International Journal of Nonlinear Analysis and Applications
A new method for solving three-dimensional nonlinear Fredholm integral equations by Haar wavelet
Document Type : Research Paper
Authors
1 Department of Mathematics, Ashtian Branch, Islamic Azad University, Ashtian, Iran
2 Department of Mathematics, Farhangian University, Tehran, Iran. Member of Young Researchers and Elite club Shahr-e-Qods, Branch Islamic Azad University, Tehran, Iran
3 Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran
Abstract
In this paper, a new iterative method of successive approximations based on Haar wavelets is proposed for solving three-dimensional nonlinear Fredholm integral equations. The convergence of the method is verified. The error estimation and numerical stability of the proposed method are provided in terms of Lipschitz condition. Conducting numerical experiments confirm the theoretical results of the proposed method and endorse the accuracy of the method.
Keywords
[1] M. A. Abdelkawy, E. A. Doha, A. H. Bhrawy, and A. Z. M. Amin, Efficient pseudospectral scheme for 3D integral equations, Roman. Acad. Ser. A. Math. Phys. Tech. Sci. Inf. Sci. 18(3) (2017) 199–206.
[2] M. Abdou, A. Badr and M. Soliman, On a method for solving a two-dimensional nonlinear integral equation of the second kind, J. Comput. Appl. Math. 235(12) (2011) 3589–3598.
[3] R. P. Agarwal, N. Hussain, and M. A. Taoudi, Fixed point theorems in ordered banach spaces and applications to nonlinear integral equations, Abst. Appl. Anal. 2012 (2012).
[4] M. Almousa, The solutions of three dimensional Fredholm integral equations using Adomian decomposition method, AIP Conference Proceedings, 020053, (2016) 020053-1-020053-2.
[5] B. Asady, F. Hakimzadegan, and R. Nazarlue, Utilizing artificial neural network approach for solving twodimensional integral equations, Math. Sci. 8(1) (2014) 1–9.
[6] M. Asgari and R. Ezzati, Using operational matrix of two-dimensional bernstein polynomials for solving twodimensional integral equations of fractional order, Appl. Math. Comput. 307 (2017) 290–298.
[7] K. Atkinson and E. Kendall, The Numerical Solution of Integral Equations of the Second Kind, Cambridge: Cambridge University Press, 2011.
[8] K. Atkinson and F. Potra, Projection and iterated projection methods for nonlinear integral equations, SIAM J. Numerical Anal. 24 (1987) 1352–1373.
[9] I. Aziz, F. Khan, et al, A new method based on haar wavelet for the numerical solution of two-dimensional nonlinear integral equations, J. Comput. Appl. Math. 272 (2014) 70–80.
[10] I. Aziz, S. Islam and W. Khan, Quadrature rules for numerical integration based on Haar wavelets and hybrid functions, Comput. Math. Appl. 61(9) (2011) 2770–2781.
[11] M. Bakhshi, M. Asghari-Larimi and M. Asghari-Larimi, Three dimensional differential transform method for solving nonlinear threedimensional Volterra integral equations, J. Math. Comput. Sci. 4(2) (2012) 246–256.
[12] M. Basseem, Degenerate kernel method for three dimension nonlinear integral equations of the second kind, Universal J. Integ. Eq. 3 (2015) 61–66.
[13] A. M. Bica , M. Curila and S. Curila, About a numerical method of successive interpolations for functional Hammerstein integral equations, J. Comput. Appl. Math. 236(2) (2012) 2005–2024.
[14] A. Boggess and F. J. Narcowich, First Course in Wavelets with Fourier Analyis, Prentice Hall, 2001.
[15] A. H. Borzabadi and O. S. Fard, A numerical scheme for a class of nonlinear fredholm integral equations of the second kind, J. Comput. Appl. Math. 232(2) (2009) 449–454.
[16] H. Brunner, Collocation Methods for Volterra Integral and Related Functional Differential Equations, Cambridge University Press, 2004.
[17] Z. Cheng, Quantum effects of thermal radiation in a Kerr nonlinear blackbody, J. Optical Soc. Amer. B 19 (2002) 1692–1705.
[18] W. C. Chew, M. S. Tong and B. Hu, Integral Equation Methods for Electromagnetic and Elastic Waves, Synthesis Lectures on Comput. Elect. 3(1) (2008) 1–241.
