. Comput. 160 (2) (2005) 517-522.
  13. M. Ghasemi, Numerical solution of linear Fredholm integral equations using sine-cosine wavelets, J. Comput. Appl. Math. 204 (2) (2007) 423-432.
  14. H. Fatahi, J. Saberi-Nadjafi, E. Shivanian, A new spectral meshless radial point interpolation (SMRPI) method for the two-dimensional Fredholm integral equations on general domains with error analysis, J. Comput. Appl. Math. 294 (2016) 196-209.
  15. H. Bin Jebreen, A novel and efficient numerical algorithm for solving 2D Fredholm integral equations, J. Math. 2020 (9) 6662721.
  16. F. Fattahzadeh, E.G. Raboky, Multivariable and scattered data interpolation for solving multivariable integral equations, J. Prime Res. Math. 8 (2012) 51-60.
  17. R. Cavoretto, A. De Rossi, D. Mezzanotte, A review of radial kernel methods for the resolution of Fredholm integral equations of the second kind, Constr. Math. Anal. 7 (2024) 142-153.
  18. Afkar Kareem Mnahi, Majid Tavassoli Kajani, Mohammed Jasim Mohammed, Masoud Allame, Numerical Solution of the Spread Model of COVID-19 by Using Muntz Functions, International Journal of Mathematical Modelling & Computations, 14 (3) (2024), 231-240.
  19. G. V. Milovanovic, Muntz orthogonal polynomials and their numerical evaluation, Applications and Computation of Orthogonal Polynomials, Springer, Oberwolfach Germany, 1998, 179-194.
  20. S. Akhlaghi, M. Tavassoli Kajani, M. Allame, Numerical Solution of Fractional Order IntegroDifferential Equations via Muntz Orthogonal Functions, Journal of Mathematics 2023 (2023) 6647128, 1-13.

Most read articles by the same author(s)