Geodesic
Symmetry reduction and invariant solutions for nonlinear fractional diffusion equation with a source term
Ufa mathematical journal, Tome 8 (2016) no. 4, pp. 111-122 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We consider a problem on constructing invariant solutions to a nonlinear fractional differential equations of anomalous diffusion with a source. On the base of an earlier made group classification of the considered equation, for each case in the classification we construct the optimal systems of one-dimensional subalgebras of Lie algebras of infinitesimal operators of the point transformations group admitted by the equation. For each one-dimensional subalgebra of each optimal system we find the corresponding form of the invariant solution and made the symmetry reduction to an ordinary differential equation. We prove that there are three different types of the reduction equations (factor equations): a second order ordinary differential equation integrated by quadratures and two ordinary nonlinear fractional differential equations. For particular cases of the latter we find exact solutions.
Keywords: symmetry, symmetry reduction
Mots-clés : fractional diffusion equation, optimal system of subalgebras, invariant solution. 
@article{UFA_2016_8_4_a7,
     author = {S. Yu. Lukashchuk},
     title = {Symmetry reduction and invariant solutions for nonlinear fractional diffusion equation with a~source term},
     journal = {Ufa mathematical journal},
     pages = {111--122},
     year = {2016},
     volume = {8},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/UFA_2016_8_4_a7/}
}
TY  - JOUR
AU  - S. Yu. Lukashchuk
TI  - Symmetry reduction and invariant solutions for nonlinear fractional diffusion equation with a source term
JO  - Ufa mathematical journal
PY  - 2016
SP  - 111
EP  - 122
VL  - 8
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/UFA_2016_8_4_a7/
LA  - en
ID  - UFA_2016_8_4_a7
ER  - 
%0 Journal Article
%A S. Yu. Lukashchuk
%T Symmetry reduction and invariant solutions for nonlinear fractional diffusion equation with a source term
%J Ufa mathematical journal
%D 2016
%P 111-122
%V 8
%N 4
%U http://geodesic.mathdoc.fr/item/UFA_2016_8_4_a7/
%G en
%F UFA_2016_8_4_a7
S. Yu. Lukashchuk. Symmetry reduction and invariant solutions for nonlinear fractional diffusion equation with a source term. Ufa mathematical journal, Tome 8 (2016) no. 4, pp. 111-122. http://geodesic.mathdoc.fr/item/UFA_2016_8_4_a7/

[1] Samko S. G., Kilbas A. A., Marichev O. I., Integraly i proizvodnye drobnogo poryadka i nekotorye ikh prilozheniya, Nauka i tekhnika, Minsk, 1987, 688 pp. | MR

[2] Kilbas A. A., Srivastava H. M., Trujillo J. J., Theory and applications of fractional differential equations, Elsevier, Amsterdam, 2006, 539 pp. | MR | Zbl

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

[4] Zaslavsky G. M., “Chaos, fractional kinetics, and anomalous transport”, Phys. Rep., 371 (2002), 461–580 | DOI | MR | Zbl

[5] Uchaikin V. V., “Avtomodelnaya anomalnaya diffuziya i ustoichivye zakony”, UFN, 173:8 (2003), 847–876 | DOI

[6] Pskhu A. V., Uravneniya v chastnykh proizvodnykh drobnogo poryadka, Nauka, M., 2005, 199 pp.

[7] Mainardi F., Pagnini G., “The Wright functions as solutions of the time-fractional diffusion equation”, App. Math. Comp., 141:1 (2003), 51–62 | DOI | MR | Zbl

[8] Mainardi F., “The fundamental solutions for the fractional diffusion-wave equation”, Appl. Math. Lett., 9:6 (2007), 23–28 | DOI | MR

[9] Ovsyannikov L. V., Gruppovoi analiz differentsialnykh uravnenii, Nauka, M., 1978, 339 pp.

[10] Ibragimov N. H. (ed.), CRC Handbook of Lie Group Analysis of Differential Equations, v. 1, CRC Press, Boca Raton, 1994, 430 pp. | MR | Zbl

[11] Buckwar E., Luchko Yu., “Invariance of a partial differential equation of fractional order under the Lie group of scaling transformations”, J. Math. Anal. Appl., 227 (1998), 81–97 | DOI | MR | Zbl

[12] Gorenflo R., Luchko Yu., Mainardi F., “Wright functions as scale-invariant solutions of the diffusion-wave equation”, J. Comp. Appl. Math., 118:1 (2000), 175–191 | DOI | MR | Zbl

[13] Gazizov R. K., Kasatkin A. A., Lukaschuk S. Yu., “Nepreryvnye gruppy preobrazovanii differentsialnykh uravnenii drobnogo poryadka”, Vestnik UGATU, 9:3(21) (2007), 125–135

[14] Gazizov R. K., Kasatkin A. A., Lukashchuk S. Yu., “Symmetry properties of fractional diffusion equations”, Physica Scripta, 136 (2009), 014016 | DOI

[15] Gazizov R. K., Kasatkin A. A., Lukaschuk S. Yu., “Uravneniya s proizvodnymi drobnogo poryadka: zameny peremennykh i nelokalnye simmetrii”, Ufimskii mat. zhurnal, 4:4 (2012), 54–68 | MR | Zbl

[16] Gazizov R. K., Kasatkin A. A., Lukashchuk S. Yu., “Linearly autonomous symmetries of the ordinary fractional differential equations”, Proc. of 2014 International Conference on Fractional Differentiation and Its Applications, ICFDA 2014, IEEE, 2014, 1–6 | DOI

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

[18] Lukaschuk S. Yu., “O postroenii zakonov sokhraneniya dlya integro-differentsialnykh uravnenii drobnogo poryadka”, TMF, 184:2 (2015), 179–199 | DOI | MR

[19] Dorodnitsyn V. A., “Ob invariantnykh resheniyakh uravneniya nelineinoi teploprovodnosti s istochnikom”, ZhVMMF, 22:6 (1982), 1393–1400 | MR | Zbl

[20] Lukashchuk S. Yu., Makunin A. V., “Group classification of nonlinear time-fractional diffusion equation with a source term”, Appl. Math. Comp., 257 (2015), 335–343 | DOI | MR | Zbl

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

[22] Belevtsov N. S., Lukaschuk S. Yu., “Gruppovaya klassifikatsiya odnogo obyknovennogo differentsialnogo uravneniya drobnogo poryadka”, Mavlyutovskie chteniya, Materialy IX Vserossiiskoi molodezhnoi nauchnoi konferentsii (28–30 oktyabrya 2015 g.), v. 3, UGATU, Ufa, 2015, 170–175