Volume 1, Issue 4, November 2016, Page: 42-49
Partial Averaging of Fuzzy Hyperbolic Differential Inclusions
Tatyana Alexandrovna Komleva, Department of Mathematics, Odessa State Academy of Civil Engineering and Architecture, Odessa, Ukraine
Irina Vladimirovna Molchanyuk, Department of Applied Mathematics, Odessa State Academy of Civil Engineering and Architecture, Odessa, Ukraine
Andrej Viktorovich Plotnikov, Department of Applied Mathematics, Odessa State Academy of Civil Engineering and Architecture, Odessa, Ukraine
Liliya Ivanovna Plotnikova, Department of Mathematics, Odessa National Polytechnic University, Odessa, Ukraine
Received: Sep. 19, 2016;       Accepted: Sep. 28, 2016;       Published: Oct. 19, 2016
DOI: 10.11648/j.ijssam.20160104.12      View  2127      Downloads  94
In this article, we considered the fuzzy hyperbolic differential inclusions (fuzzy Darboux problem), introduced the concept of R-solution and proved the existence of such a solution. Also the substantiation of a possibility of application of partial averaging method for hyperbolic differential inclusions with the fuzzy right-hand side with the small parameters is considered.
Hyperbolic Differential Inclusion, Fuzzy, Averaging, R-solution
To cite this article
Tatyana Alexandrovna Komleva, Irina Vladimirovna Molchanyuk, Andrej Viktorovich Plotnikov, Liliya Ivanovna Plotnikova, Partial Averaging of Fuzzy Hyperbolic Differential Inclusions, International Journal of Systems Science and Applied Mathematics. Vol. 1, No. 4, 2016, pp. 42-49. doi: 10.11648/j.ijssam.20160104.12
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S. Abbasbandy, T.A. Viranloo, O. Lopez-Pouso, and J. J. Nieto, “Numerical methods for fuzzy differential inclusions,” Computers & Mathematics with Applications, vol. 48, pp. 1633-1641, 2004. http://dx.doi.org/10.1016/j.camwa.2004.03.009.
N. A. Abdul Rahman, and M.Z. Ahmad, “Applications of the Fuzzy Sumudu Transform for the Solution of First Order Fuzzy Differential Equations,” Entropy, vol. 17, pp. 4582-4601, 2015.
A. Arara, M. Benchohra, S. K. Ntouyas, and A. Ouahab, “Fuzzy solutions for hyperbolic partial differential equations,” Int. J. Appl. Math. Sci., vol. 2, no. 2, pp. 181-195, 2005.
R.P. Agarwal, D. O'Regan, and V. Lakshmikantham, “A stacking theorem approach for fuzzy differential equations,” Nonlinear Anal., vol. 55, pp. 299-312, 2003, http://dx.doi.org/10.1016/S0362-546X(03)00241-4.
R. P. Agarwal, D. O'Regan, and V. Lakshmikantham, “Maximal solutions and existence theory for fuzzy differential and integral equations”, J. Appl. Anal., vol. 11, no. 2, pp. 171-186, 2005, http://dx.doi.org/10.1515/JAA.2005.171.
J.-P. Aubin, “Fuzzy differential inclusions,” Probl. Control Inf. Theory, vol. 19, no. 1, pp. 55-67, 1990.
V. A. Baidosov, “Differential inclusions with fuzzy right-hand side,” Sov. Math., vol. 40, no. 3, pp. 567-569, 1990.
V. A. Baidosov, “Fuzzy differential inclusions”, J. of Appl. Math. and Mechan., vol. 54, no. 1, pp. 8-13, 1990, http://dx.doi.org/10.1016/0021-8928(90)90080-T.
A. M. Bertone, R. M. Jafelice, L. C. de Barros, and R. C. Bassanezi, “On fuzzy solutions for partial differential equations,” Fuzzy Sets Syst., vol. 219, pp. 68-80, 2013, http://dx.doi.org/10.1016/j.fss.2012.12.002.
N. N. Bogoliubov, and Yu. A. Mitropolsky, Asymptotic methods in the theory of non-linear oscillations. New York, Gordon and Breach, 1961.
J. J. Buckley, and Th. Feuring, “Introduction to fuzzy partial differential equations,” Fuzzy analysis and related topics (Prague, 1997). Fuzzy Sets Syst., vol. 105, no. 2, pp. 241-248, 1999, http://dx.doi.org/10.1016/S0165-0114(98)00323-6.
A. Cernea, “On the set of solutions of some nonconvex nonclosed hyperbolic differential inclusions,” Czechoslovak Math. J., vol. 52, no. 1, pp. 215-224, 2002, http://dx.doi.org/10.1023/A:1021787808233.
Y. Y. Chen, Y.T. Chang, and B.S. Chen, “Fuzzy Solutions to Partial Differential Equations: Adaptive Approach,” IEEE Transactions on Fuzzy Systems, vol. 17, no. 1, pp. 116-127, 2009.
