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

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 24 (2020) no. 2, pp. 226-248.

Résumé

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.

Détail

Mots-clés : fractional filtration equation, group classification, invariant solution
Keywords: Lie point symmetry, conservation law.

Bibliographie

[1] Samko S. G., Kilbas A. A., Marichev O. I., Fractional integrals and derivatives. Theory and applications, Gordon Breach Sci. Publishers, New York, 1993, xxxvi+976 pp. | MR | Zbl

[2] Kilbas A. A., Srivastava H. M., Trujillo J. J., Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier, Amsterdam, 2006, xv+523 pp. | MR | Zbl

[3] Podlubny I., Fractional differential equations. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering, 198, Academic Press, San Diego, 1999, xxiv+340 pp. | MR | Zbl

[4] Metzler R., Klafter J., “The random walk's guide to anomalous diffusion: a fractional dynamic approach”, Phys. Rep., 339:1 (2000), 1–77 | DOI | MR | Zbl

[5] Hilfer R., Applications of fractional calculus in physics, World Scientific, Singapore, 2000, vii+463 pp. | MR | Zbl

[6] Nakhushev A. M., Drobnoe ischislenie i ego primenenie [Fractional Calculus and its Applications], Fizmatlit, Moscow, 2003, 272 pp. (In Russian) | Zbl

[7] Uchaikin V. V., Metod drobnykh proizvodnykh [Method of Fractional Derivatives], Artishok, Ul'yanovsk, 2008, 512 pp. (In Russian)

[8] Mainardi F., Fractional calculus and waves in linear viscoelasticity. An introduction to mathematical models, World Scientific, Hackensack, 2010, xx+347 pp. | MR | Zbl

[9] Goloviznin V. M., Kondratenko P. S., Matveev L. V., Anomal'naia diffuziia radionuklidov v sil'noneodnorodnykh geologicheskikh formatsiiakh [Anomalous Radionuclide Diffusion in Highly Heterogeneous Heological Formations], Nauka, Moscow, 2010, 342 pp. (In Russian)

[10] Fractional Dynamics: Recent Advances, eds. J. Klafter, S. C. Lim, R. Metzler, World Scientific, Hackensack, 2011, xiv+515 pp. | MR | Zbl

[11] Fractional kinetics in solids: Anomalous charge transport in semiconductors, dielectrics and nanosystems, eds. V. Uchaikin, R. Sibatov, CRC Press, Boca Raton, 2013, xvi+257 pp. | DOI | MR | Zbl

[12] Baleanu D., Diethelm K., Scalas E., Trujillo J. J., Fractional calculus: models and numerical methods, Series on Complexity, Nonlinearity and Chaos, 5, World Scientific, Hackensack, 2017, xxviii+448 pp. | DOI | MR | Zbl

[13] Ovsyannikov L. V., Group analysis of differential equations, Academic Press, New York, 1982, xvi+416 pp. | MR | Zbl

[14] Olver P. J., Applications of Lie groups to differential equations, Graduate Texts in Mathematics, 107, Springer, New York, 2000, xxviii+513 pp. | MR | Zbl

[15] Bluman G. W., Cheviakov A. F., Anco S. C., Applications of symmetry methods to partial differential equations, Applied Mathematical Sciences, 168, Springer, New York, 2010, xix+398 pp pp. | DOI | MR | Zbl

[16] Grigoriev Yu. N., Ibragimov N. H., Kovalev V. F., Meleshko S. V., Symmetries of integro-differential equations. With applications in mechanics and plasma physics., Lecture Notes in Physics, 806, Springer, Dordrecht, 2010, xiii+305 pp. | DOI | MR | Zbl

[17] Ibragimov N. H., Transformation groups and Lie algebras, World Scientific, Hackensack, 2013, x+185 pp. | DOI | MR | Zbl

[18] Gazizov R. K., Kasatkin A. A., Lukashchuk S. Y., “Symmetries and group invariant solutions of fractional ordinary differential equations”, Handbook of Fractional Calculus with Applications, eds. A. Kochubei, Y. Luchko, De Gruyter, Berlin, 2019, 65–90 | DOI | MR

