On integrability of the time fractional nonlinear heat conduction equation. (English) Zbl 1439.35540
Summary: Under investigation in this letter is a time fractional nonlinear heat conduction equation which usually appears in mathematics physics, integrable system, fluid mechanics and nonlinear areas, by means of applying the fractional symmetry group method with the sense of Riemann-Liouville (R-L)fractional derivative. First of all, we use the fractional symmetry group method to obtain symmetries of the time fractional nonlinear heat conduction equation. Second, according to the above find symmetries, this equation can be reduced to a fractional ordinary differential equation. Moreover, invariant solutions of the time fractional nonlinear heat conduction equation are yielded. Finally, with the aid of the Ibragimov theorem, the conservation laws are also find to the time fractional nonlinear heat conduction equation. These new results are an effective complement to existing knowledge.
MSC:
| 35R11 | Fractional partial differential equations |
| 22E70 | Applications of Lie groups to the sciences; explicit representations |
| 35K59 | Quasilinear parabolic equations |
| 35L65 | Hyperbolic conservation laws |
| 35Q51 | Soliton equations |
Keywords:
fractional symmetry group method; time fractional nonlinear heat conduction equation; invariant solutions; conservation lawsReferences:
| [1] | Asadollah, Aghajani; Pourhadi, Application of measure of noncompactness to a Cauchy problem for the fractional differential equations in Banach spaces, Fract. Calc. Appl. Anal., 16, 4, 962-977 (2013) · Zbl 1312.47095 |
| [2] | Baleanu, D.; Inc, M.; Yusuf, A.; Aliyu, A. I., Time fractional third-order evolution equation: Symmetry analysis, explicit solutions, and conservation laws, J. Comput. Nonlinear Dyn., 13, Article 021011 pp. (2018) |
| [3] | Barone, V.; De Lillo, S.; Lupo, G., Dirichlet-to-Neumann Map for a nonlinear diffusion equation, Stud. Appl. Math., 126, 2, 145-155 (2011) · Zbl 1213.35238 |
| [4] | Bluman, G. W.; Anco, S., Symmetry and Integration Methods for Differential Equations (2002), Springer-Verlag: Springer-Verlag Heidelburg · Zbl 1013.34004 |
| [5] | Buckwar, E.; Luchko, Y., Invariance of a partial differential equation of fractional order under the lie group of scaling transformations, J. Math. Anal. Appl., 227, 1, 81-97 (1998) · Zbl 0932.58038 |
| [6] | Bueno-Orovio, A.; Kay, D.; Burrage, K., Fourier spectral methods for fractional-in-space reaction – diffusion equations, BIT Numer. Math., 54, 4, 937-954 (2014) · Zbl 1306.65265 |
| [7] | Bulut, H.; Pandir, Y.; Demiray, S. T., Exact solutions of time-fractional KdV equations by using generalized Kudryashov method, Int. J. Model. Optim., 4, 4, 315 (2014) |
| [8] | Changzheng, Q., Exact solutions to nonlinear diffusion equations obtained by a generalized conditional symmetry method, IMA J. Appl. Math., 62, 3, 283-302 (1999) · Zbl 0936.35039 |
| [9] | El-Shahed, M.; Shammakh, W., Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations, Comput. Math. Appl., 59, 3, 1363-1375 (2010) · Zbl 1189.34014 |
| [10] | Erturk, V. S.; Momani, S.; Odibat, Z., Application of generalized differential transform method to multi-order fractional differential equations, Commun. Nonlinear Sci. Numer. Simul., 13, 8, 1642-1654 (2008) · Zbl 1221.34022 |
| [11] | Gao, Li-Na; Zhao, Xue-Ying; Zi, Yao-Yao; Yu, Jun; Lü, Xing, Resonant behavior of multiple wave solutions to a Hirota bilinear equation, Comput. Math. Appl., 72, 1225-1229 (2016) · Zbl 1361.35155 |
| [12] | Gazizov, R. K.; Ibragimov, N. H.; Lukashchuk, S. Y., Nonlinear self-adjointness, conservation laws and exact solutions of time-fractional Kompaneets equations, Commun. Nonlinear Sci. Numer. Simul., 23, 1-3, 153-163 (2015) · Zbl 1351.35250 |
| [13] | Gazizov, R. K.; Kasatkin, A. A.; Lukashchuk, S. Y., Continuous transformation groups of fractional differential equations, Vestn. Usatu, 9, 3, 21 (2007) |
| [14] | Gazizov, R. K.; Kasatkin, A. A.; Lukashchuk, S. Y., Symmetry properties of fractional diffusion equations, Phys. Scr., 2009, T136, Article 014016 pp. (2009) |
| [15] | Hosseini, K.; Ansari, R., New exact solutions of nonlinear conformable time-fractional boussinesq equations using the modified Kudryashov method, Waves Random Complex Media, 27, 4, 628-636 (2017) · Zbl 07659364 |
| [16] | Ibragimov, N. H., A new conservation theorem, J. Math. Anal. Appl., 333, 1, 311-328 (2007) · Zbl 1160.35008 |
| [17] | Jafari, H.; Kadkhoda, N.; Baleanu, D., Fractional lie group method of the time-fractional Boussinesq equation, Nonlinear Dynam., 81, 3, 1569-1574 (2015) · Zbl 1348.35189 |
