A new method for solving n-order fuzzy differential equation by using polynomial interpolation

Subject Areas : Statistics

Elham Ahmadi ¹ , Nazanin Ahmadi ²

1 - Department of Mathematics, Quds City Branch, Islamic Azad University, Tehran, Iran
2 - Department of Mathematics, Varamin-Pishva Branch, Islamic Azad University, Varamin, Iran

Received: 2020-07-07 Accepted : 2020-09-09 Published : 2022-06-22

Keywords: روش عددی, مشتق تعمیم یافته, معادلات دیفرانسیل فازی, چند جمله ای درونیاب,

Abstract :

A new method for solving n-order fuzzy differential equation by using polynomial interpolationA new method for solving n-order fuzzy differential equation by using polynomial interpolationGiven the importance of the role of fuzzy differential equations in science and engineering,in this paper, we study a numerical method for solving N th order fuzzy differential equations under generalized differentiability. In this method a solution of fuzzy differential equation is approximated by fuzzy polynomial in the form of piece wise fuzzy polynomials in eachsub interval of interval solution. In special case, for solving second order fuzzy differential equation under generalized differentiability, according to the type of differentiability, four cases are considered, then fuzzy polynomial approximation in each cases for solving fuzzy differential equation were constructed. The order of the piece wise fuzzy polynomial in each sub interval of solution is two .Finally this method is illustrated by solving two second order fuzzy differentialequations under generalized differentiability.

References:

[1] S. Chang and L. Zadeh, On fuzzy mapping and control, IEEE, Trans Syst Cybern 2(1972) 30-34.

 [2] M. Hukuhara, Integration des applications mesurables dont la valeur est un compact convex, Funkcial. Ekvac. 10 (1967) 205–229.

[3] M. L. Puri, D. A. Ralescu, Differentials of fuzzy functions, J. Math. Anal. Appl. 114(1986) 409-422.

[4] R. Goetschel, W. Voxman, Elementary fuzzy calculus, Fuzzy Sets and Systems 24 (1987) 31-43.

[5] D. Dubois, H. Prade,Toward fuzzy differential calculus: Part 3. Differ, Fuzzy Sets and Systems (1982) 225-233.

 

[6] S. Seikkala, On the fuzzy initial value problem, Fuzzy Sets and Systems. 24 (1987) 319-330.

 [7] T. Allahviranloo, S. Abbasbandy, N. Ahmady, E. Ahmady, Improved predictor-corrector method for solving fuzzy initial value problems, Information Sciences 179 (2009)945-955.

 [8] T. Allahviranloo, N. Ahmady, E. Ahmady, Numerical solution of fuzzy differential equations by predictor-corrector method. Inform Sci 177(7)(2007)1633-1647.

 [9] M. Ma, M. Friedman, A. Kandel, Numerical solutions of fuzzy differential equations, Fuzzy Sets and Systems 105 (1999) 133-138.

 [10] S. Abbasbandy,T. Allahviranloo,  Numerical solution of fuzzy differential equation by Taylor method, J. Comput Method Appl Math 2 (2002) 113-124.

[11]  N. Parandin, Numerical Solution of fuzzy differential equations of nth-order by Adams-Bashforth method, New Researches and Mathematics, 17 (2019) 85-94.

[12] B. Bede, S. G. Gal, Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations, Fuzzy Set and Systems 151(2005)581-599.

[13] B.Bede,L.Stefanini,Generalized differentiability of fuzzy-valued functions, Fuzzy Sets and Systems 230 (2013) 119-141.

[14] B. Bede, S. G. Gal,Remark on the new solutions of fuzzy differential equations, Chaos Solitons Fractals (2006).

[15] B. Bede, L. Stefanini, Solution of Fuzzy Differential Equations with generalized differentiability using LU-parametric representation, EUSFLAT 1(2011)785- 790.

[16] J.J. Nieto, A. Khastan, K. Ivaz, Numerical solution of fuzzy differential equation under generalized differentiability, Nonlinear Anal. Hybrid Syst. 3 (2009) 700-707. http://dx.doi.org/10.1016/j.nahs.2009.06.013.

[17] A. Khastan, F. Bahrami, K. Ivaz, New results on multiple solutions for Nth-order fuzzy differential equation under generalized differentiability, Boundary Value Problem (Hindawi Publishing Corporation)(2009) http://dx.doi.org/10.1155/2009/395714

[18] J. F. Epperson, An Introduction to Numerical Methods and Analysis, John Wiley and Sons 2007.

[19] O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems 24(1987), 301-317.

[20] L. Stefanini, B. Bede, eneralized Hukuhara differentiability of interval-valued functions and interval differential equations, Nonlinear Analysis 71 (2009) 1311-1328.

[21] A. Khastan, J. J Nieto, R. Rodríguez-López, Variation of constant formula for first order fuzzy differential equations, Fuzzy Sets and Systems 177 (2011), 20-33.

[22] T. Allahviranloo, N.Ahmady, E.Ahmadi, Approximated solution of First order Fuzzy Differential Eqations under generalized differentiability, New Researches and Mathematics, 9 (2017) 33-44.