Advertisement

The interpolating element-free Galerkin method for solving Korteweg–de Vries–Rosenau-regularized long-wave equation with error analysis

  • Mostafa Abbaszadeh
  • Mehdi DehghanEmail author
Original Paper
  • 2 Downloads

Abstract

The main target of this investigation is to develop a new numerical method for solving a class of wave models, i.e., the Korteweg–de Vries–Rosenau-regularized long-wave equation with application in plasma physics. The developed technique is concerning the interpolating element-free Galerkin method. The test and trial functions for the interpolating element-free Galerkin technique have been chosen from the interpolating moving least squares approximation. The interpolating moving least squares approximation shape functions unlike the moving least squares shape functions have the delta Kronecker property due to the use of a singular weight function. The stability and convergence of the new scheme are analyzed. Furthermore, the existence and uniqueness of solution have been proved for the full-discrete scheme. Finally, several examples in one- and two-dimensional cases have been studied to confirm the influence of the new scheme.

Keywords

Interpolating element-free Galerkin method Error estimate Korteweg–de Vries (KdV)-Rosenau-regularized long-wave equation 

Mathematics Subject Classification

65N12 65N30 65M12 

Notes

Acknowledgements

We would like to give our sincere gratitude to the reviewers for their comments and suggestions that greatly improved the manuscript.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Apolinar-Fernandez, A., Ramos, J.I.: Numerical solution of the generalized, dissipative KdV–RLW–Rosenau equation with a compact method. Commun. Nonlinear Sci. Numer. Simul. 60, 165–183 (2018)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Atouani, N., Omrani, K.: On the convergence of conservative difference schemes for the 2D generalized Rosenau–Korteweg de Vries equation. Appl. Math. Comput. 250, 832–847 (2015)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Atouani, N., Omrani, K.: Galerkin finite element method for the Rosenau-RLW equation. Comput. Math. Appl. 66, 289–303 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Atouani, N., Omrani, K.: A new conservative high-order accurate difference scheme for the Rosenau equation. Appl. Anal. 94, 2435–2455 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Belytschko, T., Lu, Y.Y., Gu, L.: Element free Galerkin methods. Int. J. Numer. Methods Eng. 37, 229–256 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Belytschko, T., Krongauz, Y., Organ, D., Fleming, M., Krysl, P.: Meshless methods: an overview and recent developments. Comput. Methods Appl. Mech. Eng. 139, 3–47 (1996)CrossRefzbMATHGoogle Scholar
  7. 7.
    Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York (2011)zbMATHGoogle Scholar
  8. 8.
    Bui, T.Q., Nguyen, M.N., Zhang, C.: A moving Kriging interpolation-based element-free Galerkin method for structural dynamic analysis. Comput. Methods Appl. Mech. Eng. 200, 1354–1366 (2011)CrossRefzbMATHGoogle Scholar
  9. 9.
    Cai, W., Sun, Y., Wang, Y.: Variational discretizations for the generalized Rosenau-type equations. Appl. Math. Comput. 271, 860–873 (2015)MathSciNetGoogle Scholar
  10. 10.
    Chen, L., Cheng, Y.M.: The complex variable reproducing kernel particle method for bending problems of thin plates on elastic foundations. Comput. Mech. 62, 67–80 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Cheng, Y.M., Bai, F., Peng, M.: A novel interpolating element-free Galerkin (IEFG) method for two-dimensional elastoplasticity. Appl. Math. Model. 38(21), 5187–5197 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Cheng, Y.M., Bai, F., Liu, C., Peng, M.: Analyzing nonlinear large deformation with an improved element-free Galerkin method via the interpolating moving least-squares method. Int. J. Comput. Mater. Sci. Eng. 5, 1650023 (2016)Google Scholar
  13. 13.
    Chung, H.J., Belytschko, T.: An error estimate in the EFG method. Comput. Mech. 21, 91–100 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Chung, S.K., Ha, S.N.: Finite element Galerkin solutions for the Rosenau equation. Appl. Anal. 54, 39–56 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Chung, S.K.: Finite difference approximate solutions for the Rosenau equation. Appl. Anal. 69(1–2), 149–156 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Chunk, S.K., Pani, A.K.: Numerical methods for the Rosenau equation. Appl. Anal. 77, 351–369 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Dai, B.D., Cheng, J., Zheng, B.: Numerical solution of transient heat conduction problems using improved meshless local Petrov–Galerkin method. Appl. Math. Comput. 219, 10044–10052 (2013)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Dai, B.D., Cheng, J., Zheng, B.: A moving Kriging interpolation-based meshless local Petrov–Galerkin method for elastodynamic analysis. Int. J. Appl. Mech. 5(1), 1350011–1350021 (2013)CrossRefGoogle Scholar
