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Volumn 65, Issue 1-2, 2011, Pages 55-63

Time-fractional KdV equation: Formulation and solution using variational methods

Author keywords

Euler Lagrange equation; Fractional KdV equation; He's variational iteration method; Riemann Liouville fractional differential operator; Riesz fractional derivative; Solitary wave

Indexed keywords

EULER-LAGRANGE EQUATIONS; FRACTIONAL DERIVATIVES; FRACTIONAL DIFFERENTIAL OPERATORS; KDV EQUATIONS; SOLITARY WAVE;

EID: 79959502347     PISSN: 0924090X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11071-010-9873-5     Document Type: Article
Times cited : (80)

References (43)
  • 3
    • 68249152474 scopus 로고    scopus 로고
    • Describing function based methods for predicting chaos in a class of fractional order differential equations
    • 1176.34051 10.1007/s11071-008-9447-y
    • M.S. Tavazoei M. Haeri 2009 Describing function based methods for predicting chaos in a class of fractional order differential equations Nonlinear Dyn. 57 3 363 373 1176.34051 10.1007/s11071-008-9447-y
    • (2009) Nonlinear Dyn. , vol.57 , Issue.3 , pp. 363-373
    • Tavazoei, M.S.1    Haeri, M.2
  • 4
    • 0001154626 scopus 로고
    • On dissipative systems and related variational principles
    • 0003.01101 10.1103/PhysRev.38.815
    • H. Bateman 1931 On dissipative systems and related variational principles Phys. Rev. 38 4 815 819 0003.01101 10.1103/PhysRev.38.815
    • (1931) Phys. Rev. , vol.38 , Issue.4 , pp. 815-819
    • Bateman, H.1
  • 5
    • 0036701004 scopus 로고    scopus 로고
    • Formulation of Euler-Lagrange equations for fractional variational problems
    • 1070.49013 10.1016/S0022-247X(02)00180-4
    • O.P. Agrawal 2002 Formulation of Euler-Lagrange equations for fractional variational problems J. Math. Anal. Appl. 272 1 368 379 1070.49013 10.1016/S0022-247X(02)00180-4
    • (2002) J. Math. Anal. Appl. , vol.272 , Issue.1 , pp. 368-379
    • Agrawal, O.P.1
  • 6
    • 15544379439 scopus 로고    scopus 로고
    • A general formulation and solution scheme for fractional optimal control problems
    • DOI 10.1007/s11071-004-3764-6
    • O.P. Agrawal 2004 A general formulation and solution scheme for fractional optimal control problems Nonlinear Dyn. 38 4 323 337 1121.70019 10.1007/s11071-004-3764-6 (Pubitemid 40400624)
    • (2004) Nonlinear Dynamics , vol.38 , Issue.1-4 , pp. 323-337
    • Agrawal, O.P.1
  • 7
    • 33746876366 scopus 로고    scopus 로고
    • Fractional variational calculus and the transversality conditions
    • DOI 10.1088/0305-4470/39/33/008, PII S0305447006218080, 008
    • O.P. Agrawal 2006 Fractional variational calculus and the transversality conditions J. Phys. A, Math. Gen. 39 10375 1097.49021 10.1088/0305-4470/39/33/ 008 (Pubitemid 44183559)
    • (2006) Journal of Physics A: Mathematical and General , vol.39 , Issue.33 , pp. 10375-10384
    • Agrawal, O.P.1
  • 8
    • 34250210234 scopus 로고    scopus 로고
    • Fractional variational calculus in terms of Riesz fractional derivatives
    • 1125.26007 10.1088/1751-8113/40/24/003
    • O.P. Agrawal 2007 Fractional variational calculus in terms of Riesz fractional derivatives J. Phys. A, Math. Theor. 40 6287 1125.26007 10.1088/1751-8113/40/24/003
    • (2007) J. Phys. A, Math. Theor. , vol.40 , pp. 6287
    • Agrawal, O.P.1
  • 9
    • 4043139312 scopus 로고    scopus 로고
