×

Lie symmetry and exact solution of the time-fractional Hirota-Satsuma Korteweg-de Vries system. (English) Zbl 1477.35227

Summary: In the present work, we consider the nonlinear time-fractional Hirota-Satsuma KdV (Korteweg-de Vries) system in the sense of the Riemann-Liouville fractional calculus and the Erdélyi-Kober fractional calculus. By appealing to Lie group analysis, we derive the symmetry groups of transformations under which the given equations remain invariant. We also construct the symmetry reductions and particular group invariant solutions for the given system of equations. Finally, in order to highlight the importance of the study, the physical significance of the solution, which is described in this paper, is investigated and illustrated graphically.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
22E70 Applications of Lie groups to the sciences; explicit representations
26A33 Fractional derivatives and integrals
35R11 Fractional partial differential equations
35R03 PDEs on Heisenberg groups, Lie groups, Carnot groups, etc.
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bluman, G. W.; Anco, S. C., Symmetries and Integration Methods for Differential Equations, Springer Series on Applied Mathematical Sciences, 154, 0 (2002) · Zbl 1013.34004
[2] Bluman, G. W.; Kumei, S., Symmetries and Differential Equations, Springer Series on Applied Mathematical Sciences, 81, 0 (1989) · Zbl 0698.35001
[3] Buckwar, E.; Luchko, Y. F., Invariance of a Partial Differential Equation of Fractional Order Under the Lie Group of Scaling Transformations, J. Math. Anal. Appl., 227, 81-97 (1998) · Zbl 0932.58038 · doi:10.1006/jmaa.1998.6078
[4] Cattani, C.; Srivastava, H. M.; Yang, X.-J., Fractional Dynamics (2015), Berlin and Warsaw: Emerging Science Publishers (De Gruyter Open), Berlin and Warsaw · Zbl 1333.00024 · doi:10.1515/9783110472097
[5] Djordjević, V. D.; Atanacković, T. M., Similarity Solutions to Nonlinear Heat Conduction and Burgers/Korteweg-deVries Fractional Equations, J. Comput. Appl. Math., 222, 701-714 (2008) · Zbl 1157.35470 · doi:10.1016/j.cam.2007.12.013
[6] Fan, E.-G.; Hon, Y.-C., A Series of Travelling Wave Solutions for Two Variant Boussinesq Equations in Shallow Water Waves, Chaos Solitons Fractals, 15, 559-566 (2003) · Zbl 1031.76008 · doi:10.1016/S0960-0779(02)00144-3
[7] Gazizov, R. K.; Kasatkin, A. A.; Lukashchuk, S. Y., Symmetry Properties of Fractional Diffusion Equations, Phys. Scripta, T136, 1-5 (2009) · doi:10.1088/0031-8949/2009/T136/014016
[8] Gepreel, K. A.; Omran, S., The Exact Solutions for the Nonlinear Partial Fractional Differential Equations, Chinese Phys. B, 21, 11, 110204-110211 (2012) · Zbl 1274.70029 · doi:10.1088/1674-1056/21/11/110204
[9] Gupta, A. K.; Ray, S. S., Comparison Between Homotopy Pertubation Method and Optimal Homotopy Asymptotic Method for the Soliton Solutions of Bossinesq-Burger Equations, Computers and Fluids, 103, 34-41 (2014) · Zbl 1391.76547 · doi:10.1016/j.compfluid.2014.07.008
[10] (Ed.), R. Hilfer, Applications of Fractional Calculus in Physics (2000), Singapore, New Jersey, London and Hong Kong: World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong · Zbl 0998.26002
[11] Hirota, R.; Satsuma, J., Soliton Solutions of a Coupled Korteweg-de Vries Equation, Phys. Lett. A, 85, 407-408 (1981) · doi:10.1016/0375-9601(81)90423-0
[12] Hu, J.-A.; Ye, Y.-J.; Shen, S.-F.; Zhang, J.-U., Lie Symmetry Analysis of the Time Fractional KDV-Type Equation, Appl. Math. Comput., 233, 439-444 (2014) · Zbl 1334.37075
