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Ufimsk. Mat. Zh., 2012, Volume 4, Issue 4, Pages 54–68 (Mi ufa168)  

This article is cited in 16 scientific papers (total in 16 papers)

Fractional differential equations: change of variables and nonlocal symmetries

R. K. Gazizov, A. A. Kasatkin, S. Yu. Lukashchuk

Ufa State Aviation Technical University, Ufa, Russia

Abstract: In the work point changes of variables in different types of fractional integrals and derivatives are considered. In a general case fractional integrodifferentiation of a function with respect to another function arises after such change. The problem of extending a group of point transformations to operators of this type is considered, corresponding prolongation formulae for the group infinitesimal operator are constructed. Usage of prolongation formulae for finding some nonlocal symmetries of the equation and checking their admittance is demonstrated as a simple example of an ordinary fractional differential equation.

Keywords: fractional derivative, prolongation formulae, nonlocal symmetry.

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UDC: 517.9
Received: 09.11.2012

Citation: R. K. Gazizov, A. A. Kasatkin, S. Yu. Lukashchuk, “Fractional differential equations: change of variables and nonlocal symmetries”, Ufimsk. Mat. Zh., 4:4 (2012), 54–68

Citation in format AMSBIB
\by R.~K.~Gazizov, A.~A.~Kasatkin, S.~Yu.~Lukashchuk
\paper Fractional differential equations: change of variables and nonlocal symmetries
\jour Ufimsk. Mat. Zh.
\yr 2012
\vol 4
\issue 4
\pages 54--68

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    This publication is cited in the following articles:
    1. S. Yu. Lukashchuk, “Constructing conservation laws for fractional-order integro-differential equations”, Theoret. and Math. Phys., 184:2 (2015), 1049–1066  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    2. Zedan H.A., Shapll S., Abdel-Malek A., “Invariance of the Nonlinear Generalized Nls Equation Under the Lie Group of Scaling Transformations”, Nonlinear Dyn., 82:4 (2015), 2001–2005  crossref  mathscinet  zmath  isi  scopus
    3. Gazizov R.K. Ibragimov N.H. Lukashchuk S.Yu., “Nonlinear Self-Adjointness, Conservation Laws and Exact Solutions of Time-Fractional Kompaneets Equations”, Commun. Nonlinear Sci. Numer. Simul., 23:1-3 (2015), 153–163  crossref  mathscinet  zmath  isi  scopus
    4. Lukashchuk S.Yu., Makunin A.V., “Group Classification of Nonlinear Time-Fractional Diffusion Equation With a Source Term”, Appl. Math. Comput., 257 (2015), 335–343  crossref  mathscinet  zmath  isi  scopus
    5. Lukashchuk S.Yu., “Conservation Laws For Time-Fractional Subdiffusion and Diffusion-Wave Equations”, Nonlinear Dyn., 80:1-2 (2015), 791–802  crossref  mathscinet  zmath  isi  scopus
    6. Malkawi E., “Spatial Rotation of the Fractional Derivative in Two-Dimensional Space”, Adv. Math. Phys., 2015, 719173  crossref  mathscinet  zmath  isi  scopus
    7. S. Yu. Lukashchuk, “Symmetry reduction and invariant solutions for nonlinear fractional diffusion equation with a source term”, Ufa Math. J., 8:4 (2016), 111–122  mathnet  crossref  isi  elib
    8. S. Yu. Lukaschuk, “Gruppovaya klassifikatsiya odnogo nelineinogo priblizhennogo uravneniya subdiffuzii”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 20:4 (2016), 603–619  mathnet  crossref  zmath  elib
    9. S. San, “Invariant Analysis of Nonlinear Time Fractional Qiao Equation”, Nonlinear Dyn., 85:4 (2016), 2127–2132  crossref  mathscinet  zmath  isi  scopus
    10. S. Yu. Lukaschuk, “Priblizhenie obyknovennykh drobno-differentsialnykh uravnenii differentsialnymi uravneniyami s malym parametrom”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 27:4 (2017), 515–531  mathnet  crossref  elib
    11. S. Rashidi, S. R. Hejazi, “Analyzing Lie Symmetry and Constructing Conservation Laws For Time-Fractional Benny-Lin Equation”, Int. J. Geom. Methods Mod. Phys., 14:12 (2017), 1750170  crossref  mathscinet  zmath  isi  scopus
    12. R. Almeida, “A Caputo Fractional Derivative of a Function With Respect to Another Function”, Commun. Nonlinear Sci. Numer. Simul., 44 (2017), 460–481  crossref  mathscinet  isi  scopus
    13. R. K. Gazizov, S. Yu. Lukashchuk, “Approximations of Fractional Differential Equations and Approximate Symmetries”, IFAC-PapersOnLine, 50:1 (2017), 14022–14027  crossref  isi  scopus
    14. I. Naeem, M. D. Khan, “Symmetry Classification of Time-Fractional Diffusion Equation”, Commun. Nonlinear Sci. Numer. Simul., 42 (2017), 560–570  crossref  mathscinet  isi  scopus
    15. Belevtsov N.S., Lukashchuk S.Yu., “Lie Group Analysis of 2-Dimensional Space-Fractional Model For Flow in Porous Media”, Math. Meth. Appl. Sci., 41:18, SI (2018), 9123–9133  crossref  mathscinet  zmath  isi  scopus
    16. Lukashchuk S.Yu., Saburova R.D., “Approximate Symmetry Group Classification For a Nonlinear Fractional Filtration Equation of Diffusion-Wave Type”, Nonlinear Dyn., 93:2 (2018), 295–305  crossref  zmath  isi  scopus
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