| [18] | Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations (2006), Elsevier: Elsevier Amsterdam · Zbl 1092.45003 |
| [19] | Liu, H., Generalized symmetry classifications, integrable properties and exact solutions to the general nonlinear diffusion equations, Commun. Nonlinear Sci. Numer. Simul., 36, 21-28 (2016) · Zbl 1470.35017 |
| [20] | Liu, J.; Yang, X. J.; Cheng, M., Abound rogue wave type solutions to the extended (3+1)-dimensional Jimbo-Miwa equation, Comput. Math. Appl. (2019) · Zbl 1442.35387 |
| [21] | Liu, J.; Zhang, Y., Construction of lump soliton and mixed lump stripe solutions of (3+1)-dimensional soliton equation, Results Phys., 10, 94-98 (2018) |
| [22] | Liu, J.; Zhang, Y.; Muhammad, I., Resonant soliton and complexiton solutions for (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation, Comput. Math. Appl., 75, 11, 3939-3945 (2018) · Zbl 1420.35321 |
| [23] | Lü, Xing; Chen, Shou-Ting; Ma, Wen-Xiu, Constructing lump solutions to a generalized Kadomtsev-Petviashvili-Boussinesq equation, Nonlinear Dynam., 86, 523-534 (2016) · Zbl 1349.35007 |
| [24] | Lukashchuk, S. Y., Conservation laws for time-fractional subdiffusion and diffusion-wave equations, Nonlinear Dynam., 80, 1-2, 791-802 (2015) · Zbl 1345.35131 |
| [25] | Ma, W. X., Abundant lumps and their interaction solutions of (3+1)-dimensional linear PDEs, J. Geom. Phys., 133, 10-16 (2018) · Zbl 1401.35261 |
| [26] | Ma, W. X., Conservation laws by symmetries and adjoint symmetries, Discrete Contin. Dyn. Syst. Ser. S, 11, 707-721 (2018) · Zbl 1386.70041 |
| [27] | Ma, W. X., Interaction solutions to the Hirota-Satsuma-Ito equation in (2+1)-dimensions, Front. Math. China (2019) · Zbl 1421.35314 |
| [28] | Ma, W. X., A search for lump solutions to a combined fourth-order nonlinear PDE in (2+1)-dimensions, J. Appl. Anal. Comput., 9, 1-15 (2019) |
| [29] | Ma, W. X.; Li, J.; Khalique, C. M., A study on lump solutions to a generalized Hirota-Satsuma-Ito equation in (2+1)-dimensions, Complexity, 2018 (2018) · Zbl 1407.35177 |
| [30] | Meerschaert, M. M.; Scheffler, H. P.; Tadjeran, C., Finite difference methods for two-dimensional fractional dispersion equation, J. Comput. Phys., 211, 249-261 (2006) · Zbl 1085.65080 |
| [31] | Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations (1993), Wiley: Wiley New York · Zbl 0789.26002 |
| [32] | Odibat, Z.; Momani, S.; Hang, X., A reliable algorithm of homotopy analysis method for solving nonlinear fractional differential equations, Appl. Math. Model., 34, 3, 593-600 (2010) · Zbl 1185.65139 |
| [33] | Olver, P. J., Applications of Lie Groups to Differential Equations (1986), Springer-Verlag: Springer-Verlag Heidelberg · Zbl 0588.22001 |
| [34] | Osler, T. J., Leibniz rule for fractional derivatives generalized and an application to infinite series, SIAM J. Appl. Math., 18, 3, 658-674 (1970) · Zbl 0201.44102 |
| [35] | Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press San Diego · Zbl 0918.34010 |
| [36] | Rehman, M. U.; Khan, R. A., The Legendre wavelet method for solving fractional differential equations, Commun. Nonlinear Sci. Numer. Simul., 92, 6, 1275-1291 (2011) · Zbl 1315.65111 |
| [37] | Rui, W.; Zhang, X., Invariant analysis and conservation laws for the time fractional foam drainage equation, Eur. Phys. J. Plus, 130, 10, 192 (2015) |
| [38] | Rui, W.; Zhang, X., Lie symmetries and conservation laws for the time fractional Derrida-Lebowitz-Speer-Spohn equation, Commun. Nonlinear Sci. Numer. Simul., 34, 38-44 (2016) · Zbl 1510.35386 |
| [39] | Sahadevan, R.; Bakkyaraj, T., Invariant analysis of time fractional generalized Burgers and Korteweg – de Vries equations, J. Math. Anal. Appl., 393, 2, 341-347 (2012) · Zbl 1245.35142 |
| [40] | Sahoo, S.; Ray, S. S., Invariant analysis with conservation laws for the time fractional Drinfeld-Sokolov-Satsuma-Hirota equations, Chaos Solitons Fractals, 104, 725-733 (2017) · Zbl 1380.35162 |
| [41] | Samko, S.; Kilbas, A. A.; Marichev, O., Fractional Integrals and Derivatives: Theory and Applications (1993), Gordon and Breach science: Gordon and Breach science Yverdon, Switzerland · Zbl 0818.26003 |
| [42] | Yang, X.-J., A new integral transform operator for solving the heat-diffusion problem, Appl. Math. Lett., 64, 193-197 (2017) · Zbl 1353.35018 |
| [43] | Yang, X.-J.; Baleanu, D.; Srivastava, H., Local fractional similarity solution for the diffusion equation defined on cantor sets, Appl. Math. Lett., 47, 54-60 (2015) · Zbl 1388.35218 |
| [44] | Yang, X.-J.; Gao, F.; Srivastava, H. M., A new computational approach for solving nonlinear local fractional PDEs, J. Comput. Appl. Math., 339, 285-296 (2018) · Zbl 1490.35530 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
