A Numerical Solution for 2D-Nonlinear Fredholm Integral Equations Based on Hybrid Functions Basis

Subject Areas : International Journal of Mathematical Modelling & Computations

Maryam Mohammadi ¹ , A. Zakeri ^{2 *} , Majid Karami ³ , Narges Taheri ⁴ , Raheleh Nouraei ⁵

1 - Department of Mathematics, Islamic Azad University, South Tehran Branch, Tehran, Iran
2 - Faculty of Mathematics, K. N. Toosi University of Technology, Tehran, Iran
3 - Department of Mathematics, Islamic Azad University, South Tehran Branch, Tehran, Iran
4 - Department of Mathematics, Islamic Azad University, South Tehran Branch, Tehran, Iran
5 - Department of Mathematics, Islamic Azad University, South Tehran Branch, Tehran, Iran

Received: 2022-06-08 Accepted : 2023-04-02 Published : 2023-03-01

Keywords: collocation method, Fredholm integral equations, Convergence analysis, Bivariate hybrid block-pulse functions, Normalized Bernstein polynomials,

Abstract :

This work considers a numerical method based on the 2D-hybrid block-pulse functions and normalized Bernstein polynomials to solve 2D-nonlinear Fredholm integral equations of the second type. These problems are reduced to a system of nonlinear algebraic equations and solved by Newton's iterative method along with the numerical integration and collocation methods. Also, the convergence theorem for this algorithm is proved. Finally, some numerical examples are given to show the effectiveness and simplicity of the proposed method.

References:

