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Numerical Solution of Euler-Lagrange Equation with Caputo Derivatives

Part of: Hamiltonian and Lagrangian mechanics Integral equations, integral transforms Real functions General theory in ordinary differential equations

Published online by Cambridge University Press:  11 October 2016

Tomasz Blaszczyk  and
Mariusz Ciesielski
Tomasz Blaszczyk*
Affiliation:
Czestochowa University of Technology, Institute of Mathematics, al. Armii Krajowej 21, 42-201 Czestochowa, Poland
Mariusz Ciesielski*
Affiliation:
Czestochowa University of Technology, Institute of Computer and Information Sciences, ul. Dabrowskiego 73, 42-201 Czestochowa, Poland
*
*Corresponding author. Email:[email protected] (T. Blaszczyk), [email protected] (M. Ciesielski)
*Corresponding author. Email:[email protected] (T. Blaszczyk), [email protected] (M. Ciesielski)
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Abstract

In this paper the fractional Euler-Lagrange equation is considered. The fractional equation with the left and right Caputo derivatives of order α ∈ (0,1] is transformed into its corresponding integral form. Next, we present a numerical solution of the integral form of the considered equation. On the basis of numerical results, the convergence of the proposed method is determined. Examples of numerical solutions of this equation are shown in the final part of this paper.

Keywords

Fractional Euler-Lagrange equationfractional integral equationnumerical solutionCaputo derivatives