    • Lagrangians with linear velocities within Riemann-Liouville fractional derivatives
    • D. Baleanu T. Avkar 2004 Lagrangians with linear velocities within Riemann-Liouville fractional derivatives Nuovo Cim. B 119 73 79 (Pubitemid 39057448)
    • (2004) Nuovo Cimento della Societa Italiana di Fisica B , vol.119 , Issue.1 , pp. 73-79
    • Baleanu, D.1    Avkar, T.2
  • 10
    • 23344444772 scopus 로고    scopus 로고
    • Lagrangian formulation of classical fields within Riemann-Liouville fractional derivatives
    • 1122.70360 10.1238/Physica.Regular.072a00119
    • D. Baleanu S.I. Muslih 2005 Lagrangian formulation of classical fields within Riemann-Liouville fractional derivatives Phys Scr. 72 119 123 1122.70360 10.1238/Physica.Regular.072a00119
    • (2005) Phys Scr. , vol.72 , pp. 119-123
    • Baleanu, D.1    Muslih, S.I.2
  • 11
    • 33745514693 scopus 로고    scopus 로고
    • Hamiltonian formulation of classical fields within Riemann-Liouville fractional derivatives
    • DOI 10.1088/0031-8949/73/5/003, PII S0031894906195883, 003
    • S.I. Muslih D. Baleanu E. Rabei 2006 Hamiltonian formulation of classical fields within Riemann-Liouville fractional derivatives Phys. Scr. 73 436 438 1165.70310 10.1088/0031-8949/73/5/003 (Pubitemid 43958970)
    • (2006) Physica Scripta , vol.73 , Issue.5 , pp. 436-438
    • Muslih, S.I.1    Baleanu, D.2    Rabei, E.3
  • 12
    • 67449115682 scopus 로고    scopus 로고
    • Fractional WKB approximation
    • 1176.70014 10.1007/s11071-008-9430-7
    • E.M. Rabei I.M.A. Altarazi S.I. Muslih D. Baleanu 2009 Fractional WKB approximation Nonlinear Dyn. 57 1-2 171 175 1176.70014 10.1007/s11071-008-9430-7
    • (2009) Nonlinear Dyn. , vol.57 , Issue.12 , pp. 171-175
    • Rabei, E.M.1    Altarazi, I.M.A.2    Muslih, S.I.3    Baleanu, D.4
  • 13
    • 77952842666 scopus 로고    scopus 로고
    • Fractional variational principles in action
    • 10.1088/0031-8949/2009/T136/014006
    • D. Baleanu 2009 Fractional variational principles in action Phys. Scr. T136 014006 10.1088/0031-8949/2009/T136/014006
    • (2009) Phys. Scr. , vol.136
    • Baleanu, D.1
  • 14
    • 70349224466 scopus 로고    scopus 로고
    • Fractional-order Euler-Lagrange equations and formulation of Hamiltonian equations
    • 1183.26006 10.1007/s11071-009-9486-z
    • M.A.E. Herzallah D. Baleanu 2009 Fractional-order Euler-Lagrange equations and formulation of Hamiltonian equations Nonlinear Dyn. 58 1-2 385 391 1183.26006 10.1007/s11071-009-9486-z
    • (2009) Nonlinear Dyn. , vol.58 , Issue.12 , pp. 385-391
    • Herzallah, M.A.E.1    Baleanu, D.2
  • 15
    • 70449529531 scopus 로고    scopus 로고
    • A new method of finding the fractional Euler-Lagrange and Hamilton equations within Caputo fractional derivatives
    • 10.1016/j.cnsns.2009.05.023
    • D. Baleanu J.I. Trujillo 2010 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 10.1016/j.cnsns.2009.05.023
    • (2010) Commun. Nonlinear Sci. Numer. Simul. , vol.15 , Issue.5 , pp. 1111-1115
    • Baleanu, D.1    Trujillo, J.I.2
  • 16
    • 19944384253 scopus 로고    scopus 로고
    • Fractional Ginzburg-Landau equation for fractal media
    • DOI 10.1016/j.physa.2005.02.047, PII S0378437105002293
    • V.E. Tarasov G.M. Zaslavsky 2005 Fractional Ginzburg-Landau equation for fractal media Phys. A, Stat. Mech. Appl. 354 249 261 10.1016/j.physa.2005.02.047 (Pubitemid 40752473)
    • (2005) Physica A: Statistical Mechanics and its Applications , vol.354 , Issue.1-4 , pp. 249-261
