
Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2020, Volume 24, Number 2, Pages 226–248
(Mi vsgtu1761)




Differential Equations and Mathematical Physics
Group classification, invariant solutions and conservation laws of nonlinear orthotropic twodimensional filtration equation with the Riemann–Liouville timefractional derivative
V. O. Lukashchuk^{}, S. Yu. Lukashchuk^{}^{†} ^{} Ufa State Aviation Technical University, Ufa, 450008, Russian Federation
Abstract:
A nonlinear twodimensional orthotropic filtration equation with the Riemann–Liouville timefractional derivative is considered. It is proved that this equation can admits only linear autonomous groups of point transformations. The Lie point symmetry group classification problem for the equation in question is solved with respect to coefficients of piezoconductivity. These coefficients are assumed to be functions of the square of the pressure gradient absolute value. It is proved that if the order of fractional differentiation is less than one then the considered equation with arbitrary coefficients admits a fourparameter group of point transformations in orthotropic case, and a fiveparameter group in isotropic case. For the powerlaw piezoconductivity, the group admitted by the equation is fiveparametric in orthotropic case, and sixparametric in isotropic case. Also, a special case of power function of piezoconductivity is determined for which there is an additional extension of admitted groups by the projective transformation. There is no an analogue of this case for the integerorder filtration equation. It is also shown that if the order of fractional differentiation $\alpha \in (1,2)$ then dimensions of admitted groups are incremented by one for all cases since an additional translation symmetry exists. This symmetry is corresponded to an additional particular solution of the fractional filtration equation under consideration.
Using the group classification results for orthotropic case, the representations of groupinvariant solutions are obtained for twodimensional subalgebras from optimal systems of symmetry subalgebras. Examples of reduced equations obtained by the symmetry reduction technique are given, and some exact solutions of these equations are presented.
It is proved that the considered timefractional filtration equation is nonlinearly selfadjoint and therefore the corresponding conservation laws can be constructed. The components of obtained conserved vectors are given in an explicit form.
Keywords:
fractional filtration equation, group classification, Lie point symmetry, invariant solution, conservation law
^{†} Author to whom correspondence should be addressed
DOI:
https://doi.org/10.14498/vsgtu1761
Full text:
PDF file (1075 kB)
(published under the terms of the Creative Commons Attribution 4.0 International License)
References:
PDF file
HTML file
Bibliographic databases:
UDC:
517.958
MSC: 35R11, 76M60, 76S05 Received: November 29, 2019 Revised: May 17, 2020 Accepted: June 1, 2020 First online: June 30, 2020
Citation:
V. O. Lukashchuk, S. Yu. Lukashchuk, “Group classification, invariant solutions and conservation laws of nonlinear orthotropic twodimensional filtration equation with the Riemann–Liouville timefractional derivative”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 24:2 (2020), 226–248
Citation in format AMSBIB
\Bibitem{LukLuk20}
\by V.~O.~Lukashchuk, S.~Yu.~Lukashchuk
\paper Group classification, invariant solutions and conservation laws of nonlinear orthotropic twodimensional filtration equation with the RiemannLiouville timefractional derivative
\jour Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.]
\yr 2020
\vol 24
\issue 2
\pages 226248
\mathnet{http://mi.mathnet.ru/vsgtu1761}
\crossref{https://doi.org/10.14498/vsgtu1761}
Linking options:
http://mi.mathnet.ru/eng/vsgtu1761 http://mi.mathnet.ru/eng/vsgtu/v224/i2/p226
Citing articles on Google Scholar:
Russian citations,
English citations
Related articles on Google Scholar:
Russian articles,
English articles

Number of views: 
This page:  97  Full text:  35  References:  4 