[19] M. Erfanian and H. Zeidabadi, Solving two-dimensional nonlinear mixed Volterra Fredholm integral equations by using rationalized Haar functions in the complex plane, J. Math. Mod. 7(4) (2019) 399–416.
[20] M. Esmaeilbeigi, F. Mirzaee and D. Moazami, Radial basis functions method for solving three-dimensional linear Fredholm integral equations on the cubic domains, Iran. J. Numerical Anal. Opt. 7(2) (2017) 15–37.
[21] S. Fazeli, G. Hojjati and H. Kheiri, A piecewise approximation for linear two-dimensional volterra integral equation by chebyshev polynomials, Int. J. Nonlinear Sci. 16(3) (2013) 255–261.
[22] A. Haar. Zur Theories der orthogonalen Funktionensystem, Math. Annal. 69 (1910) 331–371.
[23] G. Han and R. Wang, Richardson extrapolation of iterated discrete galerkin solution for two-dimensional fredholm integral equations, J. Comput. Appl. Math. 139(1) (2002) 49–63.
[24] G. Hursan and M. S. Zhdanov, Contraction integral equation method in three-dimensional electromagnetic modeling, Radio Sci. 6(37) (2002) 1–13.
[25] S. Islam, I. Aziz and F. Haq, A comparative study of numerical integration based on Haar wavelets and hybrid functions, Comput. Math. Appl. 59(6) (2010) 2026–2036.
[26] M. Kazemi and R. Ezzati, Numerical solution of two-dimensional nonlinear integral equations via quadrature rules and iterative method, Adv. Diff. Eq. Control Proc. 17 (2016) 27–56.
[27] M. Kazemi and R. Ezzati, Existence of solutions for some nonlinear Volterra integral equations via Petryshyn’s fixed point theorem, Int. J. Nonlinear Anal. Appl. 9 (2018) 1–12.
[28] M. Kazemi and R. Ezzati, Existence of solution for some nonlinear two-dimensional volterra integral equations via measures of noncompactness, Appl. Math. Comput. 275 (2016) 165–171.
[2] M. Abdou, A. Badr and M. Soliman, On a method for solving a two-dimensional nonlinear integral equation of the second kind, J. Comput. Appl. Math. 235(12) (2011) 3589–3598.
[3] R. P. Agarwal, N. Hussain, and M. A. Taoudi, Fixed point theorems in ordered banach spaces and applications to nonlinear integral equations, Abst. Appl. Anal. 2012 (2012).
[4] M. Almousa, The solutions of three dimensional Fredholm integral equations using Adomian decomposition method, AIP Conference Proceedings, 020053, (2016) 020053-1-020053-2.
[5] B. Asady, F. Hakimzadegan, and R. Nazarlue, Utilizing artificial neural network approach for solving twodimensional integral equations, Math. Sci. 8(1) (2014) 1–9.
[6] M. Asgari and R. Ezzati, Using operational matrix of two-dimensional bernstein polynomials for solving twodimensional integral equations of fractional order, Appl. Math. Comput. 307 (2017) 290–298.
[7] K. Atkinson and E. Kendall, The Numerical Solution of Integral Equations of the Second Kind, Cambridge: Cambridge University Press, 2011.
[8] K. Atkinson and F. Potra, Projection and iterated projection methods for nonlinear integral equations, SIAM J. Numerical Anal. 24 (1987) 1352–1373.
[9] I. Aziz, F. Khan, et al, A new method based on haar wavelet for the numerical solution of two-dimensional nonlinear integral equations, J. Comput. Appl. Math. 272 (2014) 70–80.
[10] I. Aziz, S. Islam and W. Khan, Quadrature rules for numerical integration based on Haar wavelets and hybrid functions, Comput. Math. Appl. 61(9) (2011) 2770–2781.
[11] M. Bakhshi, M. Asghari-Larimi and M. Asghari-Larimi, Three dimensional differential transform method for solving nonlinear threedimensional Volterra integral equations, J. Math. Comput. Sci. 4(2) (2012) 246–256.
[12] M. Basseem, Degenerate kernel method for three dimension nonlinear integral equations of the second kind, Universal J. Integ. Eq. 3 (2015) 61–66.
[13] A. M. Bica , M. Curila and S. Curila, About a numerical method of successive interpolations for functional Hammerstein integral equations, J. Comput. Appl. Math. 236(2) (2012) 2005–2024.
