A theorem on the existence of symmetries of fractional PDEs

Rosario Antonio Leo; Gabriele Sicuro; Piergiulio Tempesta

doi:10.1016/j.crma.2013.11.007

Partial differential equations/Mathematical physics

A theorem on the existence of symmetries of fractional PDEs
[Un théorème sur l'existence de symétries pour les équations aux derivées partielles fractionnaires]

Présenté par : Pierre-Louis Lions

Rosario Antonio Leo ¹ ; Gabriele Sicuro ² ; Piergiulio Tempesta ^3,⁴

¹ Dipartimento di Matematica e Fisica “Ennio De Giorgi”, Università del Salento, Via per Arnesano, 73100 Lecce, Italy
² Dipartimento di Fisica “Enrico Fermi”, Università di Pisa, Italy
³ Departamento de Fisica Teorica II, Métodos Matemáticos de la Física, Universidad Complutense de Madrid, Ciudad Universitaria, 28040, Madrid, Spain
⁴ Instituto de Ciencias Matemáticas, C/ Nicolás Cabrera, No 13-15, 28049 Madrid, Spain

Comptes Rendus. Mathématique, Volume 352 (2014) no. 3, pp. 219-222.

Résumé
Abstract

Nous proposons un théorème qui generalise la méthode classique de Lie à l'étude d'équations aux derivées partielles fractionnaires de type Riemann–Liouville en ( $1 + 1$ ) dimensions.

We propose a theorem that extends the classical Lie approach to the case of fractional partial differential equations (fPDEs) of the Riemann–Liouville type in ( $1 + 1$ ) dimensions.

Reçu le : 2013-09-24
Accepté le : 2013-11-12
Publié le : 2014-01-20

DOI : 10.1016/j.crma.2013.11.007

Affiliations des auteurs :

Rosario Antonio Leo ¹ ; Gabriele Sicuro ² ; Piergiulio Tempesta ^{3, 4}

¹ Dipartimento di Matematica e Fisica “Ennio De Giorgi”, Università del Salento, Via per Arnesano, 73100 Lecce, Italy
² Dipartimento di Fisica “Enrico Fermi”, Università di Pisa, Italy
³ Departamento de Fisica Teorica II, Métodos Matemáticos de la Física, Universidad Complutense de Madrid, Ciudad Universitaria, 28040, Madrid, Spain
⁴ Instituto de Ciencias Matemáticas, C/ Nicolás Cabrera, No 13-15, 28049 Madrid, Spain

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     title = {A theorem on the existence of symmetries of fractional {PDEs}},
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Rosario Antonio Leo; Gabriele Sicuro; Piergiulio Tempesta. A theorem on the existence of symmetries of fractional PDEs. Comptes Rendus. Mathématique, Volume 352 (2014) no. 3, pp. 219-222. doi : 10.1016/j.crma.2013.11.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.11.007/

Bibliographie
Cité par

[1] E. Buckwar; Y. Luchko Invariance of a partial differential equation of fractional order under the Lie group of scaling transformations, J. Math. Anal. Appl., Volume 227 (1998), pp. 81-97

[2] R.K. Gazizov; A.A. Kasatkin; S.Y. Lukashchuk; R.K. Gazizov; A.A. Kasatkin; S.Y. Lukashchuk, Phys. Scr. T (J.A.T. Machado; A.C.J. Luo; R.S. Barbosa; M.F. Silva; L.B. Figueiredo, eds.) (Group-invariant solutions of fractional differential equations, Nonlinear Science and Complexity), Volume 136, Springer, 2009, pp. 51-58

[3] R. Gorenflo; Yu. Luchko; F. Mainardi Wright functions as scale-invariant solutions of the diffusion-wave equation, J. Comput. Appl. Math., Volume 11 (2000), pp. 175-191

[4] Yu. Luchko; R. Gorenflo Scale-invariant solutions of a partial differential equation of fractional order, Fract. Calc. Appl. Anal., Volume 1 (1998), pp. 63-78

[5] P.J. Olver Applications of Lie Groups to Differential Equations, Springer, 1986

[6] T.J. Osler Leibniz rule for fractional derivatives generalized and an application to infinite series, SIAM J. Appl. Math., Volume 18 (1970) no. 3, pp. 658-674

[7] S.G. Samko; A.A. Kilbas; O.I. Marichev Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach Science Publishers, 1993

