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 Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.]: Year: Volume: Issue: Page: Find

 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 two-dimensional filtration equation with the Riemann–Liouville time-fractional derivative

V. O. Lukashchuk, S. Yu. Lukashchuk

Ufa State Aviation Technical University, Ufa, 450008, Russian Federation

Abstract: A nonlinear two-dimensional orthotropic filtration equation with the Riemann–Liouville time-fractional 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 four-parameter group of point transformations in orthotropic case, and a five-parameter group in isotropic case. For the power-law piezoconductivity, the group admitted by the equation is five-parametric in orthotropic case, and six-parametric 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 integer-order 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 group-invariant solutions are obtained for two-dimensional 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 time-fractional filtration equation is nonlinearly self-adjoint 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

 Funding Agency Grant Number Ministry of Science and Higher Education of the Russian Federation 1.3103.2017/4.6 This work was supported by the Ministry of Science and Higher Education of the Russian Federation (State task no. 1.3103.2017/4.6).

Author to whom correspondence should be addressed

DOI: https://doi.org/10.14498/vsgtu1761

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Bibliographic databases:

UDC: 517.958
MSC: 35R11, 76M60, 76S05
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 two-dimensional filtration equation with the Riemann–Liouville time-fractional 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 two-dimensional filtration equation with the Riemann--Liouville time-fractional 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 226--248 \mathnet{http://mi.mathnet.ru/vsgtu1761} \crossref{https://doi.org/10.14498/vsgtu1761}