[13] Jumarie, G., On the Solution of the Stochastic Differential Equation of Exponential Growth Driven by Fractional Brownian Motion, Appl. Math. Lett., 18, 817-826 (2005) · Zbl 1075.60068 · doi:10.1016/j.aml.2004.09.012
[14] Jumarie, G., Modified Riemann-Liouville Derivative and Fractional Taylor Series of Non-Differentiable Functions Further Results, Comput. Math. Appl., 51, 1367-1376 (2006) · Zbl 1137.65001 · doi:10.1016/j.camwa.2006.02.001
[15] Jumarie, G., Table of some Basic Fractional Calculus Formulae Derived from a Modified Riemann-Liouville Derivative for Non-Differentiable Functions, Appl. Math. Lett., 22, 378-385 (2009) · Zbl 1171.26305 · doi:10.1016/j.aml.2008.06.003
[16] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations. North-Holland Mathematical Studies. Vol. 204 (2006), Amsterdam, London and New York: Elsevier (North-Holland) Science Publishers, Amsterdam, London and New York · Zbl 1092.45003
[17] Liu, C.-S., On the Local Fractional Derivative of Everywhere Non-Differentiable Continuous Functions on Intervals, Commun. Nonlinear Sci. Numer. Simulat., 42, 229-235 (2017) · Zbl 1473.26007 · doi:10.1016/j.cnsns.2016.05.029
[18] Liu, H.-Z., Complete Group Classifications and Symmetry Reductions of the Fractional Fifth-Order KdV Types of Equations, Stud. Appl. Math., 131, 317-330 (2013) · Zbl 1277.35305 · doi:10.1111/sapm.12011
[19] Lü, D.-Z., Jacobi Elliptic Function Solutions For Two Variant Bossinesq Equations, Chaos Solitons Fractals, 24, 1373-1385 (2005) · Zbl 1072.35567 · doi:10.1016/j.chaos.2004.09.085
[20] Lu, D.-C.; Hong, B.-J.; Tian, L.-X., New Explicit Exact Solutions for the Generalized Coupled Hirota-Satsuma KdV System, Comput. Math. Appl., 53, 1181-1190 (2007) · Zbl 1129.35445 · doi:10.1016/j.camwa.2006.08.047
[21] Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations (1993), New York, Chichester, Brisbane, Toronto and Singapore: A Wiley-Interscience Publication, John Wiley and Sons, New York, Chichester, Brisbane, Toronto and Singapore · Zbl 0789.26002
[22] Mohamed, S. M.; Gepreel, K. A., Numerical Solutions for the Time Fractional Variant Bossinesq Equation by Homotopy Analysis Method, Sci. Res. Essays, 8, 2163-2170 (2013) · doi:10.5897/SRE2013.5460
[23] Neirameh, A., Soliton Solutions of the Time Fractional Generalized Hirota-Satsuma Coupled KdV System, Appl. Math. Inform. Sci., 9, 1847-1853 (2015)
[24] Oldham, K. B.; Spanier, J., The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order (1974), New York and London: Academic Press, New York and London · Zbl 0292.26011
[25] Olver, P. J., Applications of Lie Groups to Differential Equations. Graduate Texts in Mathematics (1986), New York, Berlin, Heidelberg and Tokyo: Springer-Verlag, New York, Berlin, Heidelberg and Tokyo · Zbl 0588.22001 · doi:10.1007/978-1-4684-0274-2
[26] Podlubny, I., Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications (1999), New York, London, Sydney, Tokyo and Toronto: Mathematics in Science and Engineering, Vol., New York, London, Sydney, Tokyo and Toronto · Zbl 0924.34008
[27] Costa, F. Silva; Mar, J. A. P. F., Similarity Solution to Fractional Nonlinear Space-Time Diffusion-Wave Equation, J. Math. Phys., 56, 1-16 (2015) · Zbl 1507.35318 · doi:10.1063/1.4915293
[28] Singh, H.; Srivastava, H. M., Numerical Simulation for Fractional-Order Bloch Equation Arising in Nuclear Magnetic Resonance by Using the Jacobi Polynomials, Appl. Sci., 10, 1-18 (0000)
