×

Numerical solution of fractional oscillator equation. (English) Zbl 1243.65090

Summary: We focus on a numerical scheme applied for a fractional oscillator equation in a finite time interval. This type of equation includes a complex form of left- and right-sided fractional derivatives. Its analytical solution is represented by a series of left and right fractional integrals and therefore is difficult in practical calculations. Here we elaborate two numerical schemes being dependent on a fractional order of the equation. The results of numerical calculations are compared with analytical solutions. Then we illustrate convergence and stability of our schemes.

MSC:

65L10 Numerical solution of boundary value problems involving ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34A08 Fractional ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Agrawal, O.P., Formulation of euler – lagrange equations for fractional variational problems, J. math. anal. appl., 272, 368-379, (2002) · Zbl 1070.49013
[2] Agrawal, O.P., A general formulation and solution scheme for fractional optimal control problems, Nonlinear dyn., 38, 323-337, (2004) · Zbl 1121.70019
[3] Agrawal, O.P., Analytical schemes for a new class of fractional differential equations, J. phys. A: math. theor., 40, 5469-5477, (2007) · Zbl 1126.26007
[4] Agrawal, O.P., A general finite element formulation for fractional variational problems, J. math. anal. appl., 337, 1-12, (2008) · Zbl 1123.65059
[5] Agrawal, O.P., Generalized variational problems and euler – lagrange equations, Comput. math. appl., 59, 1852-1864, (2010) · Zbl 1189.49029
[6] Agrawal, O.P.; Muslih, S.I.; Baleanu, D., Generalized variational calculus in terms of multi-parameters fractional derivatives, Commun. nonlinear sci. numer. simulat., 16, 4756-4767, (2011) · Zbl 1236.49030
[7] Almeida, R.; Torres, D.F.M., Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives, Commun. nonlinear sci. numer. simulat., 16, 1490-1500, (2011) · Zbl 1221.49038
[8] Baleanu, D.; Avkar, T., Lagrangians with linear velocities within riemann – liouville fractional derivatives, Nuovo ciemento B, 119, 73-79, (2004)
[9] Baleanu, D.; Muslish, S.I., Lagrangian formulation of classical fields within riemann – liouville fractional derivatives, Phys. scripta, 72, 119-121, (2005) · Zbl 1122.70360
[10] Baleanu, D., Fractional Hamiltonian analysis of irregular systems, Signal process., 86, 2632-2636, (2006) · Zbl 1172.94362
[11] Baleanu, D.; Trujillo, D.J.J., On exact solutions of a class of fractional euler – lagrange equations, Nonlinear dyn., 52, 331-335, (2008) · Zbl 1170.70328
[12] Blaszczyk, T.; Ciesielski, M., Fractional euler – lagrange equations – numerical solutions and applications of reflection operator, Sci. res. instit. math. comput. sci., 2, 9, 17-24, (2010)
[13] Cresson, J., Fractional embedding of differential operators and Lagrangian systems, J. math. phys., 48, 033504, (2007) · Zbl 1137.37322
[14] Hilfer, R., Applications of fractional calculus in physics, (2000), World Scientific Singapore · Zbl 0998.26002
[15] Jarad, F.; Abdeljawad (Maraaba), T.; Baleanu, D., Fractional variational principles with delay within Caputo derivatives, Rep. math. phys., 65, 17-28, (2010) · Zbl 1195.49030
[16] Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J., Theory and applications of fractional differential equations, (2006), Elsevier Amsterdam · Zbl 1092.45003
[17] Klimek, M., Fractional sequential mechanics-models with symmetric fractional derivative, Czech. J. phys., 51, 1348-1354, (2001) · Zbl 1064.70507
[18] Klimek, M., Lagrangian and Hamiltonian fractional sequential mechanics, Czech. J. phys., 52, 1247-1253, (2002) · Zbl 1064.70013
[19] M. Klimek, Solutions of Euler-Lagrange equations in fractional mechanics. In: P. Kielanowski, A. Odzijewicz, M. Schlichenmaier, T. Voronov (Eds.), AIP Conference Proceedings 956. XXVI Workshop on Geometrical Methods in Physics, Bialowieza, 2007, pp. 73-78. · Zbl 1221.70023
[20] M. Klimek, On Solutions of Linear Fractional Differential Equations of a Variational Type, The Publishing Office of the Czestochowa University of Technology, Czestochowa, 2009.
[21] Klimek, M., G-meijer functions series as solutions for certain fractional variational problem on a finite time interval, J. europeen des systemes automatises (JESA), 42, 653-664, (2008)
[22] Klimek, M., Existence-uniqueness result for a certain equation of motion in fractional mechanics, Bull. Pol. acad. sci.: tech. sci., 58, 573-581, (2010) · Zbl 1305.34016
[23] Klimek, M., On analogues of exponential functions for antisymmetric fractional derivatives, Comput. math. appl., 59, 1709-1717, (2010) · Zbl 1189.34012
[24] Leszczynski, J.S., Using the fractional interaction law to model the impact dynamics of multiparticle collisions in arbitrary form, Phys. rev. E, 70, 051315-1-051315-15, (2004)
[25] J.S. Leszczynski, An Introduction to Fractional Mechanics, Monograph No. 198, Publishing Office of Czestochowa University of Technology, Czestochowa, 2011.
[26] Leszczynski, J.S.; Blaszczyk, T., Modeling the transition between stable and unstable operation while emptying a silo, Granular matter, 13, 429-438, (2011)
[27] Magin, R.L., Fractional calculus in bioengineering, (2006), Begell House Inc Redding
[28] Mainardi, F.; Raberto, M.; Gorenflo, R.; Scalas, E., Fractional calculus and continuous-time finance II: the waiting-time distribution, Physica A, 287, 468-481, (2000)
[29] Muslih, S.I.; Baleanu, D., Hamiltonian formulation of systems with linear velocities within riemannliouville fractional derivatives, J. math. anal. appl., 304, 599-603, (2005) · Zbl 1149.70320
[30] T. Odzijewicz, A.B. Malinowska, D.F.M. Torres, Fractional variational calculus with classical and combined Caputo derivatives, Nonlinear Anal.: TMA, doi:10.1016/j.na.2011.01.010. · Zbl 1236.49043
[31] Podlubny, I., Fractional differential equations, (1999), Academic Press San Diego · Zbl 0918.34010
[32] Riewe, F., Nonconservative Lagrangian and Hamiltonian mechanics, Phys. rev. E, 53, 1890-1899, (1996)
[33] Riewe, F., Mechanics with fractional derivatives, Phys. rev. E, 55, 3581-3592, (1997)
[34] Samko, S.G.; Kilbas, A.A.; Marichev, O.I., Fractional integrals and derivatives, Theory and applications, (1993), Gordon and Breach Amsterdam · Zbl 0818.26003
[35] Scalas, E.; Gorenflo, R.; Mainardi, F., Fractional calculus and continuous time finance, Physica A, 284, 376-384, (2000)
[36] Youcef, F., Computation of fractional order derivative and integral via power series expansion and signal modelling, Nonlinear dyn., 46, 1-15, (2006) · Zbl 1170.94311
[37] Zaslavsky, G., Hamiltonian chaos and fractional dynamics, (2005), Oxford University Press Oxford · Zbl 1083.37002
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.