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Original Article

On the solution of a Riesz space-fractional nonlinear wave equation through an efficient and energy-invariant scheme

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Pages 337-361 | Received 28 Nov 2017, Accepted 05 Feb 2018, Published online: 20 Feb 2018

References

  • G.D. Akrivis, Finite difference discretization of the cubic Schrödinger equation, IMA J. Numer. Anal. 13 (1993), pp. 115–124. doi: 10.1093/imanum/13.1.115
  • G. Alfimov, T. Pierantozzi and L. Vázquez, Numerical study of a fractional sine-Gordon equation, in Fractional Differentiation and its Applications, A. Le Mehuate, J.A. Tenreiro Machado, L.C. Trigeassou and J. Sabatier, eds., Proceedings of the IFAC-FDA Workshop, July 2004, Vol. 4, Bordeaux, France, 2004, pp. 153–162.
  • S. Arshad, J. Huang, A.Q. Khaliq and Y. Tang, Trapezoidal scheme for time–space fractional diffusion equation with Riesz derivative, J. Comput. Phys. 350 (2017), pp. 1–15. doi: 10.1016/j.jcp.2017.08.038
  • G. Ben-Yu, P.J. Pascual, M.J. Rodriguez and L. Vázquez, Numerical solution of the sine-Gordon equation, Appl. Math. Comput. 18 (1986), pp. 1–14.
  • M. Bonforte and J.L. Vázquez, A priori estimates for fractional nonlinear degenerate diffusion equations on bounded domains, Arch. Ration. Mech. Anal. 218 (2015), pp. 317–362. doi: 10.1007/s00205-015-0861-2
  • A. Chen and C. Li, A novel compact ADI scheme for the time-fractional subdiffusion equation in two space dimensions, Int. J. Comput. Math. 93 (2016), pp. 889–914. doi: 10.1080/00207160.2015.1009905
  • A. de Pablo, F. Quirós, A. Rodríguez and J.L. Vázquez, A general fractional porous medium equation, Commun. Pure Appl. Math. 65 (2012), pp. 1242–1284. doi: 10.1002/cpa.21408
  • J. Eshaghi, H. Adibi and S. Kazem, On a numerical investigation of the time fractional Fokker–Planck equation via local discontinuous Galerkin method, Int. J. Comput. Math. 94 (2017), pp. 1916–1942. doi: 10.1080/00207160.2016.1247442
  • S. Esmaeili and R. Garrappa, A pseudo-spectral scheme for the approximate solution of a time-fractional diffusion equation, Int. J. Comput. Math. 92 (2015), pp. 980–994. doi: 10.1080/00207160.2014.915962
  • Z. Fei and L. Vázquez, Two energy conserving numerical schemes for the sine-Gordon equation, Appl. Math. Comput. 45 (1991), pp. 17–30.
  • G.S.F. Frederico and D.F.M. Torres, Fractional conservation laws in optimal control theory, Nonlinear Dyn. 53 (2008), pp. 215–222. doi: 10.1007/s11071-007-9309-z
  • A. Friedman, Foundations of Modern Analysis, Courier Corporation, New York, 1970.
  • K.M. Furati, M. Yousuf and A.Q.M. Khaliq, Fourth-order methods for space fractional reaction–diffusion equations with non-smooth data, Int. J. Comput. Math. 29 (2017), pp. 1–17. doi: 10.1080/00207160.2017.1404037
  • D. Furihata, Finite-difference schemes for nonlinear wave equation that inherit energy conservation property, J. Comput. Appl. Math. 134 (2001), pp. 37–57. doi: 10.1016/S0377-0427(00)00527-6
  • D. Furihata and T. Matsuo, Discrete Variational Derivative Method: A Structure-preserving Numerical Method for Partial Differential Equations, CRC Press, New York, 2010.
  • T.H. Gronwall, Note on the derivatives with respect to a parameter of the solutions of a system of differential equations, Ann. Math. 20 (1919), pp. 292–296. doi: 10.2307/1967124
  • M. Gülsu, Y. Öztürk and A. Anapali, Numerical solution the fractional Bagley–Torvik equation arising in fluid mechanics, Int. J. Comput. Math. 94 (2017), pp. 173–184. doi: 10.1080/00207160.2015.1099633
  • A.K. Gupta and S.S. Ray, A novel attempt for finding comparatively accurate solution for sine-Gordon equation comprising Riesz space fractional derivative, Math. Methods Appl. Sci. 39 (2016), pp. 2871–2882. doi: 10.1002/mma.3736
  • S. Johnson, P. Suarez and A. Biswas, New exact solutions for the sine-Gordon equation in 2+1 dimensions, Comput. Math. Math. Phys. 52 (2012), pp. 98–104. doi: 10.1134/S0965542512010058
  • A.Q.M. Khaliq, X. Liang and K.M. Furati, A fourth-order implicit-explicit scheme for the space fractional nonlinear Schrödinger equations, Numer. Algorithms 75 (2017), pp. 147–172. doi: 10.1007/s11075-016-0200-1
  • N. Korabel, G.M. Zaslavsky and V.E. Tarasov, Coupled oscillators with power-law interaction and their fractional dynamics analogues, Commun. Nonlinear Sci. Numer. Simul. 12 (2007), pp. 1405–1417. doi: 10.1016/j.cnsns.2006.03.015
