Skip to main content

Space–Time Fully Decoupled Wavelet-Based Solution to Nonlinear Problems

  • Chapter
  • First Online:
Wavelet Numerical Method and Its Applications in Nonlinear Problems

Part of the book series: Engineering Applications