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Numerical study of the one-dimensional coupled nonlinear sine-Gordon equations by a novel geometric meshless method

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Abstract

In the paper, we derive a geometric meshless method for coupled nonlinear sine-Gordon (CNSG) equations. Approximate solutions of the CNSG equations are supposed to be expressed as the moving Kriging (MK) shape functions in the space direction. Global weak form of the CNSG equations is obtained, and then, a system of ODEs in time coordinate is extracted after imposing the MK meshless method. Then the geometric integrator, namely group preserving scheme, is offered to approximate the solution of obtained system of ODEs. Stability analysis of the method is numerically investigated. Numerical experiments show that the proposed method is effective and accurate for the CNSG equations.

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References

  1. Khusnutdinova KR, Pelinovsky DE (2003) On the exchange of energy in coupled Klein–Gordon equations. Wave Motion 38(1):1–10

    MathSciNet  MATH  Google Scholar 

  2. Kontorova T, Frenkel J (1938) On the theory of plastic deformation and twinning. II. Zh Eksp Teor Fiz 8:1340–1348

    MATH  Google Scholar 

  3. Mohebbi A, Dehghan M (2010) High-order solution of one-dimensional sine-Gordon equation using compact finite difference and DIRKN methods. Math Comput Model 51(5–6):537–549

    MathSciNet  MATH  Google Scholar 

  4. Salas AH (2010) Exact solutions of coupled sine-Gordon equations. Nonlinear Anal Real World Appl 11(5):3930–3935

    MathSciNet  MATH  Google Scholar 

  5. Jiang Z-W, Wang R-H (2012) Numerical solution of one-dimensional sine-Gordon equation using high accuracy multiquadric quasi-interpolation. Appl Math Comput 218(15):7711–7716

    MathSciNet  MATH  Google Scholar 

  6. Wazwaz A-M (2018) Multiple complex and multiple real soliton solutions for the integrable sine-Gordon equation. Optik 172:622–627

    Google Scholar 

  7. Hosseini K, Mayeli P, Kumar D (2018) New exact solutions of the coupled sine-Gordon equations in nonlinear optics using the modified Kudryashov method. J Mod Opt 65(3):361–364

    MathSciNet  Google Scholar 

  8. Zhang C, Cheng Q, Zhang D-J (2018) Soliton solutions of the sine-Gordon equation on the half line. Appl Math Lett 86:64–69

    MathSciNet  MATH  Google Scholar 

  9. Jagtap AD, Murthy AV (2018) Higher order scheme for two-dimensional inhomogeneous sine-Gordon equation with impulsive forcing. Commun Nonlinear Sci Numer Simul 64:178–197

    MathSciNet  MATH  Google Scholar 

  10. Baccouch M (2018) A posteriori local discontinuous Galerkin error estimates for the one-dimensional sine-Gordon equation. Int J Comput Math 95(4):815–844

    MathSciNet  MATH  Google Scholar 

  11. Yin F, Tian T, Song J, Zhu M (2015) Spectral methods using legendre wavelets for nonlinear klein\(\setminus \) sine-Gordon equations. J Comput Appl Math 275:321–334

    MathSciNet  MATH  Google Scholar 

  12. Pekmen B, Tezer-Sezgin M (2012) Differential quadrature solution of nonlinear Klein–Gordon and sine-Gordon equations. Comput Phys Commun 183(8):1702–1713

    MathSciNet  MATH  Google Scholar 

  13. Jafarabadi A, Shivanian E (2018) Numerical simulation of nonlinear coupled Burgers’ equation through meshless radial point interpolation method. Eng Anal Bound Elem 95:187–199

    MathSciNet  MATH  Google Scholar 

  14. Shivanian E, Jafarabadi A (2016) More accurate results for nonlinear generalized Benjamin–Bona–Mahony–Burgers (GBBMB) problem through spectral meshless radial point interpolation (SMRPI). Eng Anal Bound Elem 72:42–54

    MathSciNet  MATH  Google Scholar 

  15. Abbasbandy S, Ghehsareh HR, Hashim I, Alsaedi A (2014) A comparison study of meshfree techniques for solving the two-dimensional linear hyperbolic telegraph equation. Eng Anal Bound Elem 47:10–20

    MathSciNet  MATH  Google Scholar 

  16. Zhu P, Zhang L, Liew K (2014) Geometrically nonlinear thermomechanical analysis of moderately thick functionally graded plates using a local Petrov–Galerkin approach with moving Kriging interpolation. Compos Struct 107:298–314

    Google Scholar 

  17. Chen L, Liew KM (2011) A local Petrov–Galerkin approach with moving Kriging interpolation for solving transient heat conduction problems. Comput Mech 47(4):455–467

    MathSciNet  MATH  Google Scholar 

  18. Zheng B, Dai B (2011) A meshless local moving Kriging method for two-dimensional solids. Appl Math Comput 218(2):563–573

    MathSciNet  MATH  Google Scholar 

  19. Dai B, Cheng J, Zheng B (2013) A moving Kriging interpolation-based meshless local Petrov–Galerkin method for elastodynamic analysis. Int J Appl Mech 5(01):1350011

    Google Scholar 

  20. Bui TQ, Zhang C (2011) Moving Kriging interpolation-based meshfree method for dynamic analysis of structures. PAMM 11(1):197–198

    Google Scholar 

  21. Thai CH, Do VN, Nguyen-Xuan H (2016) An improved moving Kriging-based meshfree method for static, dynamic and buckling analyses of functionally graded isotropic and sandwich plates. Eng Anal Bound Elem 64:122–136

