Necessary Conditions to A Fractional Variational Problem
Abstract
The fractional variational calculus is a recent fifield, where classical variational problems are considered, but in the presence of fractional derivatives. Since there are several defifinitions of fractional derivatives, it is logical to think of different types of optimality conditions. For this reason, in order to solve fractional variational problems, two theorems of necessary conditions are well known: an Euler-Lagrange equation which involves Caputo and Riemann-Liouville fractional derivatives, and other Euler-Lagrange equation that involves only Caputo derivatives. However, it is undecided which of these two methods is convenient to work with. In this article, we make a comparison solving a particular fractional variational problem with both methods to obtain some conclusions about which one gives the optimal solution.References
O.P. Agrawal, Formulation of Euler-Lagrange equations for fractional variational problems, ELSEVIER Journal of Mathematical Analysis and Applications, vol. 272, no. 1, pp. 368–379, 2002.
O.P. Agrawal, Generalized Euler-Lagrange equations and transversality conditions for FVPs in terms of Caputo derivative, SAGE Journal of Vibration and Control, vol. 13(9-10), pp. 1217–1237, 2007.
R. Almeida, R. Ferreira and D. Torres, Isoperimetric problems of the calculus of variations with fractional derivatives, ELSEVIER Acta Mathematica Scientia, vol. 32, no. 2, pp. 619–630, 2012.
R. Almeida, A. Malinowska and D. Torres, Fractional Euler-Lagrange differential equation via Caputo derivatives, SPRINGER Fractional Dynamics and Control,Part 2, pp. 109-118, 2012.
R. Almeida, A. Malinowska and D. Torres, A fractional calculus of variations for multiple integrals with application to vibrating string, AIP Journal of Mathematical Physics 51, no. 3, 033503, pp. 12, 2010.
R. Almeida and D. Torres, Neccesary and sufficient conditions for the fractional calculus of variations with Caputo derivatives, ELSEVIER Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 3, pp. 1490–1500, 2011.
P. Antunes and R. Ferreira, Analysis of a class of boundary value problems depending on left and right Caputo fractional derivatives, ELSEVIER Communications in Nonlinear Science and Numerical Simulation, vol. 48, pp. 398–413, 2017.
D. Baleanu, K. Diethelm, E. Scalas and J. Trujillo, Fractional calculus: models and numerical methods, World Scientific, vol. 3, 2012.
D. Baleanu and J. Trujillo, On exact solutions of a class of fractional Euler–Lagrange equations, SPRINGER Nonlinear Dynamics, vol. 52, no. 4, pp. 331–335, 2008.
M. Barrios and G. Reyero, An Euler-Lagrange equation only depending on derivatives of Caputo for fractional variational problems with classical derivatives, IAPRESS Statistics, Optimization & Information Computing, vol. 8, no. 2, pp. 590–601, 2020.
M. Barrios, D. Bravo and G. Reyero, Estudio de un modelo de orden fraccionario de dos especies en competencia, MECOM Mecnica Computacional, vol. XXXVIII, pp. 1407–1416, 2021.
M. Barrios, G. Reyero and M. Tidball, Harvest managment problem with a fractional logistic equation, AKJOURNALS
Mathematica Pannonica New Series, vol. 27 /NS 1/, no. 1, pp. 21–34, 2021.
N. Bastos, Calculus of variations involving Caputo-Fabrizio fractional differentiation, IAPRESS Statistics, Optimization and Information Computing, vol. 6, pp. 12–21, 2018.
N. Bastos, R. Ferreira and D. Torres, Neccesary optimality conditions for fractional difference problems of the calculus of variations, AIMS Discrete and Continuous Dynamical Systems, vol. 29, no. 2, pp. 417–437, 2011.
T. Blaszczyk and Ciesielski, Numerical solution of Euler-Lagrange equation with Caputo derivatives, CUP Advances in Applied Mathematics and Mechanics, vol. 9, no. 1, pp. 173–185, 2017.
M. Caputo, Linear models of dissipation whose Q is almost frequency independentII, BLACKWELL PUBLISHING LTD OXFORD Geophysical Journal International, vol. 13, no. 5, pp. 529–539, 1967.
