Time-fractional KdV equation: Formulation and solution using variational methods. (English) Zbl 1234.35219

Summary: The semi-inverse method has been used to derive the Lagrangian of the Korteweg-de Vries (KdV) equation. Then the time operator of the Lagrangian of the KdV equation has been transformed into fractional domain in terms of the left-Riemann-Liouville fractional differential operator. The variational of the functional of this Lagrangian leads neatly to the Euler-Lagrange equation. Via Agrawal’s method, one can easily derive the time-fractional KdV equation from this Euler-Lagrange equation. Remarkably, the time-fractional term in the resulting KdV equation is obtained in Riesz fractional derivative in a direct manner. As a second step, the derived time-fractional KdV equation is solved using He’s variational-iteration method. The calculations are carried out using initial condition depends on the nonlinear and dispersion coefficients of the KdV equation. We remark that more pronounced effects and deeper insight into the formation and properties of the resulting solitary wave by additionally considering the fractional order derivative beside the nonlinearity and dispersion terms.


35Q53 KdV equations (Korteweg-de Vries equations)
35R11 Fractional partial differential equations
35C08 Soliton solutions
35A15 Variational methods applied to PDEs
Full Text: DOI arXiv


[1] Riewe, F.: Nonconservative Lagrangian and Hamiltonian mechanics. Phys. Rev. E 53(2), 1890–1899 (1996) · doi:10.1103/PhysRevE.53.1890
[2] Riewe, F.: Mechanics with fractional derivatives. Phys. Rev. E 55(3), 3581–3592 (1997) · doi:10.1103/PhysRevE.55.3581
[3] Tavazoei, M.S., Haeri, M.: Describing function based methods for predicting chaos in a class of fractional order differential equations. Nonlinear Dyn. 57(3), 363–373 (2009) · Zbl 1176.34051 · doi:10.1007/s11071-008-9447-y
[4] Bateman, H.: On dissipative systems and related variational principles. Phys. Rev. 38(4), 815–819 (1931) · Zbl 0003.01101 · doi:10.1103/PhysRev.38.815
[5] Agrawal, O.P.: Formulation of Euler–Lagrange equations for fractional variational problems. J. Math. Anal. Appl. 272(1), 368–379 (2002) · Zbl 1070.49013 · doi:10.1016/S0022-247X(02)00180-4
[6] Agrawal, O.P.: A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dyn. 38(4), 323–337 (2004) · Zbl 1121.70019 · doi:10.1007/s11071-004-3764-6
[7] Agrawal, O.P.: Fractional variational calculus and the transversality conditions. J. Phys. A, Math. Gen. 39, 10375 (2006) · Zbl 1097.49021 · doi:10.1088/0305-4470/39/33/008
[8] Agrawal, O.P.: Fractional variational calculus in terms of Riesz fractional derivatives. J. Phys. A, Math. Theor. 40, 6287 (2007) · Zbl 1125.26007 · doi:10.1088/1751-8113/40/24/003
[9] Baleanu, D., Avkar, T.: Lagrangians with linear velocities within Riemann–Liouville fractional derivatives. Nuovo Cim. B 119, 73–79 (2004) · Zbl 1120.26001
[10] Baleanu, D., Muslih, S.I.: Lagrangian formulation of classical fields within Riemann–Liouville fractional derivatives. Phys Scr. 72, 119–123 (2005) · Zbl 1122.70360 · doi:10.1238/Physica.Regular.072a00119
[11] Muslih, S.I., Baleanu, D., Rabei, E.: Hamiltonian formulation of classical fields within Riemann–Liouville fractional derivatives. Phys. Scr. 73, 436–438 (2006) · Zbl 1165.70310 · doi:10.1088/0031-8949/73/5/003
[12] Rabei, E.M., Altarazi, I.M.A., Muslih, S.I., Baleanu, D.: Fractional WKB approximation. Nonlinear Dyn. 57(1–2), 171–175 (2009) · Zbl 1176.70014 · doi:10.1007/s11071-008-9430-7
[13] Baleanu, D.: Fractional variational principles in action. Phys. Scr. T136, 014006 (2009)
[14] Herzallah, M.A.E., Baleanu, D.: Fractional-order Euler–Lagrange equations and formulation of Hamiltonian equations. Nonlinear Dyn. 58(1–2), 385–391 (2009) · Zbl 1183.26006 · doi:10.1007/s11071-009-9486-z
[15] Baleanu, D., Trujillo, J.I.: A new method of finding the fractional Euler–Lagrange and Hamilton equations within Caputo fractional derivatives. Commun. Nonlinear Sci. Numer. Simul. 15(5), 1111–1115 (2010) · Zbl 1221.34008 · doi:10.1016/j.cnsns.2009.05.023
[16] Tarasov, V.E., Zaslavsky, G.M.: Fractional Ginzburg–Landau equation for fractal media. Phys. A, Stat. Mech. Appl. 354, 249–261 (2005) · doi:10.1016/j.physa.2005.02.047
[17] Tarasov, V.E., Zaslavsky, G.M.: Nonholonomic constraints with fractional derivatives. J. Phys. A, Math. Gen. 39(31), 9797–9815 (2006) · Zbl 1101.70011 · doi:10.1088/0305-4470/39/31/010