F. S. De Blasi, and J. Myjak, “On the structure of the set of solutions of the Darboux problem of hyperbolic equations,” Proc. Edinburgh Math. Soc. (Series 2), vol. 29, no. 1, pp. 7-14, 1986, http://dx.doi.org/10.1017/S0013091500017351.
F. S. De Blasi, and J. Myjak, “On the set of solutions of a differential inclusion,” Bull. Inst. Math. Acad. Sinica, vol. 14, no. 3, pp. 271-275, 1986.
F. S. De Blasi, G. Pianigiani, and V. Staicu, “On the solution sets of some nonconvex hyperbolic differential inclusions,” Czechoslovak Math. J., vol. 45, no. 120:1, pp. 107-116, 1995.
D. Dutta Majumder, and K.K. Majumdar, “Complexity analysis, uncertainty management and fuzzy dynamical systems: a cybernetic approach,” Kybernetes, vol. 33, no. 7, pp. 1143-1184, 2004, http://dx.doi.org/10.1108/03684920410534489.
Fuzzy partial differential equations and relational equations. Reservoir characterization and modeling. Edited by Masoud Nikravesh, Lotfi A. Zadeh and Victor Korotkikh in Studies in Fuzziness and Soft Computing. vol. 142. Springer-Verlag, Berlin, 2004.
M. Guo, X. Xue, and R. Li, “Impulsive functional differential inclusions and fuzzy population models,” Fuzzy Sets Syst., vol. 138, pp. 601-615, 2003, http://dx.doi.org/10.1016/S0165-0114(02)00522-5.
E. Hüllermeier, “An approach to modelling and simulation of uncertain dynamical system,” Int. J. Uncertain. Fuzziness Knowl.-Based Syst., vol. 5, no. 2, pp. 117-137, 1997, http://dx.doi.org/10.1142/S0218488597000117.
E. Hüllermeier, “Numerical methods for Fuzzy Initial Value problems,” Int. J. Uncertain. Fuzziness Knowl.-Based Syst., vol. 7, no. 5, pp. 439-461, 1999, http://dx.doi.org/10.1142/S0218488599000404.
R. M. Jafelice, C. G. Almeida, J. F. Meyer, and H. L. Vasconcelos, “Fuzzy parameter in a partial differential equation model for population dispersal of leaf-cutting ants,” Nonlinear Anal. Real World Appl., vol. 12, no. 6, pp. 3397-3412, 2011, http://dx.doi.org/10.1016/j.nonrwa.2011.06.003.
M. Kiselevich, “A Bogolyubov type theorem for hyperbolic equations,” Ukrainian Math. J., vol. 22, pp. 3, pp. 326-331, 1970, http://dx.doi.org/10.1007/BF01088955.
D.G. Korenevskii, “On the principle of averaging for second-order hyperbolic equations with functionally perturbed argument,” Ukrainian Math. J., vol. 23, no. 2, pp. 125-132, 1971, http://dx.doi.org/10.1007/BF01086602.
N. M. Krylov, and N. N. Bogoliubov, Introduction to nonlinear mechanics. Princeton, Princeton University Press, 1947.
V. Lakshmikantham, T. Granna Bhaskar, and J. Vasundhara Devi, Theory of set differential equations in metric spaces. Cambridge, UK, Cambridge Scientific Publishers, 2006.
V. Lakshmikantham, and R. N. Mohapatra, Theory of fuzzy differential equations and inclusions. London, UK, Taylor & Francis, 2003.
A. Lomtatidze, and J. Šremr, “Caratheodory solutions to a hyperbolic differential inequality with a non-positive coefficient and delayed arguments,” Bound. Value Probl., vol. 2014:52, pp. 1-13, 2014, http://dx.doi.org/10.1186/1687-2770-2014-52.
V. L. Lažar, “Ulam-Hyers stability for partial differential inclusions,” Electron. J. Qual. Theory Differ. Equ., vol. 21, pp. 1-19, 2012, http://dx.doi.org/10.14232/ejqtde.2012.1.21.
H. V. Long, N. T. K. Son, N. T. M. Ha, and L. H. Son, “The existence and uniqueness of fuzzy solutions for hyperbolic partial differential equations,” Fuzzy Optimization and Decision Making., vol. 13, no. 4, pp. 435-462, 2014, http://dx.doi.org/10.1007/s10700-014-9186-0.
C. V. Negoita, and D.A. Ralescu, Application of fuzzy sets to systems analysis. New York, USA, Wiley, 1975.
Norazrizal Aswad Abdul Rahman, and Muhammad Zaini Ahmad, “Fuzzy Sumudu transform for solving fuzzy partial differential equations,” J. Nonlinear Sci. Appl., vol. 9, pp. 3226-3239, 2016.
