Solving Second-Order Fuzzy Cauchy-Euler Initial Value Problems Under Generalized Differentiability
محورهای موضوعی : مجله بین المللی ریاضیات صنعتی
1 - Department of Mathematics, Savadkooh Branch, Islamic Azad University, Savadkooh, Iran.
کلید واژه: Generalized differentiability, Fuzzy differential equations, Cauchy-Euler equations, Differential Equations,
چکیده مقاله :
In this paper, we study a class of second-order fuzzy initial value problems that are known as the Cauchy-Euler differential equations, in the crisp case. This work begins by studying the structure of solution function in the crisp case and providing a requirement space of the generalized differentiable functions. In sequel, the process of production and construction of the solution formula are discussed, in details. Finally, the obtained formulas are applied and illustrated by solving some examples.
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[1] S. Abbasbandy, T. Allahviranloo, Numerical solution of fuzzy differential equation by Runge-Kutta method,Nonlinear studies 11 (2004) 117-129.
[2] MZ. Ahmad, MK. Hasan, B. De Baets, Analytical and numerical solutions of fuzzy differential equations, Information Sciences 236 (2013) 156-167.
[3] O. Akin, T. Khaniyev, O. Oruc, IB. Turksen, An algorithm for the solution of second order fuzzy initial value problems, Expert Systems with Applications 40 (2013) 953-957.
[4] T. Allahviranloo, N. Ahmady, E. Ahmady, Numerical solution of fuzzy differential equations by predictor-corrector method, Information Sciences 177 (2007) 1633-1643.
[5] T. Allahviranloo, M. Chehlabi, Solving fuzzy differential equations based on the length function properties, Soft Computing 19 (2015) 307-320.
[6] T. Allahviranloo, NA. Kiani, M. Barkhordari, Toward the existence and uniqueness of solutions of second-order fuzzy differential equations,Information Sciences 179 (2009) 1207-1215.
[7] LC. Barros, LT. Gomes, PA. Tonelli, Fuzzy differential equations: An approach via fuzzification of the derivative operator, Fuzzy Sets and Systems 230: (2013) 9-52.
[8] B. Bede, TC. Bhaskar, V. Lakshmikantham, Perspectives of fuzzy initial value problems, Communications in Applied Analysis 11 (2007) 339-358.
[9] B. Bede, SG. Gal, Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations,Fuzzy Sets and Systems 5 (2005) 581-599.
[10] B. Bede, SG. Gal, Solution of fuzzy differential equations based on generalized differentiability, Communications in Mathematical Analysis 9 (2010) 22-41.
[11] B. Bede, IJ. Rudas, AL. Bencsik, First order linear fuzzy differential equations under generalized differentiability, Information Sciences 177 (2007) 1648-1662.
[12] B. Bede, L. Stefanini, Solution of fuzzy differential equations with generalized differentiability using LU-parametric representation, EUSFLAT 11 (2011) 785-790.
[13] WE. Boyce, RC. Diprima, Elementary differential equations and boundary value problems, Printed in the United States of America, New York, (2000).
[14] JJ. Buckley, T. Feuring, Fuzzy differential equations, Fuzzy Sets and Systems 110 (2000) 43-54.
[15] Y. Chalco-Cano, H. Rom´an-Flores, On new solutions of fuzzy differential equations, Chaos, Solitons & Fractals 38 (2008) 112-119.
[16] Y. Chalco-Cano, H. Rom´an-Flores, Comparation between some approaches to solve fuzzy differential equations, Fuzzy Sets and Systems 160 (2009) 1517-1527.
[17] Y. Chalco-Cano, H. Rom´an-Flores, MD. Jimenez-Gamero, Generalized derivative and π-derivative for set-valued functions, Information Sciences 181 (2011) 2177-2188.
[18] JR. Chasnov, Differential equations with youTube examples, 1st edition, Bookboon.com, (2014).
[19] M. Chehlabi, T. Allahviranloo, Solution to a class of first-order fuzzy Cauchy-Euler differential equations, Mathematical Sciences Letters 5 (2016) 189-194.
[20] D. Dubois, H. Prade, Towards fuzzy differential calculus: Part 3, Differentiation, Fuzzy Sets and Systems 8 (1982) 225-233.
[21] NA. Gasilov, SE. Amrahov, AG. Fatullayev, A geometric approach to solve fuzzy linear systems of differential equations, Applied Mathematics & Information Sciences 5 (2011) 484-499.
[22] NA. Gasilov, S¸E. Amrahov, AG. Fatullayev, A new approach to fuzzy initial value problem, Soft Computing 18 (2014) 217-225.
[23] A. Khastan, JJ. Nieto, R. Rodr´ıguez-Lop´ez, Variation of constant formula for first order fuzzy differential equations, Fuzzy Sets and Systems 177 (2011) 20-33.
[24] A. Khastan, K. Ivaz, Numerical solution of fuzzy differential equations by Nystr¨om method, Chaos, Solitons & Fractals 41 (2009) 859-868.
[25] JJ. Nieto , A. Khastan, K. Ivaz, Numerical solution of fuzzy differential equations under generalized differentiability,Nonlinear Analysis: Hybrid Systems 3 (2009) 700-707.
[26] R . Rodr´ıguez-L´opez, On the existence of solutions to periodic boundary value problems for fuzzy linear differential equations, Fuzzy Sets and Systems 219 (2013) 1-26.
[27] S. Seikkala, On the fuzzy initial value problem, Fuzzy Sets and Systems 24 (1987) 319-330.
[28] L. Stefanini, B. Bede, Generalized Hukuhara differentiability of interval-valued functions and interval differential equations, Nonlinear Analysis: Theory, Methods & Applications 71 (2009) 1311-1328.
[29] L. Stefanini, B. Bede, Generalized fuzzy differentiability with LU-parametric representation, Fuzzy Sets and Systems 257 (2014) 184-203
[30] C. Wu, Z. Gong, On Henstock integral of fuzzy-number-valued functions I, Fuzzy Sets and Systems 120 (2001) 523-532.