[18] Heymans, N.: Fractional calculus description of non-linear viscoelastic behaviour of polymers. Nonlinear Dyn. 38(1–2), 221–231 (2004) · Zbl 1142.74312 · doi:10.1007/s11071-004-3757-5
[19] Frederico, G.S.F., Torres, D.F.M.: Fractional conservation laws in optimal control theory. Nonlinear Dyn. 53(3), 215–222 (2008) · Zbl 1170.49017 · doi:10.1007/s11071-007-9309-z
[20] Mendes, R.V.: A fractional calculus interpretation of the fractional volatility model. Nonlinear Dyn. 55(4), 395–399 (2009) · Zbl 1187.91231 · doi:10.1007/s11071-008-9372-0
[21] Tenreiro Machado, J.A.: Calculation of fractional derivatives of noisy data with genetic algorithms. Nonlinear Dyn. 57(1–2), 253–260 (2009) · Zbl 1176.94016 · doi:10.1007/s11071-008-9436-1
[22] Attari, M., Haeri, M., Tavazoei, M.S.: Analysis of a fractional order Van der Pol-like oscillator via describing function method. Nonlinear Dyn. 61(1–2), 265–274 (2010) · Zbl 1204.70018 · doi:10.1007/s11071-009-9647-0
[23] Agrawal, O.P., Tenreiro Machado, J.A, Sabatier, J. (Guest Editors): Special issue on ”Fractional Derivatives and their Applications”. Nonlinear Dyn. 38(4) (2004)
[24] Sabatier, J., Agrawal, O.P., Tenreiro Machado, J.A. (eds.): Advances in Fractional Calculus. Springer, Dordrecht (2007) · Zbl 1116.00014
[25] Baleanu, D., Tenreiro Machado, J.A. (Guest Editors): Special issue on ”Fractional Differentiation and its Applications (FDA08)”. Phys. Scr. T136 (2009)
[26] Korteweg, D.J., de Vries, G.: On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves. Philos. Mag. 39(5), 422 (1895) · JFM 26.0881.02 · doi:10.1080/14786449508620739
[27] Fung, M.K.: KdV Equation as an Euler–Poincare’ equation. Chin. J. Phys. 35(6), 789 (1997)
[28] Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006) · Zbl 1092.45003
[29] Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999) · Zbl 0924.34008
[30] Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, New York (1998)
[31] Luchko, Y., Srivastava, H.M.: The exact solution of certain differential equations of fractional order by using operational calculus. Comput. Math. Appl. 29, 73–85 (1995) · Zbl 0824.44011 · doi:10.1016/0898-1221(95)00031-S
[32] Babakhani, A., Gejji, V.D.: Existence of positive solutions of nonlinear fractional differential equations. J. Math. Anal. Appl. 278, 434–442 (2003) · Zbl 1027.34003 · doi:10.1016/S0022-247X(02)00716-3
[33] Delbosco, D.: Fractional calculus and function spaces. J. Fractal Calc. 6, 45–53 (1996) · Zbl 0829.46018
[34] Zhang, S.Q.: Existence of positive solution for some class of nonlinear fractional differential equations. J. Math. Anal. Appl. 278, 136–148 (2003) · Zbl 1026.34008 · doi:10.1016/S0022-247X(02)00583-8
[35] Saha Ray, S., Bera, R.K.: An approximate solution of a nonlinear fractional differential equation by Adomian decomposition method. Appl. Math. Comput. 167, 561–571 (2005) · Zbl 1082.65562 · doi:10.1016/j.amc.2004.07.020
[36] He, J.-H.: Approximate analytical solution for seepage flow with fractional derivatives in porous media. Comput. Methods Appl. Mech. Eng. 167, 57–68 (1998) · Zbl 0942.76077 · doi:10.1016/S0045-7825(98)00108-X
[37] Momani, S., Odibat, Z., Alawnah, A.: Variational iteration method for solving the space- and time-fractional KdV equation. Numer. Methods Part. Differ. Equ. 24(1), 261–271 (2008) · Zbl 1130.65132
[38] Molliq, R.Y., Noorani, M.S.M., Hashim, I.: Variational iteration method for fractional heat- and wave-like equations. Nonlinear Anal., Real World Appl. 10, 1854–1869 (2009) · Zbl 1172.35302 · doi:10.1016/j.nonrwa.2008.02.026
[39] Inokuti, M., Sekine, H., Mura, T.: General use of the Lagrange multiplier in non-linear mathematical physics. In: Nemat-Nasser, S. (ed.) Variational Method in the Mechanics of Solids. Pergamon Press, Oxford (1978)
[40] He, J.-H.: A new approach to nonlinear partial differential equations. Commun. Nonlinear Sci. Numer. Simul. 2(4), 230–235 (1997) · doi:10.1016/S1007-5704(97)90007-1
[41] He, J.-H.: Variational-iteration–a kind of nonlinear analytical technique: some examples. Int. J. Nonlinear Mech. 34, 699 (1999) · Zbl 1342.34005 · doi:10.1016/S0020-7462(98)00048-1
[42] He, J.-H.: Semi-inverse method of establishing generalized variational principles for fluid mechanics with emphasis on turbo-machinery aerodynamics. Int. J. Turbo Jet-Engines 14(1), 23–28 (1997)
[43] He, J.-H.: Variational principles for some nonlinear partial differential equations with variable coefficients. Chaos Solitons Fractals 19, 847–851 (2004) · Zbl 1135.35303 · doi:10.1016/S0960-0779(03)00265-0
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.