Sufficient conditions for extremum of fractional variational problems are formulated with the help of Caputo fractional derivatives. The Euler–Lagrange equation is defined in the Caputo sense and Jacobi conditions are derived using this. Again, Wierstrass integral for the considered functional is obtained from the Jacobi conditions and the transversality conditions. Further, using the Taylor’s series expansion with Caputo fractional derivatives in the Wierstrass integral, the Legendre’s sufficient condition for extremum of the fractional variational problem is established. Finally, a suitable counterexample is presented to justify the efficacy of the fresh findings.
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DOI : 10.1051/ro/2022035
Keywords: Caputo fractional derivative, Jacobi conditions, transversality conditions, Wierstrass integral, Legendre’s condition
@article{RO_2022__56_2_637_0,
author = {Pattnaik, Ashapurna and Padhan, Saroj Kumar and Mohapatra, R. N.},
title = {Sufficient conditions for extremum of fractional variational problems},
journal = {RAIRO. Operations Research},
pages = {637--648},
year = {2022},
publisher = {EDP-Sciences},
volume = {56},
number = {2},
doi = {10.1051/ro/2022035},
mrnumber = {4407590},
language = {en},
url = {https://www.numdam.org/articles/10.1051/ro/2022035/}
}
TY - JOUR AU - Pattnaik, Ashapurna AU - Padhan, Saroj Kumar AU - Mohapatra, R. N. TI - Sufficient conditions for extremum of fractional variational problems JO - RAIRO. Operations Research PY - 2022 SP - 637 EP - 648 VL - 56 IS - 2 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/ro/2022035/ DO - 10.1051/ro/2022035 LA - en ID - RO_2022__56_2_637_0 ER -
%0 Journal Article %A Pattnaik, Ashapurna %A Padhan, Saroj Kumar %A Mohapatra, R. N. %T Sufficient conditions for extremum of fractional variational problems %J RAIRO. Operations Research %D 2022 %P 637-648 %V 56 %N 2 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/ro/2022035/ %R 10.1051/ro/2022035 %G en %F RO_2022__56_2_637_0
Pattnaik, Ashapurna; Padhan, Saroj Kumar; Mohapatra, R. N. Sufficient conditions for extremum of fractional variational problems. RAIRO. Operations Research, Tome 56 (2022) no. 2, pp. 637-648. doi: 10.1051/ro/2022035
[1] , Formulation of Euler-Lagrange equations for fractional variational problems. J. Math. Anal. Appl. 272 (2002) 368–379. | MR | Zbl | DOI
[2] , Analytical schemes for a new class of fractional differential equations. J. Phys. A: Math. Theor 40 (2007) 5469–5477. | MR | Zbl | DOI
[3] , Fractional variational calculus in terms of Riesz fractional derivatives. J. Phys. A: Math. Theor 40 (2007) 6287–6303. | MR | Zbl | DOI
[4] , A general finite element formulation for fractional variational problems. J. Math. Anal. Appl. 337 (2008) 1–12. | MR | Zbl | DOI
[5] , A formulation and numerical scheme for fractional optimal control problems. J. Vib. Control 14 (2008) 1291–1299. | MR | Zbl | DOI
[6] , Generalized variational problems and Euler-Lagrange equations. Comput. Math. 59 (2010) 1852–1864. | MR | Zbl
[7] and , A Hamiltonian formulation and a direct numerical scheme for fractional optimal control problems. J. Vib. Control 13 (2007) 1269–1281. | MR | Zbl | DOI
[8] , Fractional variational problems depending on indefinite integrals and with delay. Bull. Malays. Math. Sci. Soc. 39 (2016) 1515–1528. | MR | DOI
[9] , Variational problems involving a caputo-type fractional derivative. J. Optim. Theory Appl. 174 (2017) 276–294. | MR | DOI
[10] , , Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives. Commun. Nonlinear Sci. Numer. Simul. 16 (2011) 1490–1500. | MR | Zbl | DOI
[11] , and , A new method for investigating approximate solutions of some fractional integro-differential equations involving the Caputo-Fabrizio derivative. Adv. Differ. Equ. 51 (2017) 1–12. | MR
[12] and , Application of fractional calculus in statistics. Int. J. Contemp. Math. Sci. 7 (2012) 849–856. | MR | Zbl
