The Application of Hybrid Orthonomal Bernstein and Block-Pulse Functions in Finding Numerical Solution of Fredholm Fuzzy Integral Equations

Authors: Mahmoud Zarrini, Sanaz Torkaman

Abstract:

In this paper, we have proposed a numerical method for solving fuzzy Fredholm integral equation of the second kind. In this method a combination of orthonormal Bernstein and Block-Pulse functions are used. In most cases, the proposed method leads to the exact solution. The advantages of this method are shown by an example and calculate the error analysis.

Keywords: Fuzzy Fredholm Integral Equation, Bernstein, Block-Pulse, Orthonormal.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1096527

References:

[1] L. A. Zadeh,”Linguistic variable approximate and disposition,” in Med. Inform, pp. 173-186, 1983.
[2] S. S. L. Chang and L. Zadeh, ”On fuzzy mapping and control, ” in IEEE trans system Cybernet, vol. 2, pp. 30-34, 1972.
[3] K. Maleknejad, M. Shahrezaee and H. Khatami, ”Numerical solution of integral equations system of the second kind by Block-Pulse functions,” in Applied Mathematics and Computation, vol. 166, pp. 15-24, 2005.
[4] E. Babolian and Z. Masouri, ”Direct method to solve Volterra integral equation of the first kind using operational matrix with block-pulse functions,” in Applied Mathematics and Computation, vol. 220, pp. 51-57, 2008.
[5] E. Babolian, K. Maleknejad, M. Roodaki and H. Almasieh, ”Twodimensional triangular functions and their applications to nonlinear 2D Volterra-Fredholm integral equations,” in Computer and Mathematics with Applications, vol. 60, pp. 1711-1722, 2010.
[6] F. Mirzaee and S. Piroozfar, ”Numerical solution of the linear two dimensional Fredholm integral equations of the second kind via two dimensional triangular orthogonal functions,” in Journal of King Saud University, vol. 22, pp. 185-193, 2010.
[7] Y. Ordokhani, ”Solution of nonlinear Volterra-Fredholm-Hammerstein integral equations via rationalized Haar functions,” in Applied Mathematics and Computation, vol. 180, pp. 436-443, 2006.
[8] K. Maleknejad and M. T. Kajani, ”Solving second kind integral equations by Galerkin methods with hybrid Legendre and Block-Pulse functions,” in Applied Mathematics and Computation, vol. 145, pp. 623-629, 2003.
[9] E. Hashemzadeh, K. Maleknejad and B. Basirat, ”Hybrid functions approach for the nonlinear Volterra-Fredholm integral equations,” in Procedia Computer Sience, vol.3, pp. 1189-1194, 2011.
[10] M. T. Kajani and A. H. Vencheh, ”Solving second kind integral equations with Hybrid Chebyshev and Block-Pulse functions,” in Applied Mathematics and Computation, vol. 163, pp. 71-77, 2005.
[11] X. T. Wang and Y. M. Li, ”Numerical solutions of integrodifferential systems by hybrid of general block-pulse functions and the second Chebyshev polynomials,” in Applied Mathematics and Computation, vol. 209, pp. 266- 272, 2009.
[12] K. Maleknejad and Y. Mahmoudi, ”Numerical solution of linear Fredholm integral equation by using hybrid Taylor and Block-Pulse functions,” in Applied Mathematics and Computation, vol. 149, pp. 799-806, 2004.
[13] B. Asady, M. T. Kajani, A. H. Vencheh and A. Heydari, ”Solving second kind integral equations with hybrid Fourier and block-pulse functions,” in Applied Mathematics and Computation, vol. 160, pp. 517-522, 2005.
[14] G. P. Rao, ”Piecewise Constant Orthogonal Functions and their Application to System and Control,” in Springer-Verlag, 1983.
[15] B. M. Mohan and K.B. Datta, ”Orthogonal Function in Systems and Control,” 1995.
[16] K. Maleknejad, B. Basirat and E. Hashemizadeh, ”A Bernstein operational matrix approach for solving a system of high order linear Volterra-Fredholm integro-differential equations,” in Math. Comput. Model, vol. 55, pp. 1363-1372, 2012.
[17] T. J. Rivlin, ”An introduction to the approximate of functions,” in New York, DovePublications, 1969.
[18] S. A. yousefi, M. Behroozifar, ”Operational matrrices of Bernstein polynomials and their applications,” in International Journal of System Science, vol. 41, pp. 709-716, 2010.
[19] O. Kaleva, ”Fuzzy differential equations,” in Fuzzy Sets Syst, vol. 24, pp. 301-317, 1987.
[20] M. Ma, M. Friedman and A. Kandel, ”Duality in fuzzy linear systems,” Fuzzy Sets and Systems, vol. 109, pp. 55-58, 2000.
[21] S. Gal, ”Approximation theory in fuzzy setting,” Chapter 13 in Handbook of analytic-computational methods in applied mathematics, pp. 617-666, 2000.
[22] G. A. Anastassiou, ”Fuzzy mathematcs: Approximation theory,” in Springer, Heidelberge, 2010.
[23] R. Goetschel, W. Voxman, ”Elementary fuzzy calculus,” in Fuzzy Sets and Systems, vol. 18, pp. 31-43, 1986.
[24] H. Wu, ”The fuzzy Riemann integral and its numerical integration,” in Fuzzy Sets and Systems, vol. 110, pp. 1-25, 2000.
[25] M. Friedman, M. Ming and A. Kandel, ”Fuzzy linear systems,” in FSS, vol. 96, pp. 201-20, 1998.