    • Tarasov, V.E.1    Zaslavsky, G.M.2
  • 17
    • 33746269889 scopus 로고    scopus 로고
    • Nonholonomic constraints with fractional derivatives
    • DOI 10.1088/0305-4470/39/31/010, PII S0305447006206231, 010
    • V.E. Tarasov G.M. Zaslavsky 2006 Nonholonomic constraints with fractional derivatives J. Phys. A, Math. Gen. 39 31 9797 9815 1101.70011 10.1088/0305-4470/39/31/010 (Pubitemid 44102225)
    • (2006) Journal of Physics A: Mathematical and General , vol.39 , Issue.31 , pp. 9797-9815
    • Tarasov, V.E.1    Zaslavsky, G.M.2
  • 18
    • 15744376961 scopus 로고    scopus 로고
    • Fractional calculus description of non-linear viscoelastic behaviour of polymers
    • DOI 10.1007/s11071-004-3757-5
    • N. Heymans 2004 Fractional calculus description of non-linear viscoelastic behaviour of polymers Nonlinear Dyn. 38 1-2 221 231 1142.74312 10.1007/s11071-004-3757-5 (Pubitemid 40406781)
    • (2004) Nonlinear Dynamics , vol.38 , Issue.1-4 , pp. 221-231
    • Heymans, N.1
  • 19
    • 46249125645 scopus 로고    scopus 로고
    • Fractional conservation laws in optimal control theory
    • 1170.49017 10.1007/s11071-007-9309-z
    • G.S.F. Frederico D.F.M. Torres 2008 Fractional conservation laws in optimal control theory Nonlinear Dyn. 53 3 215 222 1170.49017 10.1007/s11071-007-9309-z
    • (2008) Nonlinear Dyn. , vol.53 , Issue.3 , pp. 215-222
    • Frederico, G.S.F.1    Torres, D.F.M.2
  • 20
    • 59349109245 scopus 로고    scopus 로고
    • A fractional calculus interpretation of the fractional volatility model
    • 1187.91231 10.1007/s11071-008-9372-0
    • R.V. Mendes 2009 A fractional calculus interpretation of the fractional volatility model Nonlinear Dyn. 55 4 395 399 1187.91231 10.1007/s11071-008-9372- 0
    • (2009) Nonlinear Dyn. , vol.55 , Issue.4 , pp. 395-399
    • Mendes, R.V.1
  • 21
    • 67449164063 scopus 로고    scopus 로고
    • Calculation of fractional derivatives of noisy data with genetic algorithms
    • 1176.94016 10.1007/s11071-008-9436-1
    • J.A. Tenreiro Machado 2009 Calculation of fractional derivatives of noisy data with genetic algorithms Nonlinear Dyn. 57 1-2 253 260 1176.94016 10.1007/s11071-008-9436-1
    • (2009) Nonlinear Dyn. , vol.57 , Issue.12 , pp. 253-260
    • Tenreiro MacHado, J.A.1
  • 22
    • 77954348949 scopus 로고    scopus 로고
    • Analysis of a fractional order Van der Pol-like oscillator via describing function method
    • 1204.70018 10.1007/s11071-009-9647-0
    • M. Attari M. Haeri M.S. Tavazoei 2010 Analysis of a fractional order Van der Pol-like oscillator via describing function method Nonlinear Dyn. 61 1-2 265 274 1204.70018 10.1007/s11071-009-9647-0
    • (2010) Nonlinear Dyn. , vol.61 , Issue.12 , pp. 265-274
    • Attari, M.1    Haeri, M.2    Tavazoei, M.S.3
  • 23
    • 53149136127 scopus 로고    scopus 로고
    • Special issue on "fractional Derivatives and their Applications"
    • Agrawal, O.P., Tenreiro Machado, J.A, Sabatier, J. (Guest Editors): Special issue on "Fractional Derivatives and their Applications". Nonlinear Dyn. 38(4) (2004)
    • (2004) Nonlinear Dyn , vol.38 , Issue.4
    • Agrawal, O.P.1    Tenreiro MacHado, J.A.2    Sabatier, J.3
  • 24
    • 84885710343 scopus 로고    scopus 로고
    • J. Sabatier O.P. Agrawal J.A. Tenreiro Machado (eds). Springer Dordrecht 1116.00014
    • Sabatier, J., Agrawal, O.P., Tenreiro Machado, J.A. (eds.): Advances in Fractional Calculus. Springer, Dordrecht (2007)