[14] A. Boggess and F. J. Narcowich, First Course in Wavelets with Fourier Analyis, Prentice Hall, 2001.
[15] A. H. Borzabadi and O. S. Fard, A numerical scheme for a class of nonlinear fredholm integral equations of the second kind, J. Comput. Appl. Math. 232(2) (2009) 449–454.
[16] H. Brunner, Collocation Methods for Volterra Integral and Related Functional Differential Equations, Cambridge University Press, 2004.
[17] Z. Cheng, Quantum effects of thermal radiation in a Kerr nonlinear blackbody, J. Optical Soc. Amer. B 19 (2002) 1692–1705.
[18] W. C. Chew, M. S. Tong and B. Hu, Integral Equation Methods for Electromagnetic and Elastic Waves, Synthesis Lectures on Comput. Elect. 3(1) (2008) 1–241.
[19] M. Erfanian and H. Zeidabadi, Solving two-dimensional nonlinear mixed Volterra Fredholm integral equations by using rationalized Haar functions in the complex plane, J. Math. Mod. 7(4) (2019) 399–416.
[20] M. Esmaeilbeigi, F. Mirzaee and D. Moazami, Radial basis functions method for solving three-dimensional linear Fredholm integral equations on the cubic domains, Iran. J. Numerical Anal. Opt. 7(2) (2017) 15–37.
[21] S. Fazeli, G. Hojjati and H. Kheiri, A piecewise approximation for linear two-dimensional volterra integral equation by chebyshev polynomials, Int. J. Nonlinear Sci. 16(3) (2013) 255–261.
[22] A. Haar. Zur Theories der orthogonalen Funktionensystem, Math. Annal. 69 (1910) 331–371.
[23] G. Han and R. Wang, Richardson extrapolation of iterated discrete galerkin solution for two-dimensional fredholm integral equations, J. Comput. Appl. Math. 139(1) (2002) 49–63.
[24] G. Hursan and M. S. Zhdanov, Contraction integral equation method in three-dimensional electromagnetic modeling, Radio Sci. 6(37) (2002) 1–13.
[25] S. Islam, I. Aziz and F. Haq, A comparative study of numerical integration based on Haar wavelets and hybrid functions, Comput. Math. Appl. 59(6) (2010) 2026–2036.
[26] M. Kazemi and R. Ezzati, Numerical solution of two-dimensional nonlinear integral equations via quadrature rules and iterative method, Adv. Diff. Eq. Control Proc. 17 (2016) 27–56.
[27] M. Kazemi and R. Ezzati, Existence of solutions for some nonlinear Volterra integral equations via Petryshyn’s fixed point theorem, Int. J. Nonlinear Anal. Appl. 9 (2018) 1–12.
[28] M. Kazemi and R. Ezzati, Existence of solution for some nonlinear two-dimensional volterra integral equations via measures of noncompactness, Appl. Math. Comput. 275 (2016) 165–171.
[29] F. Khan, T. Arshad, A. Ghaffar, K.S. Nisar and D. Kumar, Numerical solutions of 2D Fredholm integral equation of first kind by discretization technique, AIMS Math. 5(3) (2020) 2295–2306.
[30] Ulo. Lepik, ¨ Application of the Haar wavelet transform to solving integral and differential equations, Proc. Estonian Acad. Sci. Phys. Math. 56(1) (2007) 28–46.
[31] J. Majak , B.S. Shvartsman, M. Kirs, M. Pohlak and H. Herranen, Convergence theorem for the Haar wavelet based discretization method, Comp. Struct. 126 (2015) 227–232.
[32] K. Maleknejad, R. Mollapourasl and K. Nouri, Study on existence of solutions for some nonlinear functional–integral equations, Nonlinear Anal. 69 (2008) 2582–2588.
[33] K. Maleknejad, J. Rashidinia and T. Eftekhari, Numerical solution of three-dimentional Voltera- Fredholm integral equations of the first and second kinds based on Bernstein’s approximation, Appl. Math. Comput. 339 (2018) 272–285.
[34] D. S. Mohamed, Shifted Chebyshev polynomials for solving three-dimensional Volterra integral equations of the second kind, arXiv preprint arXiv:1609.08539, 2016.
