Abstract
In this paper we derive a general solution for a class of nonlinear sequential fractional differential equations (SFDEs) with Riemann -Liouville (R-L) derivatives of arbitrary order. The solution of such an equation exists in arbitrary interval (0,b], provided nonlinear term obeys the respective Lipschitz condition. We prove that each pair of stationary functions of the corresponding R-L derivatives leads to a unique solution in the weighted continuous functions space.
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References
Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley & Sons, New York (1993)
Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)
Kilbas, A.A., Srivastawa, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
Klimek, M.: On Solutions of Linear Fractional Differential Equations of a Variational Type. The Publishing Office of the Czestochowa University of Technology, Czestochowa (2009)
Lakshmikantham, V., Leela, S., Vasundhara Devi, J.: Theory of Fractional Dynamic Systems. Cambridge Scientific Publishers, Cambridge (2009)
Kilbas, A.A., Trujillo, J.J.: Differential equation of fractional order: methods, results and problems. I. Appl. Anal. 78(1-2), 153–192 (2001)
Kilbas, A.A., Trujillo, J.J.: Differential equation of fractional order: methods, results and problems. II. Appl. Anal. 81(2), 435–493 (2002)
Diethelm, K.: The Analysis of Fractional Differential Equations. Springer, Berlin (2010)
Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J.J.: Fractional Calculus Models and Numerical Methods (Series on Complexity Nonlinearity and Chaos). World Scientific, Singapore (2012)
Kosmatov, N.: Integral equations and initial value problems for nonlinear differential equations of fractional order. Non. Anal. TMA 2009 70(7), 2521–2529 (2009)
Rehman, M., Khan, R.A.: Existence and uniqueness of solutions for multi-point boundary value problems for fractional differential equations. Appl. Math. Lett. 23(9), 1038–1044 (2010)
Deng, J., Ma, L.: Existence and uniqueness of solutions of initial value problems for nonlinear fractional differential equations. Appl. Math. Lett. 23(6), 676–680 (2010)
Klimek, M.: On contraction principle applied to nonlinear fractional differential equations with derivatives of order α ∈ (0,1), vol. 95, pp. 325–338. Banach Center Publ. (2011)
Klimek, M.: Sequential fractional differential equations with Hadamard derivative. Commun. Nonlinear Sci. Numer. Simulat. 16(12), 4689–4697 (2011)
Klimek, M., Błasik, M.: On application of contraction principle to solve two-term fractional differential equations. Acta Mech. Automatica 5(2), 5–10 (2011)
Klimek, M., Błasik, M.: Existence-uniqueness result for nonlinear two-term sequential FDE. In: Bernardini, D., Rega, G., Romeo, F. (eds.) Proceedings of the 7th European Nonlinear Dynamics Conference (ENOC 2011), Rome Italy, July 24-29 (2011), doi:10.3267/ENOC2011Rome
Klimek, M., Błasik, M.: Existence-uniqueness of solution for a class of nonlinear sequential differential equations of fractional order. Cent. Eur. J. Math. 10(6), 1981–1994 (2012)
Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Gordon and Breach, Amsterdam (1993)
Bielecki, A.: Une remarque sur la methode de Banach-Cacciopoli-Tikhonov dans la theorie des equations differentielles ordinaires. Bull. Acad. Polon. Sci. Cl. III(4), 261–264 (1956)
El-Raheem, Z.F.A.: Modification of the application of a contraction mapping method on a class of fractional differential equations. Appl. Math. Comput. 137(2-3), 371–374 (2003)
Baleanu, D., Mustafa, O.G.: On the global existence of solutions to a class of fractional differential equations. Comput. Math. Appl. 59(5), 1835–1841 (2010)
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Błasik, M., Klimek, M. (2013). Exact Solution of Two-Term Nonlinear Fractional Differential Equation with Sequential Riemann-Liouville Derivatives. In: Mitkowski, W., Kacprzyk, J., Baranowski, J. (eds) Advances in the Theory and Applications of Non-integer Order Systems. Lecture Notes in Electrical Engineering, vol 257. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00933-9_14
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DOI: https://doi.org/10.1007/978-3-319-00933-9_14
Publisher Name: Springer, Heidelberg
Print ISBN: 978-3-319-00932-2
Online ISBN: 978-3-319-00933-9
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