C. A. jun. Soares; F. S. Costa; J. Vanterler C. Sousa Infinitesimal prolongation for fractional derivative $Ψ$ -Caputo variable order and applications, Qualitative Theory of Dynamical Systems, Volume 24 (2025) no. 1, p. 29 (Id/No 3) | DOI:10.1007/s12346-024-01157-y | Zbl:1553.35226
Junior C. A. Soares; Felix S. Costa; J. Vanterler C. Sousa Lie symmetry analysis for fractional evolution equation with $ζ$ -Riemann-Liouville derivative, Computational and Applied Mathematics, Volume 43 (2024) no. 4, p. 25 (Id/No 159) | DOI:10.1007/s40314-024-02685-8 | Zbl:7853488
Sameerah Jamal; Reginald Champala; Suhail Khan Lie Symmetries and the Invariant Solutions of the Fractional Black–Scholes Equation under Time-Dependent Parameters, Fractal and Fractional, Volume 8 (2024) no. 5, p. 269 | DOI:10.3390/fractalfract8050269
Reginald Champala; Sameerah Jamal; Suhail Khan Fractional Pricing Models: Transformations to a Heat Equation and Lie Symmetries, Fractal and Fractional, Volume 7 (2023) no. 8, p. 632 | DOI:10.3390/fractalfract7080632
Youness Chatibi; El Hassan El Kinani; Abdelaziz Ouhadan Continuous and discrete symmetry methods for fractional differential equations, Fractional-Order Modeling of Dynamic Systems with Applications in Optimization, Signal Processing and Control (2022), p. 1 | DOI:10.1016/b978-0-32-390089-8.00006-4
Alessandra Jannelli; Maria Paola Speciale On the numerical solutions of coupled nonlinear time-fractional reaction-diffusion equations, AIMS Mathematics, Volume 6 (2021) no. 8, pp. 9109-9125 | DOI:10.3934/math.2021529 | Zbl:1485.65079
Alessandra Jannelli; Maria Paola Speciale Exact and numerical solutions of two-dimensional time-fractional diffusion–reaction equations through the Lie symmetries, Nonlinear Dynamics, Volume 105 (2021) no. 3, p. 2375 | DOI:10.1007/s11071-021-06697-5
Ewa Piotrowska; Łukasz Sajewski Analysis of an Electrical Circuit Using Two-Parameter Conformable Operator in the Caputo Sense, Symmetry, Volume 13 (2021) no. 5, p. 771 | DOI:10.3390/sym13050771
Alessandra Jannelli; Marianna Ruggieri; Maria Paola Speciale Numerical solutions of space-fractional advection-diffusion equations with nonlinear source term, Applied Numerical Mathematics, Volume 155 (2020), pp. 93-102 | DOI:10.1016/j.apnum.2020.01.016 | Zbl:1436.35022
Alessandra Jannelli; Marianna Ruggieri; Maria Paola Speciale Analytical and numerical solutions of time and space fractional advection-diffusion-reaction equation, Communications in Nonlinear Science and Numerical Simulation, Volume 70 (2019), pp. 89-101 | DOI:10.1016/j.cnsns.2018.10.012 | Zbl:1464.35396
Changzhao Li; Juan Zhang Lie symmetry analysis and exact solutions of generalized fractional Zakharov-Kuznetsov equations, Symmetry, Volume 11 (2019) no. 5, p. 12 (Id/No 601) | DOI:10.3390/sym11050601 | Zbl:1425.35216
Ruiping Wen; Peipei Zhao A medium-shifted splitting iteration method for a diagonal-plus-Toeplitz linear system from spatial fractional Schrödinger equations, Boundary Value Problems, Volume 2018 (2018), p. 17 (Id/No 45) | DOI:10.1186/s13661-018-0967-1 | Zbl:1499.35693
Alessandra Jannelli; Marianna Ruggieri; Maria Paola Speciale Exact and numerical solutions of time-fractional advection-diffusion equation with a nonlinear source term by means of the Lie symmetries, Nonlinear Dynamics, Volume 92 (2018) no. 2, pp. 543-555 | DOI:10.1007/s11071-018-4074-8 | Zbl:1398.34019
Alessandra Jannelli; Marianna Ruggieri; Maria Paola Speciale, Volume 1863 (2017), p. 530005 | DOI:10.1063/1.4992675
Rosario Antonio Leo; Gabriele Sicuro; Piergiulio Tempesta A foundational approach to the Lie theory for fractional order partial differential equations, Fractional Calculus Applied Analysis, Volume 20 (2017) no. 1, pp. 212-231 | DOI:10.1515/fca-2017-0011 | Zbl:1366.35219
Komal Singla; R. K. Gupta Generalized Lie symmetry approach for fractional order systems of differential equations. III, Journal of Mathematical Physics, Volume 58 (2017) no. 6, p. 061501 | DOI:10.1063/1.4984307 | Zbl:1372.35345
Mingyang Pan; Liancun Zheng; Chunyan Liu; Fawang Liu Symmetry analysis and conservation laws to the space-fractional Prandtl equation, Nonlinear Dynamics, Volume 90 (2017) no. 2, pp. 1343-1351 | DOI:10.1007/s11071-017-3730-8 | Zbl:1390.37113
Ramajayam Sahadevan; Thangarasu Bakkyaraj Invariant subspace method and exact solutions of certain nonlinear time fractional partial differential equations, Fractional Calculus Applied Analysis, Volume 18 (2015) no. 1, pp. 146-162 | DOI:10.1515/fca-2015-0010 | Zbl:1499.35682
J. Tenreiro Machado; Francesco Mainardi; Virginia Kiryakova Fractional calculus: quo vadimus? (where are we going?), Fractional Calculus Applied Analysis, Volume 18 (2015) no. 2, pp. 495-526 | DOI:10.1515/fca-2015-0031 | Zbl:1309.26011
Qing Huang; Shoufeng Shen Lie symmetries and group classification of a class of time fractional evolution systems, Journal of Mathematical Physics, Volume 56 (2015) no. 12, p. 123504 | DOI:10.1063/1.4937755 | Zbl:1329.35333
Ali H. Bhrawy; Taha M. Taha; José A. Tenreiro Machado A review of operational matrices and spectral techniques for fractional calculus, Nonlinear Dynamics, Volume 81 (2015) no. 3, pp. 1023-1052 | DOI:10.1007/s11071-015-2087-0 | Zbl:1348.65106

Cité par 21 documents. Sources : Crossref, zbMATH

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