  19. 19.
    Deng, Y., Liu, C., Peng, M., Cheng, Y.M.: The interpolating complex variable element-free Galerkin method for temperature field problems. Int. J. Appl. Mech. 7, 1550017 (2015)CrossRefGoogle Scholar
  20. 20.
    Dehghan, M., Salehi, R.: The solitary wave solution of the two-dimensional regularized long-wave equation in fluids and plasmas. Comput. Phys. Commun. 182, 2540–2549 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Dehghan, M., Abbaszadeh, M., Mohebbi, A.: The use of interpolating element-free Galerkin technique for solving 2D generalized Benjamin–Bona–Mahony–Burgers and regularized long-wave equations on non-rectangular domains with error estimate. J. Comput. Appl. Math. 286, 211–231 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Dehghan, M., Abbaszadeh, M.: Analysis of the element free Galerkin (EFG) method for solving fractional cable equation with Dirichlet boundary condition. Appl. Numer. Math. 109, 208–234 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Dehghan, M., Abbaszadeh, M.: Proper orthogonal decomposition variational multiscale element free Galerkin (POD-VMEFG) meshless method for solving incompressible Navier–Stokes equation. Comput. Methods Appl. Mech. Eng. 311, 856–888 (2016)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Dehghan, M., Shokri, A.: A numerical method for KdV equation using collocation and radial basis functions. Nonlinear Dyn. 50(1–2), 111–120 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Dehghan, M., Manafian-Herris, J., Saadatmandi, A.: Application of semi-analytical methods for solving the Rosenau–Hyman equation arising in the pattern formation in liquid drops. Int. J. Numer. Methods Heat Fluid Flow 23(6), 777–790 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Feng-Xin, S., Ju-Feng, W., Yu-Min, C.: An improved interpolating element-free Galerkin method for elasticity. Chin. Phys. B 22(12), 120203 (2013)CrossRefGoogle Scholar
  27. 27.
    Ghiloufi, A., Kadri, T.: Analysis of new conservative difference scheme for two-dimensional Rosenau-RLW equation. Appl. Anal. 97, 1255–1267 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Gu, L.: Moving Kriging interpolation and element-free Galerkin method. Int. J. Numer. Methods Eng. 56, 1–11 (2003)CrossRefzbMATHGoogle Scholar
  29. 29.
    Gu, Y.T., Liu, G.R.: A local point interpolation method for static and dynamic analysis of thin beams. Comput. Methods Appl. Mech. Eng. 190, 5515–5528 (2001)CrossRefzbMATHGoogle Scholar
  30. 30.
    Gu, Y.T., Liu, G.R.: A boundary point interpolation method for stress analysis of solids. Comput. Mech. 28, 47–54 (2002)CrossRefzbMATHGoogle Scholar
  31. 31.
    Gu, Y.T., Wang, W., Zhang, L.C., Feng, X.Q.: An enriched radial point interpolation method (e-RPIM) for analysis of crack tip fields. Eng. Fract. Mech. 78, 175–190 (2011)CrossRefGoogle Scholar
  32. 32.
    He, D., Pan, K.: A linearly implicit conservative difference scheme for the generalized Rosenau–Kawahara-RLW equation. Appl. Math. Comput. 271, 323–336 (2015)MathSciNetGoogle Scholar
  33. 33.
    Hu, J.S., Zheng, K.L.: Two conservative difference schemes for the generalized Rosenau equation. Bound. Value Probl. (2010). Article ID 543503Google Scholar
  34. 34.
    Lancaster, P., Salkauskas, K.: Surfaces generated by moving least squares methods. Math. Comput. 37, 141–158 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Li, D., Zhang, Z., Liew, K.M.: A numerical framework for two-dimensional large deformation of inhomogeneous swelling of gels using the improved complex variable element-free Galerkin method. Comput. Methods Appl. Mech. Eng. 274, 84–102 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Li, D., Bai, F., Cheng, Y.M., Liew, K.M.: A novel complex variable element-free Galerkin method for two-dimensional large deformation problems. Comput. Methods Appl. Mech. Eng. 233, 1–10 (2012)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Li, X.: A meshless interpolating Galerkin boundary node method for Stokes flows. Eng. Anal. Bound. Elem. 51, 112–122 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Ju-Feng, W., Feng-Xin, S., Yu-Min, C.: An improved interpolating element-free Galerkin method with a nonsingular weight function for two-dimensional potential problems. Chin. Phys. B 21(9), 090204 (2012)CrossRefGoogle Scholar
