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Research Articles

Generalized KdV equation involving Riesz time-fractional derivatives: constructing and solution utilizing variational methods

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Pages 314-321 | Received 02 Nov 2019, Accepted 11 Feb 2020, Published online: 12 Mar 2020

References

  • Riewe F. Nonconservative Lagrangian and Hamiltonian mechanics. Phys Rev E. 1996;53:1890–1899. doi: 10.1103/PhysRevE.53.1890
  • Riewe F. Mechanics with fractional derivatives. Phys Rev E. 1997;55:3581–3592. doi: 10.1103/PhysRevE.55.3581
  • Agrawal OP. Formulation of Euler–Lagrange equations for fractional variational problems. J Math Anal Appl. 2002;272:368–379. doi: 10.1016/S0022-247X(02)00180-4
  • Agrawal OP. Fractional variational calculus in terms of Riesz fractional derivatives. J Phys A Math Theor. 2007;40:6287. doi: 10.1088/1751-8113/40/24/003
  • Attari M, Haeri M, Tavazoei MS. Analysis of a fractional order Van der Pol-like oscillator via describing function method. Nonlinear Dyn. 2010;61:265–274. doi: 10.1007/s11071-009-9647-0
  • Tenreiro Machado JA. Calculation of fractional derivatives of noisy data with genetic algorithms. Non- Linear Dyn. 2009;57:253–260. doi: 10.1007/s11071-008-9436-1
  • Mendes RV. A fractional calculus interpretation of the fractional volatility model. Nonlinear Dyn. 2009;55:395–399. doi: 10.1007/s11071-008-9372-0
  • Frederico GSF, Torres DFM. Fractional conservation laws in optimal control theory. Nonlinear Dyn. 2008;53:215–222. doi: 10.1007/s11071-007-9309-z
  • Baleanu D, Trujillo JI. A new method of finding the fractional Euler–Lagrange and Hamilton equations within Caputo fractional derivatives. Commun Nonlinear Sci Numer Simul. 2010;15:1111–1115. doi: 10.1016/j.cnsns.2009.05.023
  • Yang XJ, Gao F, Srivastava HM. A new computational approach for solving nonlinear local fractional PDEs. J Comp Appl Math. 2018;339:285–296. doi: 10.1016/j.cam.2017.10.007
  • Rui W. Applications of homogenous balanced principle on investigating exact solutions to a series of time fractional nonlinear PDEs. Comm Nonlin Sci Num Simul. 2017;47:253–266. doi: 10.1016/j.cnsns.2016.11.018
  • Angstmann CN, Henry BI, Jacobs BA, et al. Integrablization of time fractional PDEs. Comput Math Appl. 2017;73:1053–1062. doi: 10.1016/j.camwa.2016.12.010
  • Sakar MG, Uludag F, Erdogan F. Numerical solution of time-fractional nonlinear PDEs with proportional delays by homotopy perturbation method. Appl Math Model. 2016;40:6639–6649. doi: 10.1016/j.apm.2016.02.005
  • Fernandez A, Baleanu D, Fokas AS. Solving PDEs of fractional order using the unified transform method. Appl Math Comp. 2018;339:738–749. doi: 10.1016/j.amc.2018.07.061
  • Zhang S, Hong S. Variable separation method for a nonlinear time fractional partial differential equation with forcing term. J Comput Appl Math. 2018;339:297–305. doi: 10.1016/j.cam.2017.09.045
  • Korteweg DJ, 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. 1895;39:422–443. doi: 10.1080/14786449508620739
  • Fung MK. Kdv equation as an Euler–Poincare’ equation. Chin J Phys. 1997;35:789–796.
  • Elmandouha AA, Ibrahim AG. Bifurcation and travelling wave solutions for a (2+ 1)-dimensional KdV equation. J Taib Univ Sci. 2020;4:139–147. doi: 10.1080/16583655.2019.1709271
  • Podlubny I. Fractional differential equations. San Diego (CA): Academic Press; 1999.
  • Samko SG, Kilabas AA, Marichev OI. Fractional integrals and derivatives: theory and applications. New York: Gordon and Breach; 1998.
