×

zbMATH — the first resource for mathematics

Lie group analysis and conservation laws for the time-fractional third order KdV-type equation with a small perturbation parameter. (English) Zbl 1448.76118
Summary: The KdV equation is one of the most famous PDEs which has a vast field of applications in fluid dynamics. The scope of our research is the Lie group analysis of time-fractional KdV-type equation including a perturbation term. Thus, the Lie group theory is extended to both cases simultaneously. Geometric vector fields of symmetries and one-dimensional optimal system are found in order to reduce the equation. Finally, approximated conservation laws are given via modified Ibragimov’s theorem.
MSC:
76M60 Symmetry analysis, Lie group and Lie algebra methods applied to problems in fluid mechanics
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76M45 Asymptotic methods, singular perturbations applied to problems in fluid mechanics
35Q53 KdV equations (Korteweg-de Vries equations)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Abdel-Gawad Hamdy, I.; Osman, M. S., On the variational approach for analyzing the stability of solutions of evolution equations, Kyungpook Math. J., 53, 661-680 (2013) · Zbl 1297.65058
[2] Baikov, V. A.; Gazizov, R. K.; Ibragimov, N. H., Perturbation methods in group analysis, J. Sov. Math., 55, 1, 1450-1490 (1991) · Zbl 0759.35003
[3] Bluman, G.; Anco, S. C., (Symmetry and Integration Methods for Differential Equations. Symmetry and Integration Methods for Differential Equations, Applied Mathematical Sciences (2002), Springer-Verlag: Springer-Verlag New York) · Zbl 1013.34004
[4] Caponetto, R.; Dongola, G.; Fortuna, L.; Petras, I., Fractional Order Systems: Modeling and Control Applications (2010), World Scientific: World Scientific Singapore
[5] Djordjevic, V. D.; Atanackovic, T. M., Similarity solutions to nonlinear heat conduction and Burgers/korteweg-devries fractional equations, J. Comput. Appl. Math., 222, 2, 701-714 (2008) · Zbl 1157.35470
[6] Euler, M.; Euler, N.; Kohler, A., On the construction of approximate solutions for a multi-dimensional nonlinear heat equation, J. Phys. A: Math. Gen., 27, 6, 2083-2092 (1994) · Zbl 0837.35065
[7] Feroze, T.; Kara, A. H., Group theoretic methods for approximate invariants and Lagrangians for some classes of \(y + \varepsilon F ( t ) y^\prime + y = f ( y , y )\), Int. J. Nonlinear Mech., 37, 2, 275-280 (2002) · Zbl 1116.34318
[8] Gazizov, R. K.; Kasatkin, A. A.; Lukashchuk, S. Y., Continuous transformation groups of fractional differential equations, Vestn. USATU, 9, 3, 125-135 (2007)
[9] Godlewski, E.; Raviart, P. A., (Numerical Approximation of Hyperbolic Systems of Conservation Laws. Numerical Approximation of Hyperbolic Systems of Conservation Laws, Applied Mathematical Sciences (1996), Springer: Springer Berlin) · Zbl 0860.65075
[10] Gutzwiller, M. C., Moon-earth-sun: The oldest three-body problem, Rev. Mod. Phys., 70, 2, 589-639 (1998)
[11] Habibi, N.; Lashkarian, E.; Dastranj, E.; Hejazi, S. R., Lie symmetry analysis conservation laws and numerical approximations of time-fractional fokker-Planck equations for special stochastic process in foreign exchange markets, Physica A, 513, 750-766 (2019)
[12] Hejazi, S. R.; Hosseinpour, S.; Lashkaian, E., Approximate symmetries, conservation laws and numerical solutions for a class of perturbed linear wave type system, Quaest. Math., 42, 10, 1393-1409 (2019) · Zbl 1427.76197
[13] Hilfer, R., Applications of Fractional Calculus in Physics (2000), World Scientific: World Scientific Singapore · Zbl 0998.26002
[14] Ibragimov, N. H., Nonlinear Self-Adjointness in Constructing Conservation Laws, Vol. 7 (2011), Archives of ALGA
[15] Ibragimov, N. H., Nonlinear self-adjointness and conservation laws, J. Phys. A, 44, 43, Article 432002 pp. (2011) · Zbl 1270.35031
[16] 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, 889-921 (2013) · Zbl 1286.35013
[17] Ibragimov, N. H.; Kovalev, V. F., (Approximate and Renormgroup Symmetries. Approximate and Renormgroup Symmetries, Nonlinear Physical Science Springer (2009), Springer: Springer New York) · Zbl 1170.22001
[18] Johnpillai, A. G.; Kara, A. H., Variational formulation of approximate symmetries and conservation laws, Internat. J. Theoret. Phys., 40, 8, 1501-1509 (2001) · Zbl 1006.81035
[19] Johnpillai, A. G.; Kara, A. H.; Mahomed, F. M., A basis of approximate conservation laws for PDEs with a small parameter, Int. J. Nonlinear Mech., 41, 6-7, 830-837 (2006) · Zbl 1160.35318
[20] Johnpillai, A. G.; Kara, A. H.; Mahomed, F. M., Approximate noether-type symmetries and conservation laws via partial Lagrangians for PDEs with a small parameter, J. Comput. Appl. Math., 223, 1, 508-518 (2009) · Zbl 1158.35306
[21] Jumarie, G., Modified Riemann-Liouville derivative and fractional taylor series of nondifferentiable functions further results, Comput. Math. Appl., 51, 9-10, 1367-1376 (2006) · Zbl 1137.65001