[13] , Differential Equations and Calculus of Variations. Mir Publication, Moscow (1970). | MR | Zbl
[14] , and , On overall behavior of Maxwell mechanical model by the combined Caputo fractional derivative. Chin. J. Phys. 66 (2020) 269–276. | MR | DOI
[15] , Fractional optimal control in the sense of Caputo and the fractional Noethers theorem. Int. Math. Forum 3 (2008) 479–493. | MR | Zbl
[16] and , A formulation of Neother’s theorem for fractional problems of the calculus of variations. J. Math. Anal. Appl. 334 (2007) 834–846. | MR | Zbl | DOI
[17] , and , A study of behaviour for immune and tumor cells in immunogenetic tumour model with non-singular fractional derivative. Chaos Solitons Fractals 133 (2020) 109619. | MR | DOI
[18] , and , Similarities in a fifth-order evolution equation with and with no singular kernel. Chaos Solitons Fractals 130 (2020) 109467. | MR | DOI
[19] , Fractional sequential mechanics – models with symmetric fractional derivative. Czech. J. Phys. 51 (2001) 1348–1354. | MR | Zbl | DOI
[20] , Lagrangean and Hamiltonian fractional sequential mechanics. Czech. J. Phys. 52 (2002) 1247–1253. | MR | Zbl | DOI
[21] , and , Modified Kawahara equation within a fractional derivative with non-singular kernel. Thermal Sci. 22 (2018) 789–796. | DOI
[22] , , and , A study of fractional Lotka-Volterra population model using Haar wavelet and Adams–Bashforth–Moulton methods. Math. Methods Appl. Sci. 43 (2020) 5564–5578. | MR | DOI
[23] , , and , Chaotic behaviour of fractional predator-prey dynamical system. Chaos Solitons Fractals 135 (2020) 109811. | MR | DOI
[24] , , , and , A new Rabotnov fractional-exponential function based fractional derivative for diffusion equation under external force. Math Meth Appl Sci. 43 (2020) 4460–4471. | MR
[25] , , and , Numerical investigations on COVID-19 model through singular and non-singular fractional operators. Numer. Methods Part. Differ. Equ. (2020) 1–27. DOI: . | DOI | MR
[26] , , and , A study on four-species fractional population competetion dynamical model. Results Phys. 24 (2021) 104089. | DOI
[27] , , and , A study on transmission dynamics of HIV/AIDS model through fractional operators. Results Phys. 22 (2021) 1–14. | DOI
[28] , , and , A study on fractional host-parasitoid population dynamical model to describe insect species. Numer. Methods Part. Differ. Equ. 37 (2021) 1673–1692. | MR | DOI
[29] and , The Legendre condition of the fractional calculus of variations. Optimization 63 (2014) 1157–1165. | MR | Zbl | DOI
[30] and , Fractional calculus of variations for a combined Caputo derivative. Fract. Calc. Appl. Anal. 14 (2011) 523–537. | MR | Zbl | DOI
[31] , and , Fractional variational calculus with classical and combined Caputo derivatives. Nonlinear Anal. 75 (2012) 1507–1515. | MR | Zbl | DOI
[32] , Fractional Differential Equations. Academic Press, New York (1999). | MR | Zbl
[33] , , and , On generalized fractional integral inequalities for the monotone weighted Chebyshev functionals. Adv. Differ. Equ. 368 (2020) 1–19. | MR
[34] , Nonconservative Lagrangian and Hamiltonian mechanics. Phys. Rev. 53 (1996) 1890–1899. | MR
[35] , Mechanics with fractional derivatives. Phys. Rev. 55 (1997) 3581–3592. | MR
[36] , The development of fractional calculus 1695–1900. Hist. Math. 4 (1977) 75–89. | MR | Zbl | DOI
[37] , and , Analysis of an El Nino-Southern Oscillation model with a new fractional derivative. Chaos Solitons Fractals 99 (2017) 109–115. | MR | DOI
[38] , Fractional Taylor series for Caputo fractional derivatives. In: Construction of Numerical Schemes. Universidad Complutense de Madrid, Spain (2007).
[39] and , Numerical methods for fractional variational problems depending on indefinite integrals. ASME. J. Comput. Nonlinear Dynam. 8 (2013) 021018. | DOI
[40] , and , Optimality conditions for fractional variational problems with Caputo-Fabrizio fractional derivatives. Adv. Differ. Equ. 357 (2017) 1–14. | MR
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