  • Luchko Y, Srivastava HM. The exact solution of certain differential equations of fractional order by using operational calculus. Comput Math Appl. 1995;29:73–85. doi: 10.1016/0898-1221(95)00031-S
  • Shah R, Li T. The thermal and laminar boundary layer flow over prolate and oblate spheroids. Int J Heat Mass Transfer. 2018;121:607–619. doi: 10.1016/j.ijheatmasstransfer.2017.12.130
  • Arshad S, Sohail A, Maqbool K. Nonlinear shallow water waves: a fractional order approach. Alex Eng J. 2016;55:525–532. doi: 10.1016/j.aej.2015.10.014
  • Ullah R, Ellahi R, Sait SM, et al. On the fractional-order model of HIV-1 infection of CD4+ T-cells under the influence of antiviral drug treatment. J Taib Univ Sci. 2020;14:50–59. doi: 10.1080/16583655.2019.1700676
  • Sohail A, Maqbool K, Ellahi R. Stability analysis for fractional-order partial differential equations by means of space spectral time Adams-Bash forth Moulton method. Num Meth Part Diff Equat. 2018;34:19–29. doi: 10.1002/num.22171
  • Khan U, Ellahi R, Khan R, et al. Extracting new solitary wave solutions of Benny–Luke equation and Phi-4 equation of fractional order by using (GI/G)-expansion method. Opt Quant Elect. 2017;49:362–376. doi: 10.1007/s11082-017-1191-4
  • Ellahi R, Mohyud-Din ST, Khan U. Exact traveling wave solutions of fractional order Boussinesq-like equations by applying Exp-function method. Res. Phys. 2018;8:114–120.
  • Babakhani A, Gejji VD. Existence of positive solutions of nonlinear fractional differential equations. J Math Anal Appl. 2003;278:434–442. doi: 10.1016/S0022-247X(02)00716-3
  • Delbosco D. Fractional calculus and function spaces. J Fractal Calc. 1996;6:45–53.
  • Zhang SQ. Existence of positive solution for some class of nonlinear fractional differential equations. J Math Anal Appl. 2003;278:136–148. doi: 10.1016/S0022-247X(02)00583-8
  • Saha Ray S, Bera RK. An approximate solution of a nonlinear fractional differential equation by Adomian decomposition method. Appl Math Comput. 2005;167:561–571.
  • He JH. A new approach to nonlinear partial differential equations. Commun Nonlinear Sci Numer Simul. 1997;2:230–235. doi: 10.1016/S1007-5704(97)90007-1
  • He JH. Variational-iteration—a kind of nonlinear analytical technique: some examples. Int J Nonlinear Mech. 1999;34:699. doi: 10.1016/S0020-7462(98)00048-1
  • He J-H. Approximate analytical solution for seepage flow with fractional derivatives in porous media. Comput Methods Appl Mech Eng. 1998;167:57–68. doi: 10.1016/S0045-7825(98)00108-X
  • Momani S, Odibat Z, Alawnah A. Variational iteration method for solving the space- and time- fractional KdV equation. Numer Methods Part Differ Equat. 2008;24:261–271.
  • Molliq RY, Noorani MSM, Hashim I. Variational iteration method for fractional heat- and wave-like equations. Nonlinear Anal Real World Appl. 2009;10:1854–1869. doi: 10.1016/j.nonrwa.2008.02.026
  • Sohail A, Maqbool K, Hayat T. Painlevé property and approximate solutions using Adomian decom position for a nonlinear KdV-like wave equation. Appl Math Comput. 2014;229:359–366.