    MathSciNet  MATH  Google Scholar 

  22. Watts G, Pradyumna S, Singha M (2017) Nonlinear analysis of quadrilateral composite plates using moving Kriging based element free Galerkin method. Compos Struct 159:719–727

    Google Scholar 

  23. Dehghan M, Abbaszadeh M (2017) Two meshless procedures: moving Kriging interpolation and element-free Galerkin for fractional PDEs. Appl Anal 96(6):936–969

    MathSciNet  MATH  Google Scholar 

  24. Gu L (2003) Moving Kriging interpolation and element-free Galerkin method. Int J Numer Methods Eng 56(1):1–11

    MATH  Google Scholar 

  25. Shokri A, Habibirad A (2016) A moving Kriging-based mlpg method for nonlinear Klein–Gordon equation. Mathematical Methods Appl Sci 39(18):5381–5394

    MathSciNet  MATH  Google Scholar 

  26. Liu C-S (2001) Cone of non-linear dynamical system and group preserving schemes. Int J Non-Linear Mech 36(7):1047–1068

    MathSciNet  MATH  Google Scholar 

  27. Hashemi M (2019) Numerical solution to the telegraph equation via the geometric moving Kriging meshfree method. Eur Physl J Plus 134(8):381

    Google Scholar 

  28. Hashemi MS, Darvishi E, Inc M (2018) A geometric numerical integration method for solving the Volterra integro-differential equations. Int J Comput Math 95(8):1654–1665

    MathSciNet  Google Scholar 

  29. Hashemi MS, Abbasbandy S (2017) A geometric approach for solving Troesch’s problem. Bull Malays Math Sci Soc 40(1):97–116

    MathSciNet  MATH  Google Scholar 

  30. Hashemi MS (2015) Constructing a new geometric numerical integration method to the nonlinear heat transfer equations. Commun Nonlinear Sci Numer Simul 22(1):990–1001

    MathSciNet  MATH  Google Scholar 

  31. Hashemi MS, Baleanu D, Parto-Haghighi M (2015) A lie group approach to solve the fractional poisson equation. Rom J Phys 60:1289–1297

    Google Scholar 

  32. Hashemi MS, Baleanu D, Parto-Haghighi M, Darvishi E (2015) Solving the time-fractional diffusion equation using a lie group integrator. Therm Sci 19:77–83

    Google Scholar 

  33. Hashemi MS, Baleanu D (2016) Numerical approximation of higher-order time-fractional telegraph equation by using a combination of a geometric approach and method of line. J Comput Phys 316:10–20

    MathSciNet  MATH  Google Scholar 

  34. Hashemi MS, Darvishi E, Baleanu D (2016) A geometric approach for solving the density-dependent diffusion Nagumo equation. Adv Differ Equ 2016(1):89

    MathSciNet  MATH  Google Scholar 

  35. Hashemi MS (2017) A novel simple algorithm for solving the magneto-hemodynamic flow in a semi-porous channel. Eur J Mech B/Fluids 65:359–367

    MathSciNet  MATH  Google Scholar 

  36. Akgül A, Hashemi MS (2017) Group preserving scheme and reproducing kernel method for the Poisson–Boltzmann equation for semiconductor devices. Nonlinear Dyn 88(4):2817–2829

    MathSciNet  MATH  Google Scholar 

  37. Akgül A, Hashemi MS, Raheem S et al (2017) Constructing two powerful methods to solve the Thomas–Fermi equation. Nonlinear Dyn 87(2):1435–1444

    MathSciNet  MATH  Google Scholar 

  38. Karami A, Abbasbandy S, Shivanian E (2019) Meshless local Petrov–Galerkin formulation of inverse Stefan problem via moving least squares approximation. Math Comput Appl 24(4):101

    MathSciNet  Google Scholar 

  39. Aslefallah M, Abbasbandy S, Shivanian E (2019) Numerical solution of a modified anomalous diffusion equation with nonlinear source term through meshless singular boundary method. Eng Anal Bound Elem 107:198–207

    MathSciNet  MATH  Google Scholar 

  40. Shivanian E, Shaban M (2019) An improved pseudospectral meshless radial point interpolation (PSMRPI) method for 3D wave equation with variable coefficients. Eng Comput 35(4):1159–1171

    Google Scholar 

  41. Ilati M, Dehghan M (2015) The use of radial basis functions (RBFs) collocation and RBF-QR methods for solving the coupled nonlinear sine-Gordon equations. Eng Anal Bound Elem 52:99–109

    MathSciNet  MATH  Google Scholar 

  42. Yomosa S (1983) Soliton excitations in deoxyribonucleic acid (DNA) double helices. Phys Rev A 27(4):2120

    MathSciNet  Google Scholar 

  43. Kumar KH, Vijesh VA (2017) Chebyshev wavelet quasilinearization scheme for coupled nonlinear sine-Gordon equations. J Comput Nonlinear Dyn 12(1):011018

    Google Scholar 

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  1. Department of Mathematics, Basic Science Faculty, University of Bonab, P.O. Box 55517-61167, Bonab, Iran

    M. S. Hashemi

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  1. M. S. Hashemi
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Hashemi, M.S. Numerical study of the one-dimensional coupled nonlinear sine-Gordon equations by a novel geometric meshless method. Engineering with Computers 37, 3397–3407 (2021). https://doi.org/10.1007/s00366-020-01001-2

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