A Hybrid Method for Numerical Solution of Fuzzy Mixed Delay Volterra-Fredholm Integral Equations System
محورهای موضوعی : Transactions on Fuzzy Sets and Systems
1 - Department of Mathematics, Associate Professor of Applied Mathematics, Lorestan University, Khorramabad, 68151-44316, Iran.
کلید واژه: Fuzzy integral equation, Block pulse function, Bernstein polynomials.,
چکیده مقاله :
A hybrid method for the numerical solution of the system of delayed linear fuzzy mixed VolterraFredholm integral equations (FMDVFIES) is introduced. Using the hybrid of Bernstein polynomials and blockpulse functions (HBBFs), an approximate solution for the equations system is provided. Firstly, the HBBFs and their operational matrices are introduced, and some of their characteristics are described. Then by applying the operational matrices on FMDVFIES convert it to the algebraic equations system. The numerical solution is obtained by solving this algebraic system. Then the convergence is investigated and some numerical examples are presented to show the effectiveness of the method.
A hybrid method for the numerical solution of the system of delayed linear fuzzy mixed VolterraFredholm integral equations (FMDVFIES) is introduced. Using the hybrid of Bernstein polynomials and blockpulse functions (HBBFs), an approximate solution for the equations system is provided. Firstly, the HBBFs and their operational matrices are introduced, and some of their characteristics are described. Then by applying the operational matrices on FMDVFIES convert it to the algebraic equations system. The numerical solution is obtained by solving this algebraic system. Then the convergence is investigated and some numerical examples are presented to show the effectiveness of the method.
[1] Zadeh LA. Fuzzy sets. Information and Control. 1965; 8 (3): 338353. DOI: https://doi.org/10.1016/S0019-9958(65)90241-X
[2] Singh H, Gupta MM, Meitzler T, Hou ZG, Garg KK, Solo AMG, Zadeh LA. Real-Life Applications of Fuzzy Logic. Advances in Fuzzy Systems. 2013. DOI: http://dx.doi.org/10.1155/2013/581879
[3] Chang SL, Zadeh LA. On fuzzy mapping and control. IEEE Transactions on Systems, Man, and Cybernetics. 1972; SMC-2(1): 30-34. DOI: http://dx.doi.org/10.1109/TSMC.1972.5408553
[4] Dubois D, Prade H. Toward fuzzy differential calculus: Part 3, differentiation. Fuzzy Sets and Systems. 1982; 8(3): 225-233. DOI: https://doi.org/10.1016/S0165-0114(82)80001-8
[5] Seikkala S. On the fuzzy initial value problem. Fuzzy Sets and Systems. 1987; 24(3): 319-330. DOI: https://doi.org/10.1016/0165-0114(87)90030-3
[6] Kaleva O. Fuzzy differential equations. Fuzzy Sets and Systems. 1987; 24(3): 301-317. DOI: https://doi.org/10.1016/0165-0114(87)90029-7
[7] Kaleva O. The Cauchy problem for fuzzy differential equations. Fuzzy Sets and Systems. 1990; 35(3): 389-396. DOI:
https://doi.org/10.1016/0165-0114(90)90010-4
[8] Buckley J, Feuring T. Fuzzy differential equations. Fuzzy Sets and Systems. 2000; 110(1): 43-54. DOI: https://doi.org/10.1016/S0165-0114(98)00141-9
[9] Dubois D, Prade H. Toward fuzzy differential calculus: Part 1: Integration of fuzzy mappings. Fuzzy Sets and Systems. 1982; 8(1): 1-17. DOI: https://doi.org/10.1016/0165-0114(82)90025-2
[10] Dubois D, Prade H. Toward fuzzy differential calculus: Part 2: Integration on fuzzy intervals. Fuzzy Sets and Systems. 1982; 8(2): 105-116. DOI: https://doi.org/10.1016/0165-0114(82)90001-X