M. Caputo and F. Mainardi, Linear models of dissipation in anelastic solids, SPRINGER La Rivista del Nuovo Cimento, vol. 1, pp. 161–198, 1971.
K. Cassel, Variational methods with applications in science and engineering, CUP, 2013.
K. Diethelm, The analysis of fractional differential equations, SPRINGER, 2004.
A. Erdlyi, W. Magnus, F. Oberhettinge and F. Tricomi, Higher transcendental functions, Vol. I, MCGRAW-HILL, New York, vol. 8, 1953.
A. Ferrari and E. Santillan Marcus, Study of a fractional-order model for HIV infection of CD4+ T-cells with treatment, Journal of Fractional Calculus and Applications, vol. 11, no. 2, pp. 12–22, 2020.
R. Herrmann, Fractional Calculus: an Introduction for Physicist, WORLD SCIENTIFIC, Singapore, 2011.
R. Hilfer, Applications of Fractional Calculus in Physics, WORLD SCIENTIFIC, River Edge, NJ, 2000.
R. Kamocki, Variational Methods for a Fractional Dirichlet Problem Involving Jumaries Derivative, HINDAWI, Mathematical Problems in Engineering, 2015.
A. Kilbas, H. Srivastava and J. Trujillo, Theory and Applications of Fractional Differential Equations, ELSEVIER, Amsterdan,
M. Lazo and D. Torres, The DuBois-Reymond fundamental lemma of the fractional calculus of variations and an Euler-Lagrange equation involving only derivatives of Caputo, SPRINGER Journal of Optimization Theory and Applications, vol. 156, pp. 56–67, 2013.
C. Li and F. Zeng, Numerical methods for fractional calculus, CRC Press, vol. 24, 2015.
A. Malinowska, T. Odzijewicz and D. Torres, Advances methods in the fractional calculus of variations, SPRINGER, 2015.
A. Malinowska and D. Torres, Generalized natural boundary conditions for fractional variational problems in terms of the Caputo derivative, ELSEVIER Computers & Mathematics with Applications, vol. 59, no. 9, pp. 3110–3116, 2010.
A. Malinowska and D. Torres, Introduction to the Fractional Calculus of Variations, IMPERIAL COLLEGE PRESS/WORLD
SCIENTIFIC, London, Singapore, 2012.
I. Malmin, A general framework for optimal control of fractional nonlinear delay systems by wavelets, IAPRESS Statistics,
Optimization & Information Computing, vol. 8, no. 4, pp. 858–875, 2020.
F. Mohammadi, L. Moradi and D. Conte, Discrete chebyshev polynomials for solving fractional variational problems, IAPRESS Statistics, Optimization & Information Computing, vol. 9, no. 3, pp. 502–515, 2021.
T. Odzijewicz and D. Torres, Calculus of variations with fractional and classical derivatives, Proceedings of FDA’10, The 4th IFAC Workshop on Fractional Differentiation and its Applications, Badajoz, Spain, article no. FDA10–076, pp. 5, 2010.
K. Oldham and J. Spanier, The Fractional Calculus, ACADEMIC PRESS, New York, 1974.
I. Podlubny, Fractional differential equations. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some their applications, ACADEMIC PRESS, San Diego-New York- London, 1999.
N. Rangaig, New aspects on the fractional Euler-Lagrange equation with non-singular kernels, Journal of Applied Mathematics and Computational Mechanics, vol. 19, no. 4, 2020.
B. Van Brunt, The calculus of variations, SPRINGER, 2004.
S. Xiao and P. Xia, Variational calculus method for passive earth pressure on rigid retaining walls with strip surcharge on backfills., Applied Mathematical Modelling, vol. 83, pp. 526–551, 2020.
M. Zelikin, Control theory and optimization I: Homogeneous spaces and the Riccati equation in the calculus of variations, SPRINGER Science & Business Media, vol. 86, 2013.
M. Zelikin and V. Borisov, Theory of chattering control: with applications to astronautics, robotics, economics, and engineering, SPRINGER Science & Business Media, vol. 86, 2012.
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