    • (2007) Advances in Fractional Calculus
  • 25
    • 77952803107 scopus 로고    scopus 로고
    • Special issue on "fractional Differentiation and its Applications (FDA08)"
    • Baleanu, D., Tenreiro Machado, J.A. (Guest Editors): Special issue on "Fractional Differentiation and its Applications (FDA08)". Phys. Scr. T136 (2009)
    • (2009) Phys. Scr. T136
    • Baleanu, D.1    Tenreiro MacHado, J.A.2
  • 26
    • 0001524186 scopus 로고
    • On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves
    • D.J. Korteweg G. de Vries 1895 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) Philos. Mag. , vol.39 , Issue.5 , pp. 422
    • Korteweg, D.J.1    De Vries, G.2
  • 27
    • 67349172205 scopus 로고    scopus 로고
    • KdV Equation as an Euler-Poincare' equation
    • M.K. Fung 1997 KdV Equation as an Euler-Poincare' equation Chin. J. Phys. 35 6 789
    • (1997) Chin. J. Phys. , vol.35 , Issue.6 , pp. 789
    • Fung, M.K.1
  • 31
    • 0029288243 scopus 로고
    • The exact solution of certain differential equations of fractional order by using operational calculus
    • 0824.44011 10.1016/0898-1221(95)00031-S
    • Y. Luchko H.M. Srivastava 1995 The exact solution of certain differential equations of fractional order by using operational calculus Comput. Math. Appl. 29 73 85 0824.44011 10.1016/0898-1221(95)00031-S
    • (1995) Comput. Math. Appl. , vol.29 , pp. 73-85
    • Luchko, Y.1    Srivastava, H.M.2
  • 32
    • 0038405057 scopus 로고    scopus 로고
    • Existence of positive solutions of nonlinear fractional differential equations
    • DOI 10.1016/S0022-247X(02)00716-3
    • A. Babakhani V.D. Gejji 2003 Existence of positive solutions of nonlinear fractional differential equations J. Math. Anal. Appl. 278 434 442 1027.34003 10.1016/S0022-247X(02)00716-3 (Pubitemid 36530147)
    • (2003) Journal of Mathematical Analysis and Applications , vol.278 , Issue.2 , pp. 434-442
    • Babakhani, A.1    Daftardar-Gejji, V.2
  • 33
    • 0011651497 scopus 로고    scopus 로고
    • Fractional calculus and function spaces
    • D. Delbosco 1996 Fractional calculus and function spaces J. Fractal Calc. 6 45 53
    • (1996) J. Fractal Calc. , vol.6 , pp. 45-53
    • Delbosco, D.1
  • 34
    • 0037308989 scopus 로고    scopus 로고
    • Existence of positive solution for some class of nonlinear fractional differential equations
    • DOI 10.1016/S0022-247X(02)00583-8
    • S.Q. Zhang 2003 Existence of positive solution for some class of nonlinear fractional differential equations J. Math. Anal. Appl. 278 136 148 1026.34008 10.1016/S0022-247X(02)00583-8 (Pubitemid 36445950)
    • (2003) Journal of Mathematical Analysis and Applications , vol.278 , Issue.1 , pp. 136-148
    • Zhang, S.1
  • 35
    • 24944478244 scopus 로고    scopus 로고
    • An approximate solution of a nonlinear fractional differential equation by Adomian decomposition method
    • DOI 10.1016/j.amc.2004.07.020, PII S0096300304005132
    • S. Saha Ray R.K. Bera 2005 An approximate solution of a nonlinear fractional differential equation by Adomian decomposition method Appl. Math. Comput. 167 561 571 1082.65562 10.1016/j.amc.2004.07.020 (Pubitemid 41316348)
    • (2005) Applied Mathematics and Computation , vol.167 , Issue.1 , pp. 561-571
    • Ray, S.S.1    Bera, R.K.2
  • 36
    • 0032307661 scopus 로고    scopus 로고
    • Approximate analytical solution for seepage flow with fractional derivatives in porous media
