Solving a Second Order Fuzzy Initial Value Problem Using The Heaviside Function

Year 2016, Volume: 4 , 16 - 25, 13.07.2016

Ömer Akın , Tahir Khaniyev , Selami Bayeğ Burhan Türkşen

Abstract

In this paper, we reformulate the algorithm in [7] to find an analytical expression for -cuts of the solution of the second order nonhomogeneous fuzzy initial value problem with fuzzy initial values and fuzzy forcing terms. Firstly, we apply Zadeh’s Extension Principle to fuzzify the crisp initial value problem. Then, we use the Heaviside function and obtain the analytical form of -cuts of the solution of the fuzzy initial value problem. Finally, we illustrate some examples using the proposed algorithm.

Keywords

Fuzzy initial value problem, fuzzy forcing function, Zadeh’s extension principle, heaviside function, breaking point, structure vector

References

• Akın, Ö., Khaniyev, T., Oruç Ö., Türkşen, I. B., An algorithm for the solution of second order fuzzy initial value problems, Expert Syst. Appl., 40(2013), 953–957.
• Allahviranloo, T., Nejatbakhsh, Y., Shafeei, M., A note on ”Fuzzy dierential equations and the extension principle”, Journal of Information Science, 179(2009), 2049–2051.
• Bahrami, F., Ivaz, K., Khastan, A., New results on multiple solutions for n-th order fuzzy dierential equations under generalized dierentiability, Boundary Value Problems, 2009(2009), Article ID 395714, 13 pages.
• Banerjee, S., Mondal, S. P., Roy, T. K., First Order Linear Homogeneous Ordinary Dierential Equation in Fuzzy Environment, Int. J. Pure Appl. Sci. Technol., 14(2013), 16–26.
• Bede, B., Gal, S., Generalizations of the dierentiability of fuzzy number valued functions with applications to fuzzy dierential equation, Fuzzy Sets and Systems, 151(2005), 581–599.
• Bede, B., Bencsik, A., Rudas, I., First order linear fuzzy dierential equations under generalized dierentiability, Information Sciences, 177(2007), 1648–1662.
• Buckley, J. J., Feuring, T., Fuzzy initial value problem for N-th order linear dierential equations, Fuzzy Sets and Systems, 121(2001), 247–255.
• Byatt,W. J., Kandel, A., Fuzzy dierential equations, In Proceedings of the International Conference on Cybernetics and Society, Tokyo, Japan, 1978.
• Congxin, W., Shiji, S., Existence theorem to the Cauchy problem of fuzzy dierential equations under compactness-type conditions, Inf. Sci., 14(1998), 123–134.
• Ding, Z., Kandel, A., Ma M., Existence of the solutions of fuzzy dierential equations with parameters, Inf. Sci., 99(1997), 205–217.
• Gasilov, N., Amrahov, S. E., Fatullayev, A. G., A geometric approach to solve fuzzy linear systems of dierential equations, Applied Mathematics and Information Sciences, 5(2011), 484–499.
• Gasilov, N., Hashimoglu, I. F., Amrahov, S. E., Fatullayev, A. G., A New Approach to Non-Homogeneous Fuzzy Initial Value Problem, CMES-Computer Modeling in Engineering & Sciences, 85(2012), 367–378.
• Gasilov, N., Fatullayev, A. G., Amrahov, S. E., Khastan, A., A new approach to fuzzy initial value problem, Soft Computing, 18(2014), 217–225.
• H¨ullermeier, E., An approach to modelling and simulation of uncertain dynamical systems, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 5(1997), 117–137.
• Kaleva, O., Fuzzy dierential equations, Fuzzy Sets and Systems, 24(1987), 301–317.
• Oberguggenberger, M., Pittschmann, S., Dierential equations with fuzzy parameters, Mathematical and Computer Modelling of Dynamical Systems, 5(1999), 181–202.
• Puri, M., Ralescu, D., Dierential of fuzzy functions, Journal of Mathematical Analysis and Applications, 91(1983), 552–558.
• Seikkala, S., On the fuzzy initial value problem, Fuzzy Sets Syst., 24(1987), 319–330.
• Zadeh, L. A., Fuzzy sets, Information and Control, 8(1965), 338–353.
• Zarei, H, Kamyad, A. V., Heydari, A. A., Fuzzy modeling and control of HIV infection, Comput. Math. Methods Med., 2012(2012), Article ID 893474, 17 pages.