×

Solving fuzzy fractional differential equations by fuzzy Laplace transforms. (English) Zbl 1245.35146

Summary: This paper deals with the solutions of fuzzy fractional differential equations (FFDEs) under Riemann-Liouville H-differentiability by fuzzy Laplace transforms. In order to solve FFDEs, it is necessary to know the fuzzy Laplace transform of the Riemann-Liouville H-derivative of \(f, (^{RL}D_{\alpha^+}^{\beta}f)(x)\). The virtue of \(\mathbf {L}[(^{RL}D_{\alpha^+}^{\beta}f)(x)]\) is that can be written in terms of \(\mathbf {L}[f(x)]\). Moreover, some illustrative examples are solved to show the efficiency and utility of Laplace transforms method.

MSC:

35R13 Fuzzy partial differential equations
35R11 Fractional partial differential equations
44A10 Laplace transform
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Abbasbandy, S.; Shirzadi, A., Homotopy analysis method for multiple solutions of the fractional sturm – liouville problems, Numer algor, 54, 521-532, (2010) · Zbl 1197.65096
[2] Allahviranloo, T.; Abbasbandy, S.; Salahshour, S.; Hakimzadeh, A., A new method for solving fuzzy linear differential equations, Computing, 92, 181-197, (2011) · Zbl 1238.34005
[3] Allahviranloo, T.; Salahshour, S., A new approach for solving first order fuzzy differential equations, Commun comput inform sci, 81, 522-531, (2010), Part 5, Part 7 · Zbl 1206.34008
[4] Allahviranloo, T.; Salahshour, S., Euler method for solving hybrid fuzzy differential equation, Soft comput, 15, 1247-1253, (2011) · Zbl 1242.65127
[5] Allahviranloo, T.; Ahmadi, M.B., Fuzzy Laplace transforms, Soft comput, 14, 235-243, (2010) · Zbl 1187.44001
[6] Allahviranloo T, Salahshour S, Abbasbandy S. Explicit solutions of fractional differential equations with uncertainty. Soft Comput. doi:10.1007/s00500-011-0743-y. · Zbl 1259.34009
[7] Agarwal, R.P.; Lakshmikantham, V.; Nieto, J.J., On the concept of solution for fractional differential equations with uncertainty, Nonlinear anal, 72, 2859-2862, (2010) · Zbl 1188.34005
[8] Arara, A.; Benchohra, M.; Hamidi, N.; Nieto, J.J., Fractional order differential equations on an unbounded domain, Nonlinear anal, 72, 580-586, (2010) · Zbl 1179.26015
[9] Arshad, S.; Lupulescu, V., On the fractional differential equations with uncertainty, Nonlinear anal, 74, 3685-3693, (2011) · Zbl 1219.34004
[10] Babenko YI. Heat and Mass Transfer, Chemia, Leningrad; 1986.
[11] Bagley, R.L., On the fractional order initial value problem and its engineering applications, (), 12-20 · Zbl 0751.73023
[12] Bede, B.; Gal, S.G., Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations, Fuzzy sets syst, 151, 581-599, (2005) · Zbl 1061.26024
[13] Bede, B.; Rudas, I.J.; Bencsik, A.L., First order linear fuzzy differential equations under generalized differentiability, Inform sci, 177, 1648-1662, (2007) · Zbl 1119.34003
[14] Beyer, H.; Kempfle, S., Definition of physically consistent damping laws with fractional derivatives, Zamm, 75, 623-635, (1995) · Zbl 0865.70014
[15] Diethelm, K.; Ford, N.J., Analysis of fractional differential equations, J math anal appl, 265, 229-248, (2002) · Zbl 1014.34003
[16] Friedman, M.; Ma, M.; Kandel, A., Numerical solution of fuzzy differential and integral equations, Fuzzy sets syst, 106, 35-48, (1999) · Zbl 0931.65076
[17] Khastan A, Nieto JJ. Rosana Rodriguez-Lopez, Variation of constant formula for first order fuzzy differential equations. Fuzzy Sets Syst; in press, doi:10.1016/j.fss.2011.02.020. · Zbl 1250.34005
[18] Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J., Theory and applications of fractional differential equations, (2006), Elsevier Science B.V Amesterdam · Zbl 1092.45003
[19] Lakshmikantham, V.; Leela, S.; Vasundhara Devi, J., Theory of fractional dynamic systems, (2009), Cambridge Scientific Pub Cambridge, UK · Zbl 1188.37002
[20] Lakshmikantham, V.; Mohapatra, R.N., Theory of fuzzy differential equations and applications, (2003), Taylor & Fracncis London · Zbl 1072.34001
[21] Lakshmikantham, V.; Vatsala, A.S., Basic theory of fractional differential equations, Nonlinear anal, 69, 2677-2682, (2008) · Zbl 1161.34001
[22] Ma, M.; Friedman, M.; Kandel, A., Numerical solution of fuzzy differential equations, Fuzzy sets syst, 105, 133-138, (1999) · Zbl 0939.65086
[23] Nieto, J.J.; Rodroguez-Lopez, R.; Franco, D., Linear first-order fuzzy differential equations, Int J uncertainty fuzziness knowledge-based syst, 14, 687-709, (2006) · Zbl 1116.34005
[24] Nieto, J.J., Maximum principles for fractional differential equations derived from mittag – leffler functions, Appl math lett, 23, 1248-1251, (2010) · Zbl 1202.34019
[25] Perfilieva, I., Fuzzy transforms: theory and applications, Fuzzy sets syst, 157, 993-1023, (2006) · Zbl 1092.41022
[26] Perfilieva I, De Meyer H, De Baets B, Cauchy problem with fuzzy initial condition and its approximate solution with the help of fuzzy transform, WCCI 2008. In: Proceedings 978-1-4244-1819-0- Hong Kong IEEE Computational Intelligence Society; 2008, p. 2285-2290.
[27] Podlubny, I., Fractional differential equation, (1999), Academic Press San Diego · Zbl 0893.65051
[28] Puri, M.L.; Ralescu, D., Fuzzy random variables, J math anal appl, 114, 409-422, (1986) · Zbl 0592.60004
[29] Salahshour, S.; Haghi, E., Solving fuzzy heat equation by fuzzy Laplace transforms, Commun comput inform sci, 81, 512-521, (2010), Part 5, Part 7 · Zbl 1201.35182
[30] Stefanini, L., A generalization of Hukuhara difference and division for interval and fuzzy arithmetic, Fuzzy sets syst, 161, (2010), p. 1564-158 · Zbl 1188.26019
[31] Stefanini, L.; Bede, B., Generalized Hukuhara differentiability of interval-valued functions and interval differential equations, Nonlinear anal: TMA, 71, 1311-1328, (2009) · Zbl 1188.28002
[32] Wu, H.C., The improper fuzzy Riemann integral and its numerical integration, Inform sci, 111, 109-137, (1999) · Zbl 0934.26014
[33] Xu, J.; Liao, Z.; Hu, Z., A class of linear differential dynamical systems with fuzzy initial condition, Fuzzy sets syst, 158, 2339-2358, (2007) · Zbl 1128.37015
[34] Xu, J.; Liao, Z.; Nieto, J.J., A class of linear differential dynamical systems with fuzzy matrices, J math anal appl, 368, 54-68, (2010) · Zbl 1193.37025
[35] Zhu, Y., Stability analysis of fuzzy linear differential equations, Fuzzy optim decision making, 9, 169-186, (2010) · Zbl 1203.34002
[36] Zimmermann, H.J., Fuzzy set theory and its applications, (1991), Kluwer Academi Publishers Dordrecht · Zbl 0719.04002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.