  • N. Laskin, Fractional Schrödinger equation, Phys. Rev. E 66 (2002), pp. 056108. doi: 10.1103/PhysRevE.66.056108
  • X. Liang, A. Khaliq, H. Bhatt and K. Furati, The locally extrapolated exponential splitting scheme for multi-dimensional nonlinear space-fractional Schrödinger equations, Numer. Algorithms 76(4) (2017), pp. 939–958. doi: 10.1007/s11075-017-0291-3
  • J.E. Macías-Díaz, A structure-preserving method for a class of nonlinear dissipative wave equations with Riesz space-fractional derivatives, J. Comput. Phys. 315 (2017), pp. 40–58. doi: 10.1016/j.jcp.2017.09.028
  • J.E. Macías-Díaz, An explicit dissipation-preserving method for Riesz space-fractional nonlinear wave equations in multiple dimensions, Commun. Nonlinear Sci. Numer. Simul. 58 (2017), pp. 67–87.
  • J.E. Macías-Díaz, Persistence of nonlinear hysteresis in fractional models of Josephson transmission lines, Commun. Nonlinear Sci. Numer. Simul. 53 (2017), pp. 31–43. doi: 10.1016/j.cnsns.2017.04.030
  • J.E. Macías-Díaz, Numerical simulation of the nonlinear dynamics of harmonically driven Riesz-fractional extensions of the Fermi–Pasta–Ulam chains, Commun. Nonlinear Sci. Numer. Simul. 55 (2018), pp. 248–264. doi: 10.1016/j.cnsns.2017.07.012
  • J.E. Macías-Díaz, A.S. Hendy and R.H. de Staelen, A pseudo energy-invariant method for relativistic wave equations with Riesz space-fractional derivatives, Comput. Phys. Commun. 224 (2017), pp. 98–107. doi: 10.1016/j.cpc.2017.11.008
  • J.E. Macías-Díaz, A.S. Hendy and R. De Staelen, A compact fourth-order in space energy-preserving method for Riesz space-fractional nonlinear wave equations, Appl. Math. Comput. 325 (2018), pp. 1–14.
  • T. Matsuo and D. Furihata, Dissipative or conservative finite-difference schemes for complex-valued nonlinear partial differential equations, J. Comput. Phys. 171 (2001), pp. 425–447. doi: 10.1006/jcph.2001.6775
  • K. Pen-Yu, Numerical methods for incompressible viscous flow, Sci. Sin. 20 (1977), pp. 287–304.
  • S.S. Ray, A new analytical modelling for nonlocal generalized Riesz fractional sine-Gordon equation, J. King Saud Univ – Sci. 28 (2016), pp. 48–54. doi: 10.1016/j.jksus.2015.03.003
  • S.S. Ray, Soliton solutions of nonlinear and nonlocal sine-Gordon equation involving Riesz space fractional derivative, Z. Naturforsch. A 70 (2015), pp. 659–667. doi: 10.1515/zna-2015-0119
  • D. Stan and J.L. Vázquez, The Fisher-KPP equation with nonlinear fractional diffusion, SIAM J. Math. Anal. 46 (2014), pp. 3241–3276. doi: 10.1137/130918289
  • W. Strauss and L. Vázquez, Numerical solution of a nonlinear Klein-Gordon equation, J. Comput. Phys. 28 (1978), pp. 271–278. doi: 10.1016/0021-9991(78)90038-4
  • V.E. Tarasov, Fractional generalization of gradient and Hamiltonian systems, J. Phys. A: Math. Gen. 38 (2005), pp. 5929–5943 doi: 10.1088/0305-4470/38/26/007
  • V.E. Tarasov, Continuous limit of discrete systems with long-range interaction, J. Phys. A: Math. Gen. 39 (2006), pp. 14895–14910. doi: 10.1088/0305-4470/39/48/005
  • V.E. Tarasov, Partial fractional derivatives of Riesz type and nonlinear fractional differential equations, Nonlinear Dyn. 86 (2016), pp. 1745–1759. doi: 10.1007/s11071-016-2991-y
  • V.E. Tarasov and G.M. Zaslavsky, Conservation laws and Hamilton's equations for systems with long-range interaction and memory, Commun. Nonlinear Sci. Numer. Simul. 13 (2008), pp. 1860–1878. doi: 10.1016/j.cnsns.2007.05.017
  • S. Tomasiello, S.K. Khattri and J. Awrejcewicz, Differential quadrature-based simulation of a class of fuzzy damped fractional dynamical systems, Int. J. Numer. Anal. Model. 14 (2017), pp. 63–75.
  • S. Tomasiello and J.E. Macías-Díaz, Note on a Picard-like method for Caputo fuzzy fractional differential equations, Appl. Math. Inform. Sci. 11 (2017), pp. 281–287. doi: 10.18576/amis/110134
  • D. Vvedensky, Partial Differential Equations with Mathematica, Addison-Wesley Longman Publishing Co., Inc., Wakingham, 1993.
  • X. Wang, F. Liu and X. Chen, Novel second-order accurate implicit numerical methods for the Riesz space distributed-order advection-dispersion equations, Adv. Math. Phys. 2015 (2015), pp. 590435.
  • Z. Wang, S. Vong and S.L. Lei, Finite difference schemes for two-dimensional time-space fractional differential equations, Int. J. Comput. Math. 93 (2016), pp. 578–595. doi: 10.1080/00207160.2015.1009902

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