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Numerical solution of non-homogenous fractional oscillator equation in integral form

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Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
In this paper, a non-homogenous fractional oscillator equation in finite time interval is considered. The fractional equation with derivatives of order α ∈ (0, 1] is transformed into its corresponding integral form. Next, a numerical solution of the integral form of the considered equation is presented. In the final part of this paper, some examples of numerical solutions of the considered equation are shown.
Rocznik
Strony
959--968
Opis fizyczny
Bibliogr. 32 poz., rys., tab.
Twórcy
  • Czestochowa University of Technology, Institute of Computer and Information Sciences, Częstochowa, Poland
  • Czestochowa University of Technology, Institute of Mathematics, Częstochowa, Poland
Bibliografia
  • 1. Agrawal O.P., 2002, Formulation of Euler-Lagrange equations for fractional variational problems, Journal of Mathematical Analysis and Applications, 272, 368-379
  • 2. Agrawal O.P., Muslih S.I., Baleanu D., 2011, Generalized variational calculus in terms of multi-parameters fractional derivatives, Communications in Nonlinear Science and Numerical Simulation, 16, 4756-4767
  • 3. Almeida R., Malinowska A.B., 2012, Generalized transversality conditions in fractional calculus of variations, Communications in Nonlinear Science and Numerical Simulation, 18, 3, 443-452
  • 4. Atanackovic T.M., Pilipovic S., Stankovic B., Zorica D., 2014, Fractional Calculus with Applications in Mechanics: Vibrations and Diffusion Processes, ISTE-Wiley, London
  • 5. Baleanu D., Trujillo J.J., 2008, On exact solutions of a class of fractional Euler-Lagrange equations, Nonlinear Dynamics, 52, 331-335
  • 6. Baleanu D., Asad J.H., Petras I., 2014, Fractional Bateman-Feshbach Tikochinsky oscillator, Communications in Theoretical Physics, 61, 221-225
  • 7. Baleanu D., Diethelm K., Scalas E., Trujillo J.J., 2012, Fractional Calculus Models and Numerical Methods, World Scientific, Singapore
  • 8. Blaszczyk T., Ciesielski M., 2014, Numerical solution of fractional Sturm-Liouville equation in integral form, Fractional Calculus and Applied Analysis, 17, 307-320
  • 9. Blaszczyk T., Ciesielski M., 2015a, Fractional oscillator equation: analytical solution and algorithm for its approximate computation, Journal of Vibration and Control (in print), http://dx.doi.org/10.1177/1077546314566836
  • 10. Blaszczyk T., Ciesielski M., 2015b, Fractional oscillator equation – transformation into integral equation and numerical solution, Applied Mathematics and Computation, 257, 428-435
  • 11. Blaszczyk T., Ciesielski M., Klimek M., Leszczynski J., 2011, Numerical solution of fractional oscillator equation, Applied Mathematics and Computation, 218, 2480-2488
  • 12. Bourdin L., Cresson J., Greff I., Inizan P., 2013, Variational integrator for fractional EulerLagrange equations, Applied Numerical Mathematics, 71, 14-23
  • 13. Ciesielski M., Leszczynski J., 2006, Numerical solutions to boundary value problem for anomalous diffusion equation with Riesz-Feller fractional operator, Journal of Theoretical and Applied Mechanics, 44, 2, 393-403
  • 14. Hilfer R., 2000, Applications of Fractional Calculus in Physics, World Scientific, Singapore
  • 15. Katsikadelis J.T., 2012, Nonlinear dynamic analysis of viscoelastic membranes described with fractional differential models, Journal of Theoretical and Applied Mechanics, 50, 3, 743-753
  • 16. Kilbas A.A., Srivastava H.M., Trujillo J.J., 2006, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam
  • 17. Klimek M., 2001, Fractional sequential mechanics – models with symmetric fractional derivative, Czechoslovak Journal of Physics, 51, 1348-1354
  • 18. Klimek M., 2009, On Solutions of Linear Fractional Differential Equations of a Variational Type, Publishing Office of Czestochowa University of Technology, Czestochowa
  • 19. Klimek M., Odzijewicz T., Malinowska A.B., 2014, Variational methods for the fractional Sturm-Liouville problem, Journal of Mathematical Analysis and Applications., 416, 402-426
  • 20. Leszczynski J.S., 2011, An Introduction to Fractional Mechanics, Publishing Office of Czestochowa University of Technology, Czestochowa
  • 21. Magin R.L., 2006, Fractional Calculus in Bioengineering, Begell House Inc, Redding
  • 22. Malinowska A.B., Torres D.F.M., 2012, Introduction to the Fractional Calculus of Variations, Imperial College Press, London
  • 23. Odzijewicz T., Malinowska A.B., Torres D.F.M., 2012, Fractional variational calculus with classical and combined Caputo derivatives, Nonlinear Analysis, Theory, Methods and Applications, 75, 1507-1515
  • 24. Oldham K.B., Spanier J., 1974, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, Academic Press, San Diego
  • 25. Podlubny I., 1999, Fractional Differential Equations, Academic Press, San Diego
  • 26. Pooseh S., Almeida R., Torres D.F.M., 2013, Discrete direct methods in the fractional calculus of variations, Computational and Applied Mathematics, 66, 668-676
  • 27. Press W.H., Teukolsky S.A., Vetterling W.T., Flannery B.P., 2007, Numerical Recipes: The Art of Scientific Computing (3rd ed.), Cambridge University Press, New York
  • 28. Riewe F., 1996, Nonconservative Lagrangian and Hamiltonian mechanics, Physical Review E, 53, 1890-1899
  • 29. Sumelka W., 2014, A note on non-associated Drucker-Prager plastic flow in terms of fractional calculus, Journal of Theoretical and Applied Mechanics, 52, 2, 571-574
  • 30. Sumelka W., Blaszczyk T., 2014, Fractional continua for linear elasticity, Archives of Mechanics, 66, 3, 147-172
  • 31. Xu Y., Agrawal O.P., 2014, Models and numerical solutions of generalized oscillator equations, Journal of Vibration and Acoustics, 136, 051005, 7pp
  • 32. Zhang Y., Chen L., Reeves D.M., Sun H.G., 2014, A fractional-order temperedstable continuity model to capture surface water runoff, Journal of Vibration and Control, http://dx.doi.org//10.1177/1077546314557554
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-b0a7f7e9-37d9-4cd5-bf4f-2159f03eba38
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