[22] Kara, A. F.; Mahomed, F. M.; Qu, C. Z., Approximate potential symmetries for partial differential equations, J. Phys. A: Math. Gen., 33, 37, 6601-6613 (2000) · Zbl 0965.35025
[23] Kara, A. H.; Mahomed, F. M.; Unal, G., Approximate symmetries and conservation laws with applications, Internat. J. Theoret. Phys., 38, 9, 2389-2399 (1999) · Zbl 0989.37076
[24] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations (2006), Elsevier: Elsevier Amsterdam, The Netherlands · Zbl 1092.45003
[25] Kiryakova, V., (Generalised Fractional Calculus and Applications. Generalised Fractional Calculus and Applications, Pitman Research Notes in Mathematics, vol. 301 (1994)) · Zbl 0882.26003
[26] Lashkarian, E.; Hejazi, S. R., Group analysis of the time fractional generalized diffusion equation, Physica A, 479, 572-579 (2017)
[27] Lashkarian, E.; Hejazi, S. R., Exact solutions of the time fractional nonlinear Schrödinger equation with two different methods, Math. Methods Appl. Sci., 41, 7, 2664-2672 (2018) · Zbl 1391.76655
[28] Lashkarian, E.; Hejazi, S. R.; Dastranj, E., Conservation laws of \(( 3 + \alpha )\)-dimensional time-fractional diffusion equation, Comput. Math. Appl., 75, 3, 740-754 (2018) · Zbl 1408.37132
[29] Lashkarian, E.; Hejazi, S. R.; Habibi, N.; Motamednezhad, A., Symmetry properties conservation laws reduction and numerical approximations of timefractional cylindrical-Burgers equation, Commun. Nonlinear Sci. Numer. Simul., 67, 176-191 (2019)
[30] Lukashchuk, S. Y., Conservation laws for time-fractional subdiffusion and diffusion-wave equations, Nonlinear Dynam., 80, 1-2, 791-802 (2015) · Zbl 1345.35131
[31] Lukashchuk, S. Y., Constructing conservation laws for fractional-order integro-differential equations, Theoret. Math. Phys., 184, 2, 1049-1066 (2015) · Zbl 1336.45004
[32] Lukashchuk, S. Y., Approximate conservation laws for fractional differential equations, Commun. Nonlinear Sci. Numer. Simul., 68, 147-159 (2019)
[33] Mahomed, F. M.; Qu, C. Z., Approximate conditional symmetries for partial differential equations, J. Phys. A: Math. Gen., 33, 2, 343-356 (2000) · Zbl 0954.35016
[34] Mainardy, F., Fractional Calculus and Waves in Linear Viscoelasticity, an Introduction to Mathematical Models (2010), Imperial College Press: Imperial College Press Singapore
[35] Meerschaert, M.; Sikorskii, A., Stochastic Models for Fractional Calculus (2012), De Gruyter: De Gruyter Berlin · Zbl 1247.60003
[36] Naderifard, A.; Hejazi, S. R.; Dastranj, E., Symmetry properties conservation laws and exact solutions of time-fractional irrigation equation, Waves Random Complex Media, 29, 1, 178-194 (2019)
[37] Olver, P. J., Application of Lie Groups to Differential Equations (1986), Springer-Verlag: Springer-Verlag New York
[38] Osman, M. S., Multiwave solutions of time-fractional (2+1)-dimensional Nizhnik-Novikov-Veselov equations, Pramana J. Phys., 88, 4, 67-75 (2017)
[39] Ouhadan, A.; Elkinani, E. H., Exact solutions of time fractional Kolmogorov equation by using lie symmetry analysis, J. Fract. Calc. Appl., 5, 1, 97-104 (2014)
[40] Podlubny, I., Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and Some of their Applications (1999), Academic Press: Academic Press New York · Zbl 0924.34008
[41] Rezazadeh, H.; Osman, M. S.; Eslami, M.; Ekici, M.; Sonmezoglu, A.; Asma, M.; Othman, W. A.M.; Wong, B. R.; Mirzazadeh, M.; Zhou, Q.; Biswas, A.; Belick, M., Mitigating internet bottleneck with fractional temporal evolution of optical solitons having quadratic-cubic nonlinearity, Optik (Stuttg), 164, 84-92 (2018)
[42] Sabatier, J.; Agrawal, O. P.; Machado, J. A.T., Advances in fractional calculus, (Theoretical Developments and Applications in Physics and Engineering (2007), Springer: Springer Dordrecht) · Zbl 1116.00014
[43] Saberi, E.; Hejazi, S. R., A comparison of conservation laws of the boussinesq system, Kragujevac J. Math., 43, 2, 173-200 (2019)
[44] 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
[45] Samko, S.; Kilbas, A.; Marichev, O., Fractional Integrals and Derivatives: Theory and Applications (1993), Gordon and Breach Sci. Publishers: Gordon and Breach Sci. Publishers London · Zbl 0818.26003
[46] Singla, K.; Gupta, R. K., Space-time fractional nonlinear partial differential equations: symmetry analysis and conservation laws, Nonlinear Dynam., 89, 321-331 (2017) · Zbl 1374.35429
[47] Sun, H.; Zhang, Y.; Baleanu, D.; Chen, W.; Chen, Y., A new collection of real world applications of fractional calculus in science and engineering, Commun. Nonlinear Sci. Numer. Simul., 64, 213-231 (2018)
[48] Wang, G.; Xi. Liu, Y.; Zhang, Y., Lie Symmetry analysis to the time fractional generalized fifth-order KDV equation, Commun. Nonlinear Sci. Numer. Simul., 18, 9, 2321-2326 (2013) · Zbl 1304.35624
[49] Wiesel, William E., Modern Astrodynamics, 107 (2010), Aphelion Press: Aphelion Press Ohio, ISBN 978-145378-1470
[50] Zhou, Y.; Wang, J.; Zhang, L., Basic Theory of Fractional Differential Equations (2016), World Scientific: World Scientific London
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.