MSC classification

Secondary: 26A33: Fractional derivatives and integrals 34A08: Fractional differential equations 65R20: Integral equations 70H03: Lagrange's equations
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 9 , Issue 1 , February 2017 , pp. 173 - 185
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Agrawal, O. P., Formulation of Euler-Lagrange equations for fractional variational problems, J. Math. Anal. Appl., 272 (2002), pp. 368–379.Google Scholar
[2] Agrawal, O. P., Hasan, M. M. and Tangpong, X. W., A numerical scheme for a class of parametric problem of fractional variational calculus, J. Comput. Nonlinear Dyn., 7 (2012), pp. 021005-1–021005-6.Google Scholar
[3] Almeida, R. and Malinowska, A. B., Generalized transversality conditions in fractional calculus of variations, Commun. Nonlinear Sci. Numer. Simulat., 18(3) (2012), pp. 443–452.Google Scholar
[4] Atanackovic, T. M., Pilipovic, S., Stankovic, B. and Zorica, D., Fractional Calculus with Applications in Mechanics: Vibrations and Diffusion Processes, ISTE–Wiley, London, 2014.Google Scholar
[5] Baleanu, D. and Trujillo, J. J., On exact solutions of a class of fractional Euler-Lagrange equations, Nonlinear Dyn., 52 (2008), pp. 331–335.Google Scholar
[6] Baleanu, D., Diethelm, K., Scalas, E. and Trujillo, J. J., Fractional Calculus Models and Numerical Methods, World Scientific, Singapore, 2012.CrossRefGoogle Scholar
[7] Baleanu, D., Asad, J. H. and Petras, I., Fractional Bateman–Feshbach Tikochinsky oscillator, Commun. Theor. Phys., 61 (2014), pp. 221–225.Google Scholar
[8] Blaszczyk, T. and Ciesielski, M., Numerical solution of fractional Sturm-Liouville equation in integral form, Fract. Calc. Appl. Anal., 17 (2014), pp. 307–320.Google Scholar
[9] Blaszczyk, T. and Ciesielski, M., Fractional oscillator equation–transformation into integral equation and numerical solution, Appl. Math. Comput., 257 (2015), pp. 428–435.Google Scholar
[10] Blaszczyk, T. and Ciesielski, M., Fractional oscillator equation-analytical solution and algorithm for its approximate computation, J. Vibration Control, 22(8) (2016), pp. 2045–2052.CrossRefGoogle Scholar
[11] Blaszczyk, T., A numerical solution of a fractional oscillator equation in a non-resisting medium with natural boundary conditions, Romanian Reports Phys., 67(2) (2015), pp. 350–358.Google Scholar
[12] Bourdin, L., Cresson, J., Greff, I. and Inizan, P., Variational integrator for fractional Euler-Lagrange equations, Appl. Numer. Math., 71 (2013), pp. 14–23.Google Scholar
[13] Ciesielski, M. and Blaszczyk, T., Numerical solution of non-homogenous fractional oscillator equation in integral form, J. Theoretical Appl. Mech., 53(4) (2015), pp. 959–968.Google Scholar
[14] Dhaigude, D. B. and Birajdar, G. A., Numerical solution of fractional partial differential equations by discrete adomian decomposition method, Adv. Appl. Math. Mech., 6(1) (2014), pp. 107–119.Google Scholar
[15] Durajski, A. P., Phonon-mediated superconductivity in compressed NbH4 compound, Euro. Phys. J. B, 87 (2014), 210.Google Scholar
[16] Durajski, A. P., In uence of hole doping on the superconducting state in graphane, Supercond. Sci. Technol., 28 (2015), Article ID 035002.Google Scholar
[17] Hilfer, R., Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.Google Scholar
[18] Kilbas, A. A., Srivastava, H. M. and Trujillo, J. J., Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.Google Scholar
[19] Klimek, M., Fractional sequential mechanics-models with symmetric fractional derivative, Czech. J. Phys., 51 (2001), pp. 1348–1354.Google Scholar
[20] Kukla, S. and Siedlecka, U., Laplace transform solution of the problem of time-fractional heat conduction in a two-layered slab, J. Appl. Math. Comput. Mech., 14(4) (2015), pp. 105–113.Google Scholar
[21] Lazo, M. J. and Torres, D. F. M., The DuBois-Reymond fundamental lemma of the fractional calculus of variations and an Euler-Lagrange equation involving only derivatives of caputo, J.Optim. Theory Appl., 156 (2013), pp. 56–67.Google Scholar
[22] Li, C. and Deng, W., Remarks on fractional derivatives, Appl. Math. Comput., 187 (2007), pp. 777–784.Google Scholar
[23] Magin, R. L., Fractional Calculus in Bioengineering, Begell House Inc, Redding, 2006.Google Scholar
[24] Malinowska, A. B. and Torres, D. F. M., Introduction to the Fractional Calculus of Variations, Imperial College Press, London, 2012.Google Scholar
[25] Oldham, K. B. and Spanier, J., The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, Academic Press, San Diego, 1974.Google Scholar
[26] Pedas, A. and Tamme, E., Piecewise polynomial collocation for linear boundary value problems of fractional differential equations, J. Comput. Appl. Math., 236 (2012), pp. 3349–3359.CrossRefGoogle Scholar
[27] Podlubny, I., Fractional Differential Equations, Academic Press, San Diego, 1999.Google Scholar
[28] Pooseh, S., Almeida, R. and Torres, D. F. M., A discrete time method to the first variation of fractional order variational functionals, Cent. Euro. J. Phys., 11(10) (2013), pp. 1262–1267.Google Scholar
[29] Pozorska, J. and Pozorski, Z., The influence of the core orthotropy on the wrinkling of sandwich panels, J. Appl. Math. Comput. Mech., 14(4) (2015), pp. 133–138.Google Scholar
[30] Press, W. H., Teukolsky, S. A., Vetterling, W. T., Flannery, B. P., Numerical Recipes: The Art of Scientific Computing (3rd ed.), Cambridge University Press, New York, 2007.Google Scholar
[31] Riewe, F., Nonconservative Lagrangian and Hamiltonian mechanics, Phys. Rev. E, 53 (1996), pp. 1890–1899.Google Scholar
[32] Siedlecki, J., Ciesielski, M. and Blaszczyk, T., Transformation of the second order boundary value problem into integral form-different approaches and a numerical solution, J. Appl. Math. Comput. Mech., 14(3) (2015), pp. 103–108.CrossRefGoogle Scholar
[33] Sumelka, W., Blaszczyk, T. and Liebold, C., Fractional Euler-Bernoulli beams: theory, numerical study and experimental validation, Euro. J. Mech. A Solids, 54 (2015), pp. 243–251.Google Scholar
[34] Wei, H., Chen, W. and Sun, H. G., Homotopy method for parameter determination of solute transport with fractional advection-dispersion equation, CMES: Comput. Model. Eng. Sci., 100(2) (2014), pp. 85–103.Google Scholar
[35] Xu, Y. and Agrawal, O. P., Models and numerical solutions of generalized oscillator equations, J. Vibration Acoustics, 136 (2014), 051005.CrossRefGoogle Scholar
[36] Yang, Y. and Huang, Y., Spectral-collocation methods for fractional pantograph delay-integrodifferential equations, Adv. Math. Phys., (2013), article ID 821327.Google Scholar
[37] Yang, Y., Chen, Y. and Huang, Y., Spectral-collocation method for fractional Fredholm integrodifferential equations, J. Korean Math. Soc., 51(1) (2014), pp. 203–224.CrossRefGoogle Scholar
[38] Zhang, Y., Chen, L., Reeves, D. M. and Sun, H. G., A fractional-order tempered-stable continuity model to capture surface water runoff, J. Vibration Control, 22(8) (2016), pp. 1993–2003.Google Scholar