Abstract
In this paper, we introduce a new collocation method, based on Boubaker polynomials, for approximate solutions of a class of two-dimensional Fredholm linear integral equations of the second kind. The properties of the two-dimensional Boubaker functions are used. The basic integration matrix is used by collocation points to reduce the answer form of the integral equation to the answer form of the algebraic equation system. The accuracy of the answer and error analysis has been studied thoroughly and structurally, and it has been emphasized that the proposed method for a variety of two-dimensional integral equations of linear Fredholm with a continuous kernel is an entirely accurate and error-free polynomial type. Moreover, the error estimation of the approximate solution and exact solution are also provided. Numerical examples are presented to illustrate and compare the results of the truncated Boubaker collocation method with the results of other methodologies to provide validity, capability and efficiency of the technique.
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References
Avazzadeh, Z., Heydari, M.: Chebyshev polynomials for solving two dimensional linear and nonlinear integral equations of the second kind. Comput. and Appl. Math. 31(1), 127–142 (2012)
Boubaker, K.: Boubaker polynomials expansion scheme (BPES) solution to Boltzmann diffusion equation in the case of strongly anisotropic neutral particles forward-backward scattering. Ann. Nucl. Energy 38, 1715–1717 (2011)
Boubaker, K., Zhang, L.: Fermat-linked relations for the Boubaker polynomial sequences via Riordan matrices analysis. J. Assoc. Arab Univ. Basic Appl. Sci. 12, 74–78 (2012)
Boubaker, K., Chaouachi, A., Amlouk, M., Bouzouita, H.: Enhancement of pyrolysis spray disposal performance using thermal time-response to precursor uniform deposition. Eur. Phys. I. Appl. Phys. 37, 105–109 (2007)
Delves, L.M., Mohamed, J.L.: Computational Method for Integral Equations. Cambridge University Press, New York (1985)
Guoqiang, H., Jiong, W.: Extrapolation of Nystrom solution for two-dimensional nonlinear Fredholm integral equations. J. Comput. and Appl. Math. 134, 259–268 (2001)
Guoqiang, H., Wang, R.: Richardson extrapolation of iterated discrete Galerkin solution for two-dimensional Fredholm integral equations. J. Comput. and Appl. Math. 139, 49–63 (2002)
Hildebrand, F.B.: Introduction to Numerical Analysis, 2nd edn. McGraw-Hill, New York (1974)
Karem Ben Mahmoud, B.: Temperature 3D profiling in cryogenic cylindrical devices using Boubaker polynomials expansion scheme (BPES). Cryogenics 49, 217–220 (2009)
Labiadh, H., Boubaker, K.: A Sturm-Liouville shaped characteristic differential equation as a guide to establish a quasi-polynomial expansion to the Boubaker polynomials. Diff. Eq. and Cont. Proc. 2, 117–133 (2007)
Lin, Q., Sloan, I.H., Xie, R.: Extrapolation of the iteration collocation method for integral equations of the second kind. SIAM J. Numer. Anal. 27, 1535–1541 (1990)
Ma, Y., Huang, J., Li, H.: A novel numerical method of two-dimensional Fredholm integral equations of the second kind. Math. Probl. Eng. (2015). https://doi.org/10.1155/2015/625013
McLean, W.: Asymptotic error expansions for numerical solutions of integral equations. IMA J. Numer. Anal. 9, 373–384 (1989)
Mirzaee, F., Hadadiyan, E.: Numerical solution of linear Fredholm integral equations via two-dimensional modification of hat functions. Appl. Math. Comput. 250, 805–816 (2015)
Mirzaei, S.M., Amirfakhrian, M.: A Multidimensional Reverse Interpolation Method and its Application in Solving the Multidimensional Fredholm Integral Equations. Int. J. Appl. Comput. Math 7, 160 (2021). https://doi.org/10.1007/s40819-021-01096-1
Rahimi, M.Y., Shahmorad, S., Talati, F., Tari, A.: An Operational Method for The Numerical Solution of Two Dimensional Linear Fredholm Integral Equations with an Error Estimation. Bulletin of the Iranian Mathematical Society 36(2), 119–132 (2010)
Tohidi, E.: Taylor matrix method for solving linear two-dimensional Fredholm integral equations with Piecewise Intervals. Computational and Applied Mathematics 34(3), 1117–1130 (2015)
Zhao, T.G., Naing, L., Yue, W.X.: Some New Features of Boubaker Polynomials Expansion Scheme BPES. Math. Notes 87(2), 165–168 (2010)
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Mehdifar, F., Khani, A. A Specific Numerical Method for Two-Dimensional Linear Fredholm Integral Equations of the Second Kind by Boubaker Polynomial Bases. Int. J. Appl. Comput. Math 8, 240 (2022). https://doi.org/10.1007/s40819-022-01417-y
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DOI: https://doi.org/10.1007/s40819-022-01417-y
Keywords
- Two-dimensional integral equations
- Fredholm integral equations
- Collocation points
- Matrix methods
- Boubaker polynomial series