  39. 39.
    Li, X., Wang, Q.: Analysis of the inherent instability of the interpolating moving least squares method when using improper polynomial bases. Eng. Anal. Bound. Elem. 73, 21–34 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Li, X.: Error estimates for the moving least-square approximation and the element-free Galerkin method in n-dimensional spaces. Appl. Numer. Math. 99, 77–97 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Li, X.: Meshless Galerkin algorithms for boundary integral equations with moving least square approximations. Appl. Numer. Math. 61(12), 1237–1256 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Li, X., Chen, H., Wang, Y.: Error analysis in Sobolev spaces for the improved moving least-square approximation and the improved element-free Galerkin method. Appl. Math. Comput. 262, 56–78 (2015)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Liew, K.M., Cheng, Y.M.: Complex variable boundary element-free method for two-dimensional elastodynamic problems. Comput. Methods Appl. Mech. Eng. 198, 3925–3933 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Liu, F., Cheng, Y.M.: The improved element-free Galerkin method based on the nonsingular weight functions for inhomogeneous swelling of polymer gels. Int. J. Appl. Mech. 10(4), 1850047 (2018)CrossRefGoogle Scholar
  45. 45.
    Meng, Z.J., Cheng, H., Ma, L.D., Cheng, Y.M.: The dimension split element-free Galerkin method for three-dimensional potential problems. Acta Mech. Sin./Lixue Xuebao 34(3), 462–474 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Mittal, R.C., Jain, R.K.: Numerical solution of general Rosenau-RLW equation using quintic B-splines collocation method. Commun. Numer. Anal. 2012, 1–19 (2012)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Pan, X., Zhang, L.: On the convergence of a conservative numerical scheme for the usual Rosenau-RLW equation. Appl. Math. Model. 36, 3371–3378 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Pan, X., Zheng, K., Zhang, L.: Finite difference discretization of the Rosenau-RLW equation. Appl. Anal. 92, 2578–2589 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations. Springer, New York (1997)zbMATHGoogle Scholar
  50. 50.
    Ren, H., Cheng, Y.: The interpolating element-free Galerkin (IEFG) method for two-dimensional potential problems. Eng. Anal. Bound. Elem. 36(5), 873–880 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Ren, H.P., Zhang, W.: An improved boundary element-free method (IBEFM) for two-dimensional potential problems. Chin. Phys. B 18(10), 4065–4073 (2009)CrossRefGoogle Scholar
  52. 52.
    Ren, H.P., Cheng, Y.M., Zhang, W.: An interpolating boundary element-free method (IBEFM) for elasticity problems. Sci. China Phys. Mech. Astron. 53(4), 758–766 (2010)CrossRefGoogle Scholar
  53. 53.
    Ren, H.P., Cheng, J., Huang, A.: The complex variable interpolating moving least-squares method. Appl. Math. Comput. 219, 1724–1736 (2012)MathSciNetzbMATHGoogle Scholar
  54. 54.
    Shokri, A., Dehghan, M.: A meshless method using the radial basis functions for numerical solution of the regularized long wave equation. Numer. Methods Part. Differ. 26, 807–825 (2010)MathSciNetzbMATHGoogle Scholar
  55. 55.
    Sun, F., Wang, J., Cheng, Y.M.: An improved interpolating element-free Galerkin method for elasticity. Chin. Phys. B 22(12), 120203 (2013)CrossRefGoogle Scholar
  56. 56.
    Sun, F., Wang, J., Cheng, Y.M., Huang, A.: Error estimates for the interpolating moving least-squares method in n-dimensional space. Appl. Numer. Math. 98, 79–105 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Tongsuk, P., Kanok-Nukulchai, W.: Further investigation of element free Galerkin method using moving Kriging interpolation. Int. J. Comput. Methods 01, 345–365 (2004)CrossRefzbMATHGoogle Scholar
  58. 58.
    Wang, J., Wang, J., Sun, F., Cheng, Y.M.: An interpolating boundary element-free method with nonsingular weight function for two-dimensional potential problems. Int. J. Comput. Methods 10, 1350043 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Wang, J., Sun, F., Cheng, Y.M., Huang, A.: Error estimates for the interpolating moving least-squares method. Appl. Math. Comput. 245, 321–342 (2014)MathSciNetzbMATHGoogle Scholar
  60. 60.
    Wazwaz, A.M.: The tanh method for traveling wave solutions of nonlinear equations. Appl. Math. Comput. 154(3), 713–723 (2004)MathSciNetzbMATHGoogle Scholar
  61. 61.