[1] Atkinson, K.: The Numerical Solutions of Integral Equations of the Second Kind, Cambridge University Press, (1997)
[2] Atkinson, K., Potra, F. A.: Projection and iterated projection methods for nonlinear integral equations, SIAM J. Numer. Anal. 24, 1352-1373, (1987)
[3] Brunner, H.: Collocation Methods for Volterra Integral and Related Functional Differential Equations.
Cambridge University Press, Cambridge, (2004)
[4] Baker, C.: The Numerical Treatment of Integral Equations. Clarendon Press, Oxford, (1978)
[5] Wazwaz, A.M.: Linear and Nonlinear Integral Equations, Vol. 639. Springer, Berlin, (2011)
[6] Brunner, H.: Collocation methods for Volterra integral and related functional differential equations.
Cambridge University Press, Cambridge, (2004)
[7] Linz, P.: Analytical and numerical methods for Volterra equations. SIAM, Philadelphia, (1985)
[8] Hamoud, A., Mohammed, N. M., Ghadle, K. P.: A study of some effective techniques for solving
Volterra-Fredholm integral equations. Mathematical Analysis Vol. 26, 389-406 (2019)
[9] Jafari Behbahani, Z., Roodaki, M.: Two-dimensional Chebyshev hybrid functions and their applications to integral equations. J. basic. Appl. Vol. 4, 134-141, (2015)
https://doi.org/10.1016/j.bjbas.2015.05.005
[10] Stenger, F.: Numerical Methods Based on Sinc and Analytic Functions. Springer, New York, (1993)
[11] Stenger, F.: Summary of Sinc numerical methods. J. Comput. Appl. Math. 121, 379-420, (2000)
[12] Sugihara, M., Matsua, T.: Recent developments of the Sinc numerical methods. J. Comput. Appl.
Math. 165, 673-689, (2004)
[13] Yousefi, S., Razzaghi, M.: Legendre wavelets method for the nonlinear Volterra-Fredholm integral
equations. Math. Comput. Simulation. Vol. 70, 1-8 (2005)
https://doi.org/10.1016/j.matcom.2005.02.035
[14] Amiri, S., Hajipour, M., Baleanu, D.: A spectral collocation method with piecewise trigonometric
basis functions for nonlinear Volterra-Fredholm integral equations. Math. Comput. Appl. Vol. 370,
(2020)
https://doi.org/10.1016/j.amc.2019.124915
[15] Bhrawy, A. H., Tohidi, E., Soleymani, F.: A new Bernoulli matrix method for solving high-order linear
and nonlinear Fredholm integro-differential equations with piecewise intervals. Appl. Math. Comput.
Vol. 219, 482-497 (2012)
https://doi.org/10.1016/j.amc.2012.06.020
M. Mohammad et al./ IJM2C, 13 - 01 (2023) 0-14. 11
[16] Bazm, S.: Bernoulli polynomials for the numerical solution of some classes of linear and nonlinear
integral equations. Math. Comput. Appl. Vol. 275, 44-60 (2015)
https://doi.org/10.1016/j.cam.2014.07.018
[17] Mirzaee, F., Samadyar, N.: Explicit representation of orthonormal Bernoulli polynomials and its
application for solving Volterra-Fredholm-Hammerstein integral equations. SeMA. J, Vol. 77,81-96
(2019)
https://doi.org/10.1007/s40324-019-00203-z
[18] Mirzaee, F., Alipour, S.: Solving two-dimensional non-linear quadratic integral equations of fractional
order via Operational matrix method. Multidiscipline Modeling in Materials and Structures. Vol. 15,
No. 6, 1136-1151 (2019)
https://doi.org/10.1108/MMMS-10-2018-0168
[19] Mustafa, M., Muhammad, M.: Numerical Solution of Linear Volterra-Fredholm Integro-Differential
Equations Using Lagrange Polynomials. Mathematical Theory and Modeling, Vol. 4, No. 9, 158-166
(2014)
https://www.researchgate.net/publication/321058742
[20] Mustafa, M., Ghanim, I.: Numerical Solution of Linear Volterra-Fredholm Integral Equations Using
Lagrange Polynomials. Mathematical Theory and Modeling, Vol. 4, No. 5, 137-146 (2014)
https://www.researchgate.net/publication/320716709
[21] Hamoud, A., Ghadle, K.: The reliable modified of Adomian decomposition method for solving integrodifferential equations. J. Ch. Math. Society. Vol. 32, No. 4, 409-420 (2019)
http://dx.doi.org/10.14403/jcms.2019.32.4.409
[22] Fariborzi Araghi, M. A., Sadigh Behzadi, Sh.: Solving nonlinear Volterra Fredholm integrodifferential
equations using the modified Adomian decomposition method. Math. Comput. Appl. Vol, 9, No. 4,
321-331 (2009)
https://doi.org/10.2478/cmam-2009-0020
[23] Altrk, A.: The Regularization-Homotopy Method for the two-dimensional Fredholm integral equations
of the first kind. Math. Comput. Appl. (MDPI), Vol. 21, (2016)
https://doi.org/10.3390/mca21020009
[24] Katani, K., Mckee, S.: A hybrid Legendre block-pulse method for mixed Volterra-Fredholm integral
equations. Math. Comput. Appl. Vol. 376, (2020)
https://doi.org/10.1016/j.cam.2020.112867
[25] Maleknejad, K., Basirat, B., Hashemizadeh, E.: Hybrid Legendre polynomials and Block-Pulse functions approach for nonlinear Volterra-Fredholm integro-differential equations. J. Math. Comput. Appl.
Vol. 61, 2821-2828 (2011)
https://doi.org/10.1016/j.camwa.2011.03.055
[26] Hui Hsiao, C.: Hybrid function method for solving Fredholm and Volterra integral equations of the
second kind. J. Comput. Appl. Math. Vol. 230, 59-68 (2009)
https://doi.org/10.1016/j.cam.2008.10.060
[27] Ramadan, M.A., Ali, M.R.: An efficient hybrid method for solving Fredholm integral equations using
triangular functions. NTMSCI 5, No. 1, 213-224 (2017)
http://dx.doi.org/10.20852/ntmsci.2017.140
[28] Maleknejad, K., Shahabi, M.: Application of hybrid functions operational matrices in the numerical
solution of two-dimensional nonlinear integral equations. J. Math. Comput. Appl. Vol. 136, 46-65
(2019)
https://doi.org/10.1016/j.apnum.2018.09.014
[29] Mohammadi, M., Zakeri, A., Karami, M.: An approximate solution of bivariate nonlinear Fredholm
integral equations using hybrid blockpulse functions with Chebyshev polynomials. Math Sci. Vol. 15,
19 (2021)
https://doi.org/10.1007/s40096-020-00336-7
[30] Maleknejad, K., Saeedipoor, E.: Hybrid function method and convergence analysis for two-dimensional
nonlinear integral equations. J. Comput. Appl. Math. 322, 96-108 (2017)
https://doi.org/10.1016/j.cam.2017.03.012
[31] Behiry, S. H.: Solution of nonlinear Fredholm integro-differential equations using a hybrid of blockpulse functions and normalized Bernstein polynomials. J. Comput. Appl. Math. Vol. 260, 258-265
(2014)
https://doi.org/10.1016/j.cam.2013.09.036
[32] Hesameddini, E., Shahbazi, M.: Solving system of VolterraFredholm integral equations with Bernstein
polynomials and hybrid Bernstein Block-Pulse functions. J. Comput. Appl. Math. Vol. 315, 182-194
(2017)
https://doi.org/10.1016/j.cam.2016.11.004
[33] Mirzaee, F., Samadyar, N.: Numerical solution based on two-dimensional orthonormal Bernstein polynomials for solving some classes of two-dimensional nonlinear integral equations of fractional order.
J. Comput. Appl. Math. Volumes. 344-345, 191-203 (2019)
https://doi.org/10.1016/j.amc.2018.10.020
[34] Kenneth, I. J.: Bernstein Polynomials. University of California, Davis, (2000)
[35] Mohamadi, M., Babolian E., Yousefi, S.A.: A Solution For Volterra Integral Equations of the First
Kind Based on Bernstein Polynomials. Int. J. Industrial Mathematics, /IJIM Vol. 10, No. 1, 19-27
(2018)
http://ijim.srbiau.ac.ir/
[36] Maleknejad, K., Hashemizadeh, E., Ezzati, R.: A new approach to the numerical solution of Volterra
integral equations by using Bernsteins approximation. Commun. Nonlinear Sci. Numer. Simul. Vol.
16, 647-655 (2011)
https://doi.org/10.1016/j.cnsns.2010.05.006
[37] Ramadan, M. A., Ali, M. R.: Solution of integral and Integro-Differential equations system using hybrid orthonormal Bernstein and block-pulse functions. J. Abstract and Computational Mathematics.
12 M. Mohammad et al./ IJM2C, 13 - 01 (2023) 0-14.
NTMSCI 2, No. 1, 35-48 (2017)
[38] Mirzaee, F., Samadyar, N.: Convergence of 2D-orthonormal Bernstein collocation method for solving
2D-mixed Volterra-Fredholm integral equations. Transactions of A. Razmadze Mathematical Institute.
Vol. 172, 631-641 (2018)
https://doi.org/10.1016/j.trmi.2017.09.006
[39] Mirzaee, F., Hoseini, S. F.: Hybrid functions of Bernstein polynomials and block-pulse functions for
solving optimal control of the nonlinear Volterra integral equations. Indagationes Mathematicae, Vol.
27, 835-849 (2016)
https://doi.org/10.1016/j.indag.2016.03.002
[40] Kenneth, I. J.: Bernstein polynomials. Visualization and Graphics Research Group Department of
Computer Science University of California, (1996)