[29] Singh, K.; Gupta, R. K., Exact Solutions of a Variant Boussinesq System, Internat. J. Engrg. Sci., 44, 1256-1268 (2006) · Zbl 1213.35362 · doi:10.1016/j.ijengsci.2006.07.009
[30] Srivastava, H. M., The Zeta and Related Functions: Recent Developments, J. Adv. Engrg. Comput., 3, 329-354 (2019) · doi:10.25073/jaec.201931.229
[31] Srivastava, H. M., Some General Families of the Hurwitz-Lerch Zeta Functions and Their Applications: Recent Developments and Directions for Further Researches, Proc. Inst. Math. Mech. Nat. Acad. Sci. Azerbaijan, 45, 234-269 (2019) · Zbl 1436.11111
[32] Srivastava, H. M., Fractional-Order Derivatives and Integrals: Introductory Overview and Recent Developments, Kyungpook Math. J., 60, 73-116 (2020) · Zbl 1453.26008
[33] Srivastava, H. M., Operators of Basic (or q-) Calculus and Fractional q-Calculus and Their Applications in Geometric Function Theory of Complex Analysis, Iran. J. Sci. Technol. Trans. A: Sci., 44, 327-344 (2020) · doi:10.1007/s40995-019-00815-0
[34] Srivastava, H. M.; Abdel-Gawad, H. I.; Saad, K. M., Stability of Traveling Waves Based upon the Evans Function and Legendre Polynomials, Appl. Sci., 10, 1-16 (2020) · doi:10.3390/app10030846
[35] Srivastava, H. M.; Baliarsingh, P., The Leibniz and Chain Rules for Fractional Derivatives, Appl. Anal. Optim., 3, 343-357 (2019) · Zbl 1485.26009
[36] Srivastava, H. M.; Saad, K. M., Some New Models of the Time-Fractional Gas Dynamics Equation, Adv. Math. Models Appl., 3, 1, 5-17 (2018)
[37] Srivastava, H. M.; Saad, K. M., New Approximate Solution of the Time-Fractional Nagumo Equation Involving Fractional Integrals Without Singular Kernel, Appl. Math. Inform. Sci., 14, 1-8 (2020) · doi:10.18576/amis/140101
[38] Srivastava, H. M.; Saad, K. M.; Al-Sharif, E. H. F., New Analysis of the Time-Fractional and Space-Time Fractional-Order Nagumo Equation, J. Inform. Math. Sci., 10, 545-561 (2018)
[39] Srivastava, H. M.; Saad, K. M.; Gómez-Aguilar, J. F., A Fractional Quadratic Autocatalysis Associated with chemical Clock Reactions Involving Linear Inhibition, Chaos Solitons Fractals, 132, 1-9 (2020) · Zbl 1444.92145
[40] Srivastava, H. M.; Jena, R. M.; Chakraverty, S.; Jena, S. K., Dynamic Response Analysis of Fractionally-Damped Generalized Bagley-Torvik Equation Subject to External Loads, Russ. J. Math. Phys., 27, 254-268 (2020) · Zbl 1440.65272 · doi:10.1134/S1061920820020120
[41] Srivastava, H. M.; Saad, K. M., A Comparative Study of the Fractional-Order Clock Chemical Model, Mathematics, 8, 1-14 (2020)
[42] Xu, M.-Y.; Tan, W.-C., Intermediate Processes and Critical Phenomena: Theory, Method and Progress of Fractional Operators and Their Applications to Modern Mechanics, Science in China Ser. G Phys. Mech. Astronom., 49, 257-272 (2006) · Zbl 1109.26005 · doi:10.1007/s11433-006-0257-2
[43] Yan, Z.-Y.; Zhang, H.-Q., New Explicit and Exact Travelling Wave Solutions for a System of Variant Boussinesq Equations in Mathematical Physics, Phys. Lett. A, 252, 291-296 (1999) · Zbl 0938.35130 · doi:10.1016/S0375-9601(98)00956-6
[44] Yang, X.-J.; Baleanu, D.; Srivastava, H. M., Local Fractional Integral Transforms and Their Applications (2016), Amsterdam, London and New York: Elsevier Science Publishers (Academic Press), Amsterdam, London and New York · Zbl 1336.44001
[45] Yang, X.-J.; Hristov, J.; Srivastava, H. M.; Ahmad, B., Modelling Fractal Waves on Shallow Water Surfaces via Local Fractional Korteweg-de Vries Equation, Abstr. Appl. Anal., 2014, 1-10 (2014) · Zbl 1468.76016
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.