[11] Nanda S. On integration of fuzzy mapping. Fuzzy Sets and Systems. 1989; 32(1): 95-101. DOI: https://doi.org/10.1016/0165-0114(89)90090-0
[12] Friedman M, Ma M, Kandel A. Numerical solutions of fuzzy differential and integral equations. Fuzzy Sets and Systems. 1999; 106(1): 35-48. DOI: https://doi.org/10.1016/S0165-0114(98)00355-8
[13] Abbasbandy S, Allahviranloo T. Numerical solution of fuzzy differential equation by RungeKutta method. Mathematical and Computational Applications. 2004; 11(1): 117-129. DOI: https://doi.org/10.3390/mca16040935
[14] Fariborzi Araghi MA, Parandin N. Numerical solution of fuzzy Fredholm integral equations by the Lagrange interpolation based on the extension principle. Soft Computing. 2011; 15: 2449-2456. DOI: https://doi.org/10.1007/s00500-011-0706-3
[15] Ezzati R, Ziari S. Numerical solution of two-dimensional fuzzy Fredholm integral equation of the second kind using fuzzy bivariate Bernstein polynomials. International Journal of Fuzzy Systems. 2013; 15(1): 84-89.
[16] Shafiee M, Abbasbandy S, Allahviranloo T. Predictor corrector method for nonlinear fuzzy Volterra integral equations. Australian Journal of Basic and Applied Sciences. 2011; 5(12): 2865-2874. https://ajbasweb.com/old/ajbas/2011/December-2011/2865-2874.pdf
[17] Amin R, Shah K, Asif M, Khan I. Efficient numerical technique for solution of delay Volterra-Fredholm integral equations using Haar wavelet. Heliyon. 2020; 6(10): e05108. DOI: https://doi.org/10.1016/j.heliyon.2020.e05108
[18] Baghmisheh M, Ezzati R. Application of hybrid Bernstein polynomials and block-pulse functions for solving nonlinear fuzzy fredholm integral equations. Fuzzy Information and Engineering. 2023; 15(1): 69-86. DOI: https://doi.org/10.26599/FIE.2023.9270006
[19] Mirzaee F, Yari MK, Hoseini SF. A computational method based on hybrid of Bernstein and block-pulse functions for solving linear fuzzy Fredholm integral equations system. Journal of Taibah University for Science. 2015; 9(2): 252-263. DOI: https://doi.org/10.1016/j.jtusci.2014.07.008
[20] Entezari M, Abbasbandy S, Babolian E. Hybrid of block-pulse and orthonormal Bernstein functions for fractional differential equations. Iranian Journal of Numerical Analysis and Optimization. 2022; 12(2): 315333. DOI: https://doi.org/10.22067/ijnao.2021.72492.1056
[21] He J, Taha MH, Ramadan MA, Moatimid GM. A combination of Bernstein and improved block-pulse functions for solving a system of linear Fredholm integral equations. Mathematical Problems in Engineering. 2022. DOI: https://doi.org/10.1155/2022/6870751
[22] Tachev G. Pointwise approximation by Bernstein polynomials. Bulletin of the Australian Mathematical Society. 2012; 85(3): 353-358. DOI: doi:10.1017/S0004972711002838
[23] Goetschel R, Voxman W. Elementary fuzzy calculus. Fuzzy Sets and Systems. 1986; 18(1): 31-43. DOI: https://doi.org/10.1016/0165-0114(86)90026-6
[24] Jiang Z, Schaufelberger W. Block Pulse Functions and Their Applications in Control Systems. Berlin: Springer-Verlag; 1992. DOI: https://doi.org/10.1007/BFb0009162
[25] Zhang J, Tang Y, Liu F, Jin Z, Lu Y. Solving fractional differential equation using block-pulse functions and Bernstein polynomials. Mathematical Methods in the Applied Sciences. 2021; 44(7): 5501-5519. DOI: https://doi.org/10.1002/mma.7126
