Abstract
We provide a general theoretical framework allowing us to extend the classical Lie theory for partial differential equations to the case of equations of fractional order. We propose a general prolongation formula for the study of Lie symmetries in the case of an arbitrary finite number of independent variables. We also prove the Lie theorem in the case of fractional differential equations, showing that the proper space for the analysis of these symmetries is the infinite dimensional jet space.
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References
E. Barkai, and R. Metzler, and J. Klafter, From continuous time random walks to the fractional Fokker–Planck equation. Phys. Rev. E 61 (2000), 132–138.
G.W. Bluman, and S. Kumei, Symmetries and Differential Equations Ser. Applied Mathematical Sciences. 81. World Publishing Co. (1989).
M. Bologna, and C. Tsallis, and P. Grigolini, Anomalous diffusion associated with nonlinear fractional derivative Fokker–Planck-like equation: Exact time-dependent solutions. Phys. Rev. E 62, No 2 (2000), 2213–2218.
E. Buckwar, and Y. Luchko, Invariance of a partial differential equation of fractional order under the Lie group of scaling transformations. J. Math. Anal. Appl. 227, No 1 (1998), 81–97.
A.V. Chechkin, and V.Yu. Gonchar, and M. Szydłowski, Fractional kinetics for relaxation and superdiffusion in a magnetic field. Phys. Plasmas 9, No 1 (2002), 78–88.
M. Concezzi, and R. Garra, and R. Spigler, Fractional relaxation and fractional oscillation models involving Erdéiyi–Kober integrals. Fract. Calc. Appl. Anal. 18, No 5 (2015), 1212–1231 10.1515/fca-2015-0070 https://www.degruyter.com/view/j/fca.2015.18.issue-5/issue-files/fca.2015.18.issue-5.xml
D. del Castillo-Negrete, and B.A. Carreras, and V.E. Lynch, Front dynamics in reaction-diffusion systems with Lévy flights: A fractional diffusion approach. Phys. Rev. Lett. 91 (July 2003), 018302.
A. Erdélyi, On fractional integration and its application to the theory of Hankel transforms. Quart. J. Math. OS-11, No 1 (1940), 293–303.
R.K. Gazizov, and A.A. Kasatkin, and S.Y. Lukashchuk, Symmetry properties of fractional diffusion equations. Phys. Scripta T136 (Oct. 2009), 014016.
R.K. Gazizov, and A.A. Kasatkin, and S.Y. Lukashchuk, Group-invariant solutions of fractional differential equations In: J.A. Tenreiro Machado, and A.C.J. Luo, and R.S. Barbosa, and M.F. Silva, and L.B. Figueiredo, Nonlinear Science and Complexity. Springer Sci. & Business Media, 2011), 51–59.
Q. Huang, and S. Shen, Lie symmetries and group classification of a class of time fractional evolution systems. J. Math. Phys. 56, No 12 (2015), 123504.
Q. Huang, and R. Zhdanov, Symmetries and exact solutions of the time fractional Harry-Dym equation with Riemann–Liouville derivative. Physica A 409 (2014), 110–118.
N.H. Ibragimov, Transformation Groups Applied to Mathematical Physics Ser. Mathematics and its Applications. Springer Netherlands (2001).
A.A. Kilbas, and H.M. Srivastava, and J.J. Trujillo, Theory and Applicationes of Fractional Differential Equations Ser. North-Holland Mthematics Studies. Elsevier (2006).
H. Kober, On fractional integrals and derivatives. Quart. J. Math. OS-11, No 1 (1940), 193–211.
J. Krasil’shchik, and A. Verbovetsky, Geometry of jet spaces and integrable systems. J. Geometry Phys. 61, No 9 (2011), 1633–1674.
E.K. Lenzi, and R.S. Mendes, and K.S. Fa, and L.S. Moraes, and L.R. da Silva, and L.S. Lucena, Nonlinear fractional diffusion equation: Exact results. J. Math. Phys. 46, No 8 (2005), 083506.
R.A. Leo, and G. Sicuro, and P. Tempesta, A theorem on the existence of symmetries of fractional PDEs. C. R. Math. 352, No 3 (2014), 219–222.
Benoit B Mandelbrot, and John W Van Ness, Fractional brownian motions, fractional noises and applications. SIAM Review 10, No 4 (1968), 422–437.
R. Metzler, and E. Barkai, and J. Klafter, Anomalous diffusion and relaxation close to thermal equilibrium: a fractional Fokker-Planck equation approach. Phys. Rev. Lett. 82 (May 1999), 3563–3567.
K.B. Oldham, and J. Spanier, The Fractional Calculus Ser. Mathematics in Science and Engineering. Academic Press (1974)
P.J. Olver, Applications of Lie Groups to Differential Equations Ser. Graduate Texts in Mathematics. Springer New York, 2000.
T.J. Osler, Leibniz rule for fractional derivatives generalized and an application to infinite series. SIAM J. Appl. Math. 18, No 3 (1970), 658–674.
B. Ross, The development of fractional calculus 1695–1900. Historia Mathematica 4, No 1 (1977), 75–89.
J. Sabatier, and O.P. Agrawal, and J.A.T. Machado, Advances in Fractional Calculus Springer (2007)
R. Sahadevan, and T. Bakkyaraj, Invariant analysis of time fractional generalized Burgers and Korteweg–de Vries equations. J. Math. Anal. Appl. 393, No 2 (2012), 341–347.
S. Samko, and A.A. Kilbas, and O. Marichev, Fractional Integrals and Derivatives Taylor & Francis (1993)
D.J. Saunders, The Geometry of Jet Bundles Cambridge University Press Cambridge, (1989).
E. Scalas, and R. Gorenflo, and F. Mainardi, Fractional calculus and continuous-time finance. Physica A 284, No 1–4 (2000), 376–384.
A.M. Vinogradov, Symmetries and conservation laws of partial differential equations: Basic notions and results. Acta Appl. Math. 15, No 1-2 (1989), 3–21.
A.M. Vinogradov, and J. Krasi’shchik, Symmetries and Conservation Laws for Differential Equations of Mathematical Physics American Mathematical Society (1999).
X.-B. Wang, and S.-F. Tian, and C.-Y. Qin, and T.-T. Zhang, Lie symmetry analysis, conservation laws and exact solutions of the generalized time fractional Burgers equation. EPL (Europhysics Letters) 114, No 2 (2016), 20003.
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Leo, R.A., Sicuro, G. & Tempesta, P. A Foundational Approach to the Lie Theory for Fractional Order Partial Differential Equations. FCAA 20, 212–231 (2017). https://doi.org/10.1515/fca-2017-0011
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DOI: https://doi.org/10.1515/fca-2017-0011