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Solution existence for non-autonomous variable-order fractional differential equations. (English) Zbl 1255.34008

Summary: We discuss the existence of the solution for a generalized fractional differential equation with non-autonomous variable order operators. In contrast to constant order fractional calculus, some standard relations including composition and sequential derivative rules do not remain correct under this generalization. Therefore, solving such a generalized fractional differential equation requires a different methodology, essential modifications, and generalizations for the basic concepts such as existence and uniqueness of the solution. The main goal of this paper is the proof of existence for the solution of a variable order fractional differential equation which is achieved by presenting four theorems. It is shown that if Lebesgue measurability, the continuity of the nonlinear term, and the conditions of differintegration operation are satisfied, then a solution for the variable order fractional differential equation exists.

MSC:

34A08 Fractional ordinary differential equations
45J05 Integro-ordinary differential equations
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[1] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations (2006), Elsevier · Zbl 1092.45003
[2] Li, C.; Deng, W., Remarks on fractional derivatives, Applied Mathematics and Computation, 187, 777-784 (2007) · Zbl 1125.26009
[3] Cafagna, D., Fractional calculus: a mathematical tool from the past for the present engineer, IEEE Transactions on Industrial Electronic, 35-40 (2007)
[4] Flandoli, F.; Tudor, C. A., Brownian and fractional Brownian stochastic currents via Malliavin calculus, Journal of Functional Analysis, 258, 279-306 (2010) · Zbl 1196.60100
[5] Jumarie, G., New stochastic fractional models for Malthusian growth, the Poissonian birth process and optimal management of populations, Mathematical and Computer Modelling, 44, 231-254 (2006) · Zbl 1130.92043
[6] Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations (1993), John Wiley and Sons · Zbl 0789.26002
[7] Lyakhov, L. N.; Gots, G., Inversion of the Kipriyanov-radon transform via fractional derivatives in a one dimensional parameter, Journal of Mathematical Sciences, 158, 235-240 (2009) · Zbl 1178.44002
[8] Vityuk, A. N.; Mikhailenko, A. V., On one class of differential equations of fractional order, Nonlinear Oscillations, 11, 307-319 (2008) · Zbl 1277.26013
[9] Sun, HongGuang; Chen, Wen; Chen, YangQuan, Variable-order fractional differential operators in anomalous diffusion modeling, Physica A, 388, 4586-4592 (2009)
[10] Coimbra, C. F.M., Mechanics with variable-order differential operators, Annalen der Physik, 12, 692-703 (2003) · Zbl 1103.26301
[11] Sheng, Hu; Sun, Hongguang; Chen, YangQuan; Qiu, TianShuang, Synthesis of multifractional Gaussian noises based on variable-order fractional operators, Signal Processing, 91, 1645-1650 (2011) · Zbl 1213.94049
[12] Tseng, C. C., Design of variable and adaptive fractional order FIR differentiators, Signal Processing, 86, 2554-2566 (2006) · Zbl 1172.94495
[13] H. Sheng, H. Sun, C. Coopmans, Y.Q. Chen, G.W. Bohannan, Physical experimental study of variable-order fractional integrator and differentiator, in: Proceedings of FDA’10. The 4th IFAC Workshop Fractional Differentiation and its Applications, 2010.; H. Sheng, H. Sun, C. Coopmans, Y.Q. Chen, G.W. Bohannan, Physical experimental study of variable-order fractional integrator and differentiator, in: Proceedings of FDA’10. The 4th IFAC Workshop Fractional Differentiation and its Applications, 2010.
[14] Sun, H. G.; Chen, W.; Wei, H.; Chen, Y. Q., A comparative study of constant-order and variable-order fractional models in characterizing memory property of systems, Eur. Phys. J. Special Topics Perspectives on Fractional Dynamics and Control, 193, 185-192 (2011)
[15] T.T. Hartley, C.F. Lorenzo, Fractional system identification: An approach using continuous order distributions, NASA/TM-1999-209640, 1999.; T.T. Hartley, C.F. Lorenzo, Fractional system identification: An approach using continuous order distributions, NASA/TM-1999-209640, 1999.
[16] Bagley, R. L.; Torvik, P. J., On the existence of the order domain and the solution of distributed order equations, part I, International Journal of Applied Mathematics, 2, 865-882 (2000), part II, 2 (2000) pp. 965-987 · Zbl 1100.34006
[17] Almeida, A.; Samko, S., Fractional and hypersingular operators in variable exponent spaces on metric measure space, Mediterranean Journal of Mathematics, 99, 1-18 (2008)
[18] Rafeiro, H.; Samko, S., On a class of fractional type integral equations in variable exponent spaces, Fractional Calculus and Applied Analysis, 10, 399-421 (2007) · Zbl 1152.45003
[19] Chang, Y. K.; Nieto, J. J., Some new existence results for fractional differential inclusions with boundary conditions, Mathematical and Computer Modelling, 49, 605-609 (2009) · Zbl 1165.34313
[20] Rivero, M.; Rodriguez-Germa, L.; Trujillo, J. J., Linear fractional differential equations with variable coefficients, Applied Mathematics Letters, 21, 892-897 (2008) · Zbl 1152.34305
[21] Muslim, M., Existence and approximation of solutions to fractional differential equations, Mathematical and Computer Modelling, 49, 1164-1172 (2009) · Zbl 1165.34304
[22] Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T., Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on R3, Communications on Pure and Applied Mathematics, 57, 987-1014 (2004) · Zbl 1060.35131
[23] Caffarelli, L.; Silvestre, L., Regularity theory for fully nonlinear integro-differential equations, Communications on Pure and Applied Mathematics, 62, 597-638 (2009) · Zbl 1170.45006
[24] Daftardar-Gejji, V.; Babakhani, A., Analysis of a system of fractional differential equations, Journal of Mathematical Analysis and Applications, 293, 511-522 (2004) · Zbl 1058.34002
[25] Lin, W., Global existence theory and chaos control of fractional differential equations, Journal of Mathematical Analysis and Applications, 332, 709-726 (2007) · Zbl 1113.37016
[26] Chang, Y. K.; Kavitha, V.; Arjunan, M. M., Existence and uniqueness of mild solutions to a semilinear integrodifferential equation of fractional order, Nonlinear Analysis TMA, 71, 5551-5559 (2009) · Zbl 1179.45010
[27] Mophou, G. M., Existence and uniqueness of mild solutions to impulsive fractional differential equations, Nonlinear Analysis TMA, 72, 1604-1615 (2010) · Zbl 1187.34108
[28] Hunter, J. K.; Nachtergaele, B., Applied Analysis (2001), World Scientific · Zbl 0981.46002
[29] Aoun, M.; Malti, R.; Levron, F.; Oustaloup, A., Numerical simulations of fractional systems: an overview of existing methods and improvements, Nonlinear Dynamics, 38, 117-131 (2004) · Zbl 1134.65300
[30] Ferdi, Y., Computation of fractional order derivative and integral via power series expansion and signal modeling, Nonlinear Dynamics, 46, 1-15 (2006) · Zbl 1170.94311
[31] Momani, S.; Odibat, Z., Numerical comparison of methods for solving linear differential equations of fractional order, Chaos, Solitons & Fractals, 31, 1248-1255 (2007) · Zbl 1137.65450
[32] Kumar, P.; Agrawal, O. P., An approximate method for numerical solution of fractional differential equations, Signal Processing, 86, 2602-2610 (2006) · Zbl 1172.94436
[33] Sweilam, N. H.; Khader, M. M.; Al-Bar, R. F., Numerical studies for a multi-order fractional differential equation, Physics Letters A, 371, 26-33 (2007) · Zbl 1209.65116
[34] Diethelm, K.; Ford, N. J.; Freed, A. D., A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynamics, 29, 3-22 (2002) · Zbl 1009.65049
[35] Valério, D.; Costa, J. S., Variable-order fractional derivatives and their numerical approximations, Signal Processing, 91, 470-483 (2011) · Zbl 1203.94060
[36] Hartley, T.; Lorenzo, C. F., Dynamics and control of initialized fractional-order systems, Nonlinear Dynamics, 29, 201-233 (2002) · Zbl 1021.93019
[37] Bartle, R. G., The Elements of Real Analysis (1967), John Wiley · Zbl 0146.28201
[38] Folland, G. B., Real Analysis, Modern Techniques and Their Applications (1999), John Wiley and sons · Zbl 0924.28001
[39] Rudin, W., Real and Complex Analysis (2006), Tata McGraw-Hill
[40] Lorenzo, C. F.; Hartley, T. T., Variable order and distributed order fractional operators, Nonlinear Dynamics, 29, 57-98 (2002) · Zbl 1018.93007
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