Solving Fuzzy Integral Equations of the Second Kind by using the Reproducing Kernel Hilbert Space Method
الموضوعات :صدیقه فرزانه جوان 1 , سعید عباسبندی 2 , محمدعلی فریبرزی عراقی 3
1 - گروه ریاضی، دانشکده علوم پایه، واحد علوم و تحقیقات، دانشگاه آزاد اسلامی، تهران، ایران
2 - گروه ریاضی کاربردی، دانشکده علوم پایه، دانشگاه بین المللی امام خمینی (ره)، قزوین، ایران
3 - گروه ریاضی، دانشکده علوم پایه، واحد تهران مرکزی، دانشگاه آزاد اسلامی، تهران، ایران
الکلمات المفتاحية: Fuzzy numbers, Gram-Schmidt process, Reproducing Kernel Hilbert Space, Fuzzy integral equations,
ملخص المقالة :
In this study, a new approach based on the Reproducing Kernel Hilbert Space Method is proposed to approximate the solution of the second kind fuzzy linear integral equations. For this purpose, at first by applying the concept of parametric form, the fuzzy integral equation is converted to a system of crisp integral equations. Then, this system is solved by using the reproducing kernel method free of the Gram-Schmidt orthogonalization process. Also, two numerical algorithms are proposed based on applying the Gram-Schmidt process and without using it. The general form of numerical solution accordingly the reproducing kernel method is introduced and the convergence theorem of solution of the proposed scheme to the exact solution is proved. Finally, a sample fuzzy integral equation is solved by means of both suggested algorithms and the results are compared for differents points and levels. Due to the difficulties in applying the Gram-Schmidt process, the obtained results of the new algorithm are satisfactory.
[1] S. Abbasbandy and T. Allahviranloo, “The Adomian decomposition method applied to the fuzzy system of Fredholm integral equations of the second kind,” International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, vol.14,no.1,(2006) pp. 101–110.
[2] E. Babolian, H. Sadeghi Goghary, S. Abbasbandy, Numerical solution of linear Fredholm fuzzy integral equations of the second kind by Adomian method, Applied Mathematics and Computation, 161(3) (2005) 733–744.
[3] M. Barkhordari Ahmadi, M. Khezerloo, Fuzzy bivariate Chebyshev method for solving fuzzy Volterra-Fredholm integral equations, Internat. J. Industrial Math. 3(2) (2011) 67–77.
[4] M. Cui, Y. Lin, Nonlinear Numerical Analysis in the Reproducing Kernel Space, Nova Science, New York, NY, USA, (2009).
[5] R. Ezzati, S.M. Sadatrasoul., On numerical solution of two-dimensional nonlinear Urysohn fuzzy integral equations based on fuzzy Haar wavelets, Fuzzy Sets Syst. 309 (2017) 145–164.
[6] M.A. Fariborzi Araghi, N. Parandin, Numerical solution of fuzzy Fredholm integral equations by the Lagrange interpolation based on the extension principle, Soft Comput. 15 (2011)2449–2456.
[7] M. Friedman, M. Ma, A. Kandel, Numerical solution of fuzzy differential and integral equations, Fuzzy Sets Syst. 106 (1999) 35–48.
[8] S. Farzaneh Javan, S. Abbasbandy, and M. Fariborzi Araghi, Application of Reproducing Kernel Hilbert Space Method for Solving a Class of Nonlinear Integral Equations, Mathematical Problems in Engineering, (2017), 1-11.
[9] G. Gumah, K. Moaddy, M. Al-Smadi, and I. Hashim, Solutions to Uncertain Volterra Integral Equations by Fitted Reproducing Kernel Hilbert Space Method, Journal of Function Spaces (2016), 1-12.
[10] M. Ghanbari, “Numerical solution of fuzzy linear Volterra integral equations of the second kind by Homotopy analysis method,” International Journal Industrial Mathematics, vol. 2, pp. 73–87, (2010).
[11] N. Gholami, T. Allahviranloo, S. Abbasbandy, N. Karamikabir., Fuzzy reproducing kernel space method for solving fuzzy boundary value problems, Mathematical Sciences, (2019) 1-7.
[12] J. Hamaydi, N. Qatanani, Computational Methods for Solving Linear Fuzzy Volterra Integral Equation, J. Appl. Math. 2017 (2017), Article ID 2417195, 12 pages.
[13] F. Mirzaee, M. Paripour, M. Komak yari, Numrical solution of Fredholm fuzzy integral equations of the second kind via direct method using Triangular function. J Hyper (2012); 1(2):46–60.
[14] F. Mirzaee, M.K. Yari, E. Hadadiyan, Numerical solution of two-dimensional fuzzy Fredholm integral equations of the second kind using triangular functions, Beni-Suef Univ. J. Basic Appl. Sci. 4 (2015) 109–118.
[15] A. Molabahrami, A. Shidfar, A.
Ghyasi., An alalytical method for solving linear Fredholm fuzzy integral equations of the second kind, Computers and Mathematics with Applications 61 (2011) 2754–2761.
[16] M. Najariyan, M. H. Farahi, Optimal control of fuzzy linear controlled system with fuzzy initial conditions, Iranian Journal of Fuzzy Systems (2013) 10: 3, 21-35.
[17] N. Parandin, M.A. Fariborzi Araghi, The numerical solution of linear fuzzy Fredholm integral equations of the second kind by using finite and divided differences methods, Soft Comput. 15 (2010) 729–741.
[18] J. Y. Park and H. K. Han, “Existence and uniqueness theorem for a solution of fuzzy Volterra integral equations,” Fuzzy Sets and Systems, vol. 105, no. 3, pp. 481–488, 1999.
[19] S.M. Sadatrasoul, R. Ezzati, Numerical solution of two-dimensional nonlinear Hammerstein fuzzy integral equations based on optimal fuzzy quadrature formula, J. Comput. Appl. Math. 292 (2016) 430–446.
[20] S. Salahshour, T. Allahviranloo, Application of fuzzy differential transform method for solving fuzzy Volterra integral equations, Appl. Math. Modelling 37 (2013) 1016–1027.
[21] S. Salahshour, M. Khezerloo, S. Hajighasemi, M. Khorasany, Solving fuzzy integral equations of the second kind by fuzzy Laplace transform method, Internat. J. Industrial Math. 4(2012) 21–9.
[22] S. Salahshour, M. Khezerloo, S. Hajighasemi, and M. Khorasany, “Solving fuzzy integral equations of the second kind by fuzzy Laplace transform method,” International Journal Industrial Mathematics, vol. 1, pp. 21–29, (2012).
[23] M. Al-Smadi, Reliable Numerical Algorithm for Handling Fuzzy Integral Equations of Second Kind in Hilbert Spaces, Published by Faculty of Sciences and Mathematics, University of Nis, Serbia 33:2 (2019), 583–597.
[24] M. Sugeno, Theory of fuzzy integrals and its application, Ph.D. Dissertation, Tokyo Institute of Technology, (1974).
[25] S. Ziari, A.M. Bica, New error estimate in the iterative numerical method for nonlinear fuzzy Hammerstein-Fredholm integral equations, Fuzzy Sets Syst. 295 (2016) 136–152.
[26] H. J. Zimmermann., Fuzzy set theory and its applications, Kluwer Academic, Boston, (1991)
