Invariance properties of a one-dimensional diffusion equation with a fractal derivative in time
Following the group analysis of differential equations, we consider the symmetry properties of equations with fractal derivatives defined in the framework of the F a-calculus. Analogues of the prolongation of transformations of independent and dependent variables are discussed. The infinitesimal invariance of equations with fractal derivatives is studied on the example of the Lie symmetries of a one-dimensional diffusion equation with a fractal time derivative.
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Keywords
analysis on fractals, fractal diffusion equation, group approach, continuation, invariance, Lie symmetriesAuthors
| Name | Organization | |
| Shapovalov A.V. | National Research Tomsk State University; National Research Tomsk Polytechnic University | shpv@phys.tsu.ru |
| Brons R. | National Research Tomsk State University | robbrons35@gmail.com |
References
Самко С.Г., Килбас А.А., Маричев О.И. Интегралы и производные дробного порядка и некоторые их приложения. - Минск: Наука и техника, 1987. - 688 с.
Metzler R., Glockle W.G., and Nonnenmacher T.F. // Physica A. - 1994. - V. 211. - No. 1. - P. 13-24.
Metzler R., Barkai E., and Kafter J. // Phys. Rev. Lett. - 1999. - V. 83. - P. 3563.
Учайкин В.В. // УФН. - 2003. - Т. 173. - № 8. - С. 847-875.
Fractional Dynamics /eds. J. Klafter, S.C. Lim, and R. Metzler. - New Jersey; London; Hong Kong: World Scientific, 2011. - 532 p.
Parvate A. and Gangal A.D. // Fractals. - 2009. - V. 17. - P. 53-148.
Parvate, A. and Gangal A.D. // Fractals. - 2011. - V. 19. - P. 271-290.
Parvate A., Satin S., and Gangal A.D. // Fractals. - 2011. - V. 19. - P. 15-27.
Golmankhaneh A.R., Fernandez A., Golmankhaneh A.K., and Baleanu D. // Entropy. - 2018. - V. 20. - P. 1-13.
Овсянников Л.В. Групповой анализ дифференциальных уравнений. - М.: Наука, 1978.
Ibragimov N.H. Transformation Groups Applied to Mathematical Physics, Mathematics and its Applications (Soviet Series). - Dordrecht: D. Reidel Publishing, 1985.
Olver P. Applications of Lie Groups to Differential Equations. - N.Y.: Springer, 1986.
Gazizov R.K., Kasatkin A.A., and Lukashchuk S.Yu. // Phys. Scripta. - 2009. - V. 136. - No. 014016. - 5 p.
Gazizov R.K., Kasatkin A.A., and Lukashchuk S.Yu. // Ufimsk. Mat. Zh. - 2012. - V. 4. - No. 4. - P. 54-68.
Golmankhaneh A.R. and Tunc C. // Fractal and Fract. - 2019. - V. 3. - No. 25. - 15 p.
Толстов Г.П. Мера и интеграл. - М.: Наука, 1976.