    Wazwaz, A.M.: A sine-cosine method for handling nonlinear wave equations. Math. Comput. Model. 40(5–6), 499–508 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    Wazwaz, A.M.: New solitary wave solutions to the Kuramoto–Sivashinsky and the Kawahara equations. Appl. Math. Comput. 182(2), 1642–1650 (2006)MathSciNetzbMATHGoogle Scholar
  63. 63.
    Wazwaz, A.M.: Partial Differential Equations and Solitary Waves Theory. Higher Education Press, Beijin (2009)CrossRefzbMATHGoogle Scholar
  64. 64.
    Zaky, M.A.: A Legendre collocation method for distributed-order fractional optimal control problems. Nonlinear Dyn. 91, 2667–2681 (2018)CrossRefzbMATHGoogle Scholar
  65. 65.
    Zaky, M.A.: An improved tau method for the multi-dimensional fractional Rayleigh–Stokes problem for a heated generalized second grade fluid. Comput. Math. Appl. 75, 2243–2258 (2018)MathSciNetCrossRefGoogle Scholar
  66. 66.
    Zaky, M.A., Ezz-Eldien, S.S., Doha, E.H., Machado, J.T., Bhrawy, A.H.: An efficient operational matrix technique for multi-dimensional variable-order time fractional diffusion equations. J. Comput. Nonlinear Dyn. 11(1–8), 061002 (2016)CrossRefGoogle Scholar
  67. 67.
    Zhang, Z., Hao, S.Y., Liew, K.M., Cheng, Y.M.: The improved element-free Galerkin method for two-dimensional elastodynamics problems. Eng. Anal. Bound. Elem. 37, 1576–1584 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  68. 68.
    Zhang, Z., Liew, K.M., Cheng, Y.: Coupling of the improved element-free Galerkin and boundary element methods for two-dimensional elasticity problems. Eng. Anal. Bound. Elem. 32, 100–107 (2008)CrossRefzbMATHGoogle Scholar
  69. 69.
    Zhang, L.W., Deng, Y.J., Liew, K.M.: An improved element-free Galerkin method for numerical modeling of the biological population problems. Eng. Anal. Bound. Elem. 40, 181–188 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  70. 70.
    Zhang, L.W., Deng, Y.J., Liew, K.M., Cheng, Y.M.: The improved complex variable element free Galerkin method for two-dimensional Schrödinger equation. Comput. Math. Appl. 68(10), 1093–1106 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  71. 71.
    Zhang, Z., Liew, K.M., Cheng, Y., Lee, Y.Y.: Analyzing 2D fracture problems with the improved element free Galerkin method. Eng. Anal. Bound. Elem. 32, 241–250 (2008)CrossRefzbMATHGoogle Scholar
  72. 72.
    Zhang, L., Deng, Y., Liew, K.M.: An improved element-free Galerkin method for numerical modeling of the biological population problems. Eng. Anal. Bound. Elem. 40, 181–188 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  73. 73.
    Zhang, L., Deng, Y., Liew, K.M., Cheng, Y.: The improved complex variable element-free Galerkin method for two-dimensional Schrödinger equation. Comput. Math. Appl. 68(10), 1093–1106 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  74. 74.
    Zhao, N., Ren, H.: The interpolating element-free Galerkin method for 2D transient heat conduction problems. Math. Probl. Eng. (2014). Article ID 712834Google Scholar
  75. 75.
    Zheng, B., Dai, B.D.: A meshless local moving Kriging method for two-dimensional solids. Appl. Math. Comput. 218, 563–573 (2011)MathSciNetzbMATHGoogle Scholar
  76. 76.
    Zhu, P., Zhang, L.W., Liew, K.M.: Geometrically nonlinear thermo-mechanical analysis of moderately thick functionally graded plates using a local Petrov–Galerkin approach with moving Kriging interpolation. Compos. Struct. 107, 298–314 (2014)CrossRefGoogle Scholar
  77. 77.
    Zuo, J.M., Zhang, Y.M., Zhang, T.D.: A new conservative difference scheme for the general Rosenau-RLW equation. Bound. Value Probl. 13, 516260 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  78. 78.
    Zuo, D.W., Jia, H.X., Shan, D.M.: Dynamics of the optical solitons for a (2+1)-dimensional nonlinear Schrödinger equation. Superlattices Microdtructers 101, 522–528 (2017)CrossRefGoogle Scholar
  79. 79.
    Zuo, D.W., Jia, H.X.: Interaction of the nonautonomous soliton in the optical fiber. Optik 127, 11282–11287 (2016)CrossRefGoogle Scholar
  80. 80.
    Zuo, D.W., Mo, H.X., Zhou, H.P.: Multi-soliton solutions of the generalized Sawada–Kotera equation. Z. Naturfors. A 71, 305–309 (2016)CrossRefGoogle Scholar
  81. 81.
    Zuo, D.W.: Modulation instability and breathers synchronization of the nonlinear Schrödinger Maxwell–Bloch equation. Appl. Math. Lett. 79, 182–186 (2018)MathSciNetCrossRefGoogle Scholar
  82. 82.
    Zuo, D.W., Gao, Y.T., Xue, L., Feng, Y.J., Sun, Y.H.: Rogue waves for the generalized nonlinear Schrödinger–Maxwell–Bloch system in optical-fiber communication. Appl. Math. Lett. 40, 78–83 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Applied Mathematics, Faculty of Mathematics and Computer SciencesAmirkabir University of TechnologyTehranIran

Personalised recommendations