Nayyar Mehmood, and Akbar Azam, “Existence Results for Fuzzy Partial Differential Inclusions,” Journal of Function Spaces, Article ID 6759294, 8 pages, 2016, http://dx.doi.org/10.1155/2016/6759294.
A. I. Panasyuk, and V.I. Panasyuk, “An equation generated by a differential inclusion,” Mathematical notes of the Academy of Sciences of the USSR, vol. 27, no. 3, pp. 213-218, 1980, http://dx.doi.org/10.1007/BF01140170.
U. M. Pirzada, and D.C. Vakaskar, “Solution of fuzzy heat equations using Adomian Decomposition method,” Int. J. Adv. Appl. Math. and Mech., vol. 3, no. 1, pp. 87-91, 2015.
A. V. Plotnikov, “A procedure of complete averaging for fuzzy differential inclusions on a finite segment,” Ukrainian Math. J., vol. 67, no. 3, pp. 421-430, 2015, http://dx.doi.org/10.1007/s11253-015-1090-4.
A. V. Plotnikov, and T. A. Komleva, “The partial averaging of fuzzy differential inclusions on finite interval,” International Journal of Differential Equations, Article ID 307941, 5 pages, 2014, http://dx.doi.org/10.1155/2014/307941.
A. V. Plotnikov, and T. A. Komleva, “The Averaging Of Fuzzy Linear Differential Inclusions On Finite Interval,” Dynamics of Continuous, Discrete and Impulsive Systems, Series B: Applications & Algorithms, vol. 23, no. 1, pp. 1-9, 2016.
A. V. Plotnikov, T. A. Komleva, and L.I. Plotnikova, “The partial averaging of differential inclusions with fuzzy right-hand side,” J. Adv. Res. Dyn. Control Syst., vol. 2, no. 2, pp. 26-34, 2010.
V. A. Plotnikov, A. V. Plotnikov, and A. N. Vityuk, Differential equations with a multivalued right-hand side. Asymptotic methods. Odessa, Ukraine, AstroPrint, 1999.
M. L. Puri, and D. A. Ralescu, “Fuzzy random variables,” J. Math. Anal. Appl., vol. 114, pp. 409-422, 1986, http://dx.doi.org/10.1016/0022-247X(86)90093-4.
J. A. Sanders, and F. Verhulst, Averaging methods in nonlinear dynamical systems. Appl. Math. Sci., vol. 59, New York, Springer-Verlag, 1985.
J. Šremr, “Absolutely continuous functions of two variables in the sense of Caratheodory,” Electron. J. Diff. Equ., vol. 2010, no. 154, pp. 1-11, 2010.
V. Staicu, “On a non-convex hyperbolic differential inclusion,” Proc. Edinburgh Math. Soc. (Series 2), vol. 35, no. 3, pp. 375-382, 1992, http://dx.doi.org/10.1017/S0013091500005666.
G. Teodoru, “Approximation of the solution of Darboux problem for the equation ,” Itinerant seminar on functional equations, approximation and convexity (Cluj-Napoca, 1985). Preprint 85-6. Univ. ''Babes-Bolyai'', Cluj-Napoca, pp. 215-220, 1985.
G. Teodoru, “On the Darboux problem for the equation ,” An. Stiint. Univ. Al. I. Cuza Iasi Sect. I a Mat., vol. 32, no. 3, pp. 41-49, 1986.
G. Teodoru, “Continuous selections for multifunctions satisfying the Caratheodory type conditions. The Darboux problem associated to a multivalued equation,” Itinerant Seminar on Functional Equations, Approximation and Convexity (Cluj-Napoca, 1987). Preprint 87-6. Univ. ''Babes-Bolyai'', Cluj-Napoca, pp. 281-286, 1987.
G. Teodoru, “A characterization of the solutions of the Darboux problem for the equation ,” An. Stiint. Univ. Al. I. Cuza Iasi Mat., vol. 33, no. 1, pp. 33-38, 1987.
A. N. Vityuk, “Properties of solutions of hyperbolic differential equations with many-valued right-hand sides,” Mat. Fiz. Nelin. Mekh., vol. 15, pp. 59-62, 1991.
A. N. Vityuk, “Equation of the integral funnel of a partial differential inclusion,” Dokl. Ukr. Akad. Nauk, Ser. A., vol. 9, pp. 19-20, 1992.
A. N. Vityuk, “On solutions of hyperbolic differential inclusions with nonconvex right-hand side,” Ukrainian Math. J., vol. 47, no. 4, pp. 617-621, 1995, http://dx.doi.org/10.1007/BF01056048.
A. N. Vityuk, “On an R-solution generated by a differential inclusion of hyperbolic type,” Differential Equations, vol. 30, no. 10, pp. 1578-1586, 1994.
A. N. Vityuk, “Continuous dependence of the R-solution generated by a differential hyperbolic inclusion on parameters,” Ukrainian Math. J., vol. 47, no. 10, pp. 1625-1631, 1995, http://dx.doi.org/10.1007/BF01060163.
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