  • Arqub OA, Maayah B. Fitted fractional reproducing kernel algorithm for the numerical solutions of ABC–fractional Volterra integro-differential equations. Chao Solit Fract. 2019;126:394–402. doi: 10.1016/j.chaos.2019.07.023
  • Arqub OA, Maayah B. Modulation of reproducing kernel Hilbert space method for numerical solutions of Riccati and Bernoulli equations in the Atangana-Baleanu fractional sense. Chao Solit Fract. 2019;125:163–170. doi: 10.1016/j.chaos.2019.05.025
  • Arqub OA, Al-Smadi M. An adaptive numerical approach for the solutions of fractional advection–diffusion and dispersion equations in singular case under Riesz’s derivative operator. Phys A Stat Mech Appl. 2020;540:123257. doi: 10.1016/j.physa.2019.123257
  • Arqub OA. Application of residual power series method for the solution of time-fractional Schrodinger equations in one-dimensional space. Fund Inform. 2019;166:87–110. doi: 10.3233/FI-2019-1795
  • Arqub OA. Numerical algorithm for the solutions of fractional order systems of Dirichlet function types with comparative analysis. Fund Inform. 2019;166:111–137. doi: 10.3233/FI-2019-1796
  • El-Wakil SA, Abulwafa EM, Zahran MA, et al. Time-fractional KdV equation: formulation and solution using variational methods. Nonlinear Dyn. 2011;65:55–63. doi: 10.1007/s11071-010-9873-5
  • Zhang Y. Formulation and solution to time-fractional generalized Korteweg-de Vries equation via variational methods. Adv Differ Equ. 2014;2014:65–77. doi: 10.1186/1687-1847-2014-65
  • Khater AH, Moussa MHM, Abdul-Aziz SF. Invariant variational principles and conservation laws for some nonlinear partial differential equations with constant coefficients-II. Chaos Solitons Fractals. 2003;15:1–13. doi: 10.1016/S0960-0779(02)00059-0
  • He JH. Semi-inverse method of establishing generalized variational principles for fluid mechanics with emphasis on turbo-machinery aerodynamics. Int J Turbo Jet-Engines. 1997;14:23–28.
  • He JH. Variational principles for some nonlinear partial differential equations with variable coefficients. Chao Solit Fract. 2004;19:847–851. doi: 10.1016/S0960-0779(03)00265-0
  • Kilbas AA, Srivastava HM, Trujillo JJ. Theory and applications of fractional differential equations. Amsterdam: Elsevier; 2006.
  • Agrawal OP. Formulation of Euler–Lagrange equations for fractional variational problems. J Math Anal Appl. 2002;272:368–379. doi: 10.1016/S0022-247X(02)00180-4
  • Agrawal OP. A general formulation and solution scheme for fractional optimal control problems. Non- Linear Dyn. 2004;38:323–337. doi: 10.1007/s11071-004-3764-6
  • Agrawal OP. Fractional variational calculus and the transversality conditions. J Phys A Math Gen. 2006;39:10375. doi: 10.1088/0305-4470/39/33/008
  • He JH. Variational iteration method for autonomous ordinary differential systems. Appl Math Comput. 2000;114(2–3):115–123.
  • He JH, Wu XH. Construction of solitary solution and compaction-like solution by variational iteration method. Chao Solit Fract. 2006;29:108–113. doi: 10.1016/j.chaos.2005.10.100
  • He JH. A generalized variational principle in micromorphic thermoelasticity. Mech Res Comm. 2005;3291:93–98. doi: 10.1016/j.mechrescom.2004.06.006
  • He JH. Variational iteration method—some recent results and new interpretations. J Comput Appl Math. 2007;207:3–17. doi: 10.1016/j.cam.2006.07.009
  • Finlayson BA. The method of weighted residuals and variational principles. New York: Academic Press; 1972.
  • Inokvti M, Sekine H, Mura T. General use of the Lagrange multiplier in nonlinear mathematical physics. In: Nemat-Nassed S, editor. Variational method in the mechanics of solids. U.S.A.: Pergamon Press; 1978. p. 156–162.
  • Abulwafa EM, Abdou MA, Mahmoud AA. The solution of nonlinear coagulation problem with mass loss. Chao Solit Fract. 2006;29:313–330. doi: 10.1016/j.chaos.2005.08.044
  • Momani S, Odibat Z. Analytical approach to linear fractional partial differential equations arising in fluid mechanics. Phys Lett A. 2006;1:1–9.
  • Momani S, Abusaad S. Application of He’s variational-iteration method to Helmholtz equation. Chao Solit Fract. 2005;27:1119–1123. doi: 10.1016/j.chaos.2005.04.113
  • Abdou MA, Soliman AA. Variational iteration method for solving Burgers’ and coupled Burgers’ equation. J Comput Appl Math. 2005;181:245–251. doi: 10.1016/j.cam.2004.11.032
  • Adomian G. Solving Frontier problems of physics: the decomposition method. Boston (MA): Kluwer; 1994.
  • Adomian G. A review of the decomposition method in applied mathematics. J Math Anal Appl. 1988;135:501–544. doi: 10.1016/0022-247X(88)90170-9