[35] F. Mirzaee, E. Hadadiyan and S. Bimesl, Numerical solution for three-dimensional nonlinear mixed VolterraFredholm integral equations via three-dimensional block-pulse functions, Appl. Math. Comput. 237 (2014) 168–175.
[36] F. Mirzaee and E. Hadadiyan, A computational method for nonlinear mixed Volterra-Fredholm integral equations, Caspian J. Math. Sci. 2 (2) (2014) 113–123.
[37] F. Mirzaee and E. Hadadiyan, Three-dimensional triangular functions and their applications for solving nonlinear mixed Volterra-Fredholm integral equations, Alexandria Engin. J. 3(55) (2016) 2943–2952.
[38] B. G. Pachpatte, Multidimensional Integral Equations and Inequalities, Springer Science, Business Media, 2011.
[39] K. Sadri, A. Amini and C. Cheng, Low cost numerical solution for three-dimensional linear and nonlinear integral equations via three-dimensional Jacobi polynomials, J. Comput. Appl. Math. 319 (2017) 493–513.
[40] Qi. Tang and D. Waxman, An integral equation describing an asexual population in a changing environment, Nonlinear Anal. Theo. Meth. Appl. 53(5) (2003) 683–699.
[41] S. M. Torabi and A. Tari, Two-step collocation methods for two-dimensional Volterra integral equations of the second kind, J. Appl. Anal. 25(1) (2019) 1–11.
[42] A. Ziqan, S. Armiti and I. Suwan, Solving three- dimensional Volterra integral equation by the reduced differential transform method, Int. J. Appl. Math. Res. 5(2) (2016) 103–106.
[30] Ulo. Lepik, ¨ Application of the Haar wavelet transform to solving integral and differential equations, Proc. Estonian Acad. Sci. Phys. Math. 56(1) (2007) 28–46.
[31] J. Majak , B.S. Shvartsman, M. Kirs, M. Pohlak and H. Herranen, Convergence theorem for the Haar wavelet based discretization method, Comp. Struct. 126 (2015) 227–232.
[32] K. Maleknejad, R. Mollapourasl and K. Nouri, Study on existence of solutions for some nonlinear functional–integral equations, Nonlinear Anal. 69 (2008) 2582–2588.
[33] K. Maleknejad, J. Rashidinia and T. Eftekhari, Numerical solution of three-dimentional Voltera- Fredholm integral equations of the first and second kinds based on Bernstein’s approximation, Appl. Math. Comput. 339 (2018) 272–285.
[34] D. S. Mohamed, Shifted Chebyshev polynomials for solving three-dimensional Volterra integral equations of the second kind, arXiv preprint arXiv:1609.08539, 2016.
[35] F. Mirzaee, E. Hadadiyan and S. Bimesl, Numerical solution for three-dimensional nonlinear mixed VolterraFredholm integral equations via three-dimensional block-pulse functions, Appl. Math. Comput. 237 (2014) 168–175.
[36] F. Mirzaee and E. Hadadiyan, A computational method for nonlinear mixed Volterra-Fredholm integral equations, Caspian J. Math. Sci. 2 (2) (2014) 113–123.
[37] F. Mirzaee and E. Hadadiyan, Three-dimensional triangular functions and their applications for solving nonlinear mixed Volterra-Fredholm integral equations, Alexandria Engin. J. 3(55) (2016) 2943–2952.
[38] B. G. Pachpatte, Multidimensional Integral Equations and Inequalities, Springer Science, Business Media, 2011.
[39] K. Sadri, A. Amini and C. Cheng, Low cost numerical solution for three-dimensional linear and nonlinear integral equations via three-dimensional Jacobi polynomials, J. Comput. Appl. Math. 319 (2017) 493–513.
[40] Qi. Tang and D. Waxman, An integral equation describing an asexual population in a changing environment, Nonlinear Anal. Theo. Meth. Appl. 53(5) (2003) 683–699.
[41] S. M. Torabi and A. Tari, Two-step collocation methods for two-dimensional Volterra integral equations of the second kind, J. Appl. Anal. 25(1) (2019) 1–11.
[42] A. Ziqan, S. Armiti and I. Suwan, Solving three- dimensional Volterra integral equation by the reduced differential transform method, Int. J. Appl. Math. Res. 5(2) (2016) 103–106.
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Volume 12, Issue 2
November 2021Pages 115-133
- Receive Date: 12 July 2020
- Revise Date: 16 August 2020
- Accept Date: 28 September 2020