[19] Gazizov R. K., Kasatkin A. A., Lukashchuk S. Y., “Symmetries, conservation laws and group invariant solutions of fractional PDEs”, Handbook of Fractional Calculus with Applications, eds. A. Kochubei, Y. Luchko, De Gruyter, Berlin, 2019, 353–382 | DOI | MR

[20] Raghavan R., Chen C., “Fractional diffusion in rocks produced by horizontal wells with multiple, transverse hydraulic fractures of finite conductivity”, J. Petrol. Sci. Eng., 109 (2013), 133–143 | DOI

[21] Obembe A. D. Al-Yousef H. Y., Hossain M. E., Abu-Khamsin S. A., “Fractional derivatives and their applications in reservoir engineering problems: A review”, J. Petrol. Sci. Eng., 157 (2017), 312–327 | DOI

[22] Gazizov R. K., Lukashchuk S. Yu., “Fractional differentiation approach to modeling of fluid filtration processes in complex heterogeneous porous media”, Vestnik UGATU, 21:4 (2017), 104–112 (In Russian)

[23] Babkov O. K., “On group classification of some evolution equations”, Book of abstracts of International conference MOGRAN-16 (October 28 – November 2, 2013), Ufa State Aviation Technical Univ., Ufa, 2013, 6–7

[24] Ovsyannikov L. V., “On the property of $x$-autonomy”, Dokl. Math., 47:3 (1993), 581–584 | MR | Zbl

[25] Chirkunov Y. A., “Linear autonomy conditions for the basic Lie algebra of a system of linear differential equations”, Dokl. Math., 79:3 (2009), 415–417 | DOI | MR | Zbl | Zbl

[26] Gazizov R. K., Kasatkin A. A., Lukashchuk S. Yu., “Fractional differential equations: change of variables and nonlocal symmetries”, Ufimsk. Mat. Zh., 4:4 (2012), 54–68 (In Russian) | MR | Zbl

[27] Ibragimov N. H., “A new conservation theorem”, J. Math. Anal. Appl., 333:1 (2007), 311–328 | DOI | MR | Zbl

[28] Ibragimov N. H., “Nonlinear self-adjointness and conservation laws”, J. Phys. A: Math. Theor., 44 (2011), 432002 | DOI | Zbl

[29] Patera J., Winternitz P., “Subalgebras of real three- and four-dimensional Lie algebras”, J. Math. Phys., 18:7 (1977), 1449–1455 | DOI | MR | Zbl

[30] Lukashchuk S. Yu., “Symmetry reduction and invariant solutions for nonlinear fractional diffusion equation with a source term”, Ufa Math. J., 8:4 (2016), 111–122 | DOI | MR

[31] Ovsyannikov L. V., “On optimal systems of subalgebras”, Dokl. Math., 48:3 (1994), 645–649 | MR | Zbl

[32] Chirkunov Yu. A., Khabirov S. V., Elementy simmetriinogo analiza differentsial'nykh uravnenii mekhaniki sploshnoi sredy [Elements of Symmetry Snalysis of Differential Equations of Continuum Mechanics], Novosibirsk State Tecnical Univ., Novosibirsk, 2012, 659 pp. (In Russian)

[33] Lukashchuk S. Yu., “Conservation laws for time-fractional subdiffusion and diffusion-wave equations”, Nonlinear Dyn., 80:1–2 (2015), 791–802 | DOI | MR | Zbl

[34] Gazizov R. K., Ibragimov N. H., Lukashchuk S. Yu., “Nonlinear self-adjointness, conservation laws and exact solutions of time-fractional Kompaneets equations”, Commun. Nonlin. Sci. Numer. Simulat., 23:1-3 (2015), 153–163 | DOI | MR | Zbl

[35] Lukashchuk S. Y., “Constructing conservation laws for fractional-order integro-differential equations”, Theoret. and Math. Phys., 184:2 (2015), 1049–1066 | DOI | DOI | MR | Zbl

[36] Ibragimov N. H., Avdonina E. D., “Nonlinear self-adjointness, conservation laws, and the construction of solutions of partial differential equations using conservation laws”, Russian Math. Surveys, 68:5 (2013), 889–921 | DOI | DOI | MR | Zbl