    • 0942.76077 10.1016/S0045-7825(98)00108-X
    • J.-H. He 1998 Approximate analytical solution for seepage flow with fractional derivatives in porous media Comput. Methods Appl. Mech. Eng. 167 57 68 0942.76077 10.1016/S0045-7825(98)00108-X
    • (1998) Comput. Methods Appl. Mech. Eng. , vol.167 , pp. 57-68
    • He, J.-H.1
  • 37
    • 38049162125 scopus 로고    scopus 로고
    • Variational iteration method for solving the space- and time-fractional KdV equation
    • S. Momani Z. Odibat A. Alawnah 2008 Variational iteration method for solving the space- and time-fractional KdV equation Numer. Methods Part. Differ. Equ. 24 1 261 271
    • (2008) Numer. Methods Part. Differ. Equ. , vol.24 , Issue.1 , pp. 261-271
    • Momani, S.1    Odibat, Z.2    Alawnah, A.3
  • 38
    • 60549112414 scopus 로고    scopus 로고
    • Variational iteration method for fractional heat- and wave-like equations
    • 1172.35302 10.1016/j.nonrwa.2008.02.026
    • R.Y. Molliq M.S.M. Noorani I. Hashim 2009 Variational iteration method for fractional heat- and wave-like equations Nonlinear Anal., Real World Appl. 10 1854 1869 1172.35302 10.1016/j.nonrwa.2008.02.026
    • (2009) Nonlinear Anal., Real World Appl. , vol.10 , pp. 1854-1869
    • Molliq, R.Y.1    Noorani, M.S.M.2    Hashim, I.3
  • 39
    • 0006996396 scopus 로고
    • General use of the Lagrange multiplier in non-linear mathematical physics
    • S. Nemat-Nasser (eds). Pergamon Press Oxford
    • 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)
    • (1978) Variational Method in the Mechanics of Solids
    • Inokuti, M.1    Sekine, H.2    Mura, T.3
  • 40
    • 0007140364 scopus 로고    scopus 로고
    • A new approach to nonlinear partial differential equations
    • 10.1016/S1007-5704(97)90007-1
    • J.-H. He 1997 A new approach to nonlinear partial differential equations Commun. Nonlinear Sci. Numer. Simul. 2 4 230 235 10.1016/S1007-5704(97)90007-1
    • (1997) Commun. Nonlinear Sci. Numer. Simul. , vol.2 , Issue.4 , pp. 230-235
    • He, J.-H.1
  • 41
    • 0000092673 scopus 로고    scopus 로고
    • Variational-iteration-a kind of nonlinear analytical technique: Some examples
    • 05137891 10.1016/S0020-7462(98)00048-1
    • J.-H. He 1999 Variational-iteration-a kind of nonlinear analytical technique: some examples Int. J. Nonlinear Mech. 34 699 05137891 10.1016/S0020-7462(98)00048-1
    • (1999) Int. J. Nonlinear Mech. , vol.34 , pp. 699
    • He, J.-H.1
  • 42
    • 0031441505 scopus 로고    scopus 로고
    • Semi-inverse method of establishing generalized variational principles for fluid mechanics with emphasis on turbomachinery aerodynamics
    • J.-H. He 1997 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 10.1515/TJJ.1997.14.1.23 (Pubitemid 127362293)
    • (1997) International Journal of Turbo and Jet Engines , vol.14 , Issue.1 , pp. 23-28
    • He, J.-H.1
  • 43
    • 0041621600 scopus 로고    scopus 로고
    • Variational principles for some nonlinear partial differential equations with variable coefficients
    • 1135.35303 10.1016/S0960-0779(03)00265-0
    • J.-H. He 2004 Variational principles for some nonlinear partial differential equations with variable coefficients Chaos Solitons Fractals 19 847 851 1135.35303 10.1016/S0960-0779(03)00265-0
    • (2004) Chaos Solitons Fractals , vol.19 , pp. 847-851
    • He, J.-H.1


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