Abstract
One of the effective methods to find explicit solutions of differential equations is the method based on the operator representation of solutions. The essence of this method is to construct a series, whose members are the relevant iteration operators acting to some classes of sufficiently smooth functions. This method is widely used in the works of B. Bondarenko for construction of solutions of differential equations of integer order. In this paper, the operator method is applied to construct solutions of linear differential equations with constant coefficients and with Caputo fractional derivatives. Then the fundamental solutions are used to obtain the unique solution of the Cauchy problem, where the initial conditions are given in terms of the unknown function and its derivatives of integer order. Comparison is made with the use of Mikusinski operational calculus for solving similar problems.
Acknowledgement
This work has been partially supported by the Ministry of Higher and Secondary Special Education of Uzbekistan under Research Grant F4-FAF010, FEDER and by Ministerio de Ciencia y Tecnología, Spain, and FEDER, Projects MTM2010-15314 and MTM2013-43014-P, and by the Ministry of Education and Science of the Republic of Kazakhstan through the project 0819/GF4.
We would also like to express our special thanks to the reviewers for their remarks, which considerably improved the content of this paper.
References
1 Yu.I. Babenko, Heat and Mass Transfer. Chemia, Leningrad (1986) (In Russian).Search in Google Scholar
2 R.L. Bagley, On the fractional order initial value problem and its engineering applications. In:Fractional Calculus and Its Applications (Ed. K. Nishimoto), College of Engineering, Nihon University, Tokyo (1990), 12–20.Search in Google Scholar
3 H. Beyer, S. Kempfle, Definition of physically consistent damping laws with fractional derivatives. ZAMM75 (1995), 623–635.10.1002/zamm.19950750820Search in Google Scholar
4 B.A. Bondarenko, Operator Algorithms in Differential Equations. “FAN” Publishers, Tashkent (1984) (In Russian).Search in Google Scholar
5 R. Caponetto, G. Dongola, L. Fortuna, I. Petraš, Fractional Order Systems: Modeling and Control Applications. Ser. on Nonlinear Science 72, World Scientific, Singapore (2010).10.1142/7709Search in Google Scholar
6 J.Fa Cheng, Y.M. Chu, Solution to the linear fractional differential equation using Adomian decomposition method. Mathematical Problems in Engineering2011 (2011), Article ID 587068, 14 p.10.1155/2011/587068Search in Google Scholar
7 K. Diethelm, The Analysis of Fractional Differential Equations. Lecture Notes in Mathematics, Springer, Berlin-Heidelberg (2010).10.1007/978-3-642-14574-2Search in Google Scholar
8 M.M. Dzerbashyan, A.B. Nersesyan, Fractional derivatives and the Cauchy problem for differential equations of fractional order. Izv. Akad. Nauk Armyan. SSR, Ser. Mat.3 (1968), 3–29 (In Russian).Search in Google Scholar
9 R. Hilfer, Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000).10.1142/3779Search in Google Scholar
10 R. Hilfer, Fractional time evolution. In: Applications of Fractional Calculus in Physics (Ed. R. Hilfer), World Scientific, Singapore (2000), 87–130.10.1142/9789812817747_0002Search in Google Scholar
11 R. Hilfer, Experimental evidence for fractional time evolution in glass forming materials, Chem. Phys.284 (2002), 399–408.10.1016/S0301-0104(02)00670-5Search in Google Scholar
12 R. Hilfer, Y. Luchko, Z. Tomovski, Operational method for the solution of fractional differential equations with generalized Riemann-Liouville fractional derivatives. Fract. Calc. Appl. Anal12, No 3 (2009), 299– 318; at http://www.math.bas.bg/ ∼fcaa.http://www.math.bas.bg/∼fcaaSearch in Google Scholar
13 Y. Hu, Y. Luo, Z. Lu, Analytical solution of the linear fractional differential equation by Adomian decomposition method. J. of Computational and Applied Mathematics215 (2008), 220–229.10.1016/j.cam.2007.04.005Search in Google Scholar
14 V.V. Karachik, Method for constructing solutions of linear ordinary differential equations with constant coeffcients. Computational Mathematics and Mathematical Physics52, No 2 (2012), 219–234; DOI: 10.1134/S0965542512020108.10.1134/S0965542512020108Search in Google Scholar
15 A.A. Kilbas, New trends on fractional integral and differential equations. Kazan. Gos. Univ. Uchen. Zap. Ser. Fiz.-Mat. Nauki147, No 1 (2005), 72–106 (In Russian).Search in Google Scholar
16 A.A. Kilbas and S.A. Marzan, Cauchy problem for differential equation with Caputo derivative. Fract. Calc. Appl. Anal.7, No 3 (2004), 297– 321; at http://www.math.bas.bg/ ∼fcaa.http://www.math.bas.bg/∼fcaaSearch in Google Scholar
17 A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier, North Holland (2006).Search in Google Scholar
18 M.H. Kim, G.C. Ri, Hyong-Chol O, Operational method for solving multi-term fractional differential equations with the generalized fractional derivatives. Fract. Calc. Appl. Anal.17, No 1 (2014), 79–95; DOI: 10.2478/s13540-014-0156-6; http://www.degruyter.com/view/j/ fca.2014.17.issue-1/issue-files/fca.2014.17.issue-1.xml.http://www.degruyter.com/view/j/fca.2014.17.issue-1/issue-files/fca.2014.17.issue-1.xmlSearch in Google Scholar
19 Yu. Luchko, Operational method in fractional calculus. Fract. Calc. Appl. Anal.2, No 4 (1999), 463–488; http://www.math.bas.bg/ ∼fcaa.http://www.math.bas.bg/∼fcaaSearch in Google Scholar
20 Yu. Luchko, R. Gorenflo, An operational method for solving fractional differential equations. Acta Mathematica Vietnamica24 (1999), 207– 234.Search in Google Scholar
21 Yu. Luchko, H.M. Srivastava, The exact solution of certain differential equations of fractional order by using operational calculus. Comput. Math. Appl.29 (1995), 73–85.10.1016/0898-1221(95)00031-SSearch in Google Scholar
22 Yu. Luchko, S.B. Yakubovich, An operational method for solving some classes of integro-differential equations. Differential Equations30 (1994), 247–256.Search in Google Scholar
23 C.A. Monje, Y. Chen, B.M. Vinagre, D. Xue, V. Feliu, Fractional– Order Systems and Controls: Fundamentals and Applications. Ser. Advances in Industrial Control, Springer, London (2010).10.1007/978-1-84996-335-0Search in Google Scholar
24 I. Podlubny, Fractional Differential Equations. Academic Press, San Diego (1999).Search in Google Scholar
25 A.V. Pskhu, Initial-value problem for a linear ordinary differential equation of noninteger order. Sbornik: Mathematics202 (2011), 571– 582.10.1070/SM2011v202n04ABEH004156Search in Google Scholar
26 M. Rivero, L. Rodriguez-Germa, J.J. Trujillo, Linear fractional differential equations with variable coeffcients. Appl. Math. Letters21 (2008), 892–897.10.1016/j.aml.2007.09.010Search in Google Scholar
27 J. Sabatier, O.P. Agrawal, J.A. Tenreiro Machado, Advances in Fractional Calculus. Ser. Theoretical Developments and Applications in Phys. and Eng., Springer, Berlin (2007).10.1007/978-1-4020-6042-7Search in Google Scholar
28 S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives, Theory and Applications. Gordon and Breach, Switzerland (1993).Search in Google Scholar
29 D. Shantanu, Functional Fractional Calculus for System Identification and Controls. Springer, Berlin (2008).Search in Google Scholar
30 K.M. Shinaliyev, B.Kh. Turmetov, S.R. Umarov, A fractional operator algorithm method for construction of solutions of fractional order differential equations. Fract. Calc. Appl. Anal.15, No 2 (2012), 267–281; DOI:10.2478/s13540-012-0020-5; http://www.degruyter.com/view/j/ fca.2012.15.issue-2/issue-files/fca.2012.15.issue-2.xml.http://www.degruyter.com/view/j/fca.2012.15.issue-2/issue-files/fca.2012.15.issue-2.xmlSearch in Google Scholar
Please cite to this paper as published in:
Fract. Calc. Appl. Anal., Vol. 19, No 1 (2016), pp. 229–252, DOI: 10.1515/fca-2016-0013
© 2016 Diogenes Co., Sofia
