Login Register Cart 0
Generic placeholder image

Journal of Fuzzy Logic and Modeling in Engineering

Editor-in-Chief

ISSN (Print): 2666-2949
ISSN (Online): 2666-2957

Back
Journal Home About Journal Editorial Board Current Issue Volumes/Issues
Subscribe
Research Article

Fuzzy Form of Euler Method to Solve Fuzzy Differential Equations

Author(s): U.M. Pirzada* and S. Rama Mohan

Volume 1, Issue 1, 2022

Published on: 11 March, 2022

Article ID: e010621189942 Pages: 7

DOI: 10.2174/2666294901666210105142449

Abstract

Background: The Euler method is a elementary method to solve fuzzy differential equations numerically. Several authors have explored the Euler method by applying various approaches and derivative concepts.

Methods: This paper proposes a fuzzy form of the Euler method to solve fuzzy initial value problems. Novelty of this approach is that the method developed based on fuzzy arithmetic. The solution by this method is readily available in the form of fuzzy-valued function. The method does not require to re-write fuzzy differential equation into a system of two crisp ordinary differential equations.

Results: The algorithm of proposed method and local error expression are discussed. An illustration and solution of the fuzzy Riccati equation are provided for the applicability of the method.

Conclusion: The proposed Euler method is a natural generalization of crisp Euler method. It is very efficient in solving linear and nonlinear fuzzy differential equations. One of the advantage of this method is that the solution obtained by the method is always a fuzzy-valued function due to well-defined fuzzy arithmetic.

Keywords: Fuzzy-valued function, fuzzy differential equation, euler method, fuzzy riccati equation.

[1]
S. Abbasbandy, and T. Allah Viranloo, "Numerical solutions of fuzzy differential equations by Taylor method", Comput. Methods Appl. Math., vol. 2, no. 2, pp. 113-124, 2002.
[http://dx.doi.org/10.2478/cmam-2002-0006]
[2]
A. Ahmadian, S. Salahshour, C.S. Chan, and D. Baleanu, "Numerical solutions of fuzzy differential equationsby an efficient Runge Kutta method with generalized differentiability", Fuzzy Sets Syst., vol. 331, pp. 47-67, 2018.
[http://dx.doi.org/10.1016/j.fss.2016.11.013]
[3]
M.Z. Ahmad, and M.K. Hasan, "A new fuzzy version of Eulers method for solving differential equations with fuzzy initial values", Sains Malays., vol. 40, no. 6, pp. 651-657, 2011.
[4]
M.Z. Ahmada, M.K. Hasan, and B.D. Baets, "Analytical and numerical solutions of fuzzy differential equations", Inf. Sci., vol. 236, pp. 156-167, 2013.
[http://dx.doi.org/10.1016/j.ins.2013.02.026]
[5]
T. Allahviranloo, and S. Salahshour, "Euler method for solving hybrid fuzzy differential equation", Soft Comput., vol. 15, pp. 1247-1253, 2011.
[http://dx.doi.org/10.1007/s00500-010-0659-y]
[6]
B. Bede, and S.G. Gal, "Generalizations of the differentiability of fuzzy number value functions with applications to fuzzy differential equations", Fuzzy Sets Syst., vol. 151, pp. 581-599, 2005.
[http://dx.doi.org/10.1016/j.fss.2004.08.001]
[7]
J. Buckley, and T. Feuring, "Introduction to fuzzy partial differential equations", Fuzzy Sets Syst., vol. 105, pp. 241-248, 1999.
[http://dx.doi.org/10.1016/S0165-0114(98)00323-6]
[8]
P. Darabi, S. Moloudzadeh, and H. Khandani, "Anumerical methodforsolving first-order fully fuzzy differential equation under strongly generalized H-differentiability", Soft Comput., vol. 20, no. 10, pp. 4085-4098, 2016.
[http://dx.doi.org/10.1007/s00500-015-1743-0]
[9]
D. Dubois, and H. Prade, "Towards fuzzy differential calculus part 3: Differentiation", Fuzzy Sets Syst., vol. 8, pp. 225-233, 1982.
[http://dx.doi.org/10.1016/S0165-0114(82)80001-8]
[10]
R. Goetschel, and W. Voxman, "Elementary fuzzy calculus", Fuzzy Sets Syst., vol. 18, pp. 31-43, 1986.
[http://dx.doi.org/10.1016/0165-0114(86)90026-6]
[11]
O. Kaleva, "Fuzzy differential equations", Fuzzy Sets Syst., vol. 24, pp. 301-317, 1987.
[http://dx.doi.org/10.1016/0165-0114(87)90029-7]
[12]
M. Ma, M. Friedman, and A. Kandel, "Numerical solution of fuzzy differential equations", Fuzzy Sets Syst., vol. 105, pp. 133-138, 1999.
[13]
J.J. Nieto, A. Khastan, and K. Ivaz, "Numerical solution of fuzzy differential equations undergeneralizeddifferentiability", Nonlinear Anal. Hybrid Syst., vol. 3, no. 4, pp. 700-707, 2009.
[http://dx.doi.org/10.1016/j.nahs.2009.06.013]
[14]
M.L. Puri, and D. Ralescu, "Differential for fuzzy function", J. Math. Anal. Appl., vol. 91, pp. 552-558, 1983.
[http://dx.doi.org/10.1016/0022-247X(83)90169-5]
[15]
S. Seikkala, "On the fuzzy initial value problem", Fuzzy Sets and Systems, vol. 24, pp. 319-330, 1987.
[http://dx.doi.org/10.1016/0165-0114(87)90030-3]
[16]
S. Tapaswini, and S. Chakraverty, "Euler-based new solution method for fuzzy initial value problems", Int. J. Artif. Intell. Soft Comput., vol. 4, no. 1, pp. 58-79, 2014.
[17]
C. You, and Y. Hao, "Fuzzy Euler approximation and its local convergence", J. Comput. Appl. Math., vol. 343, no. 1, pp. 55-61, 2018.

Rights & Permissions Print Cite
Article Metrics
30
1
© 2024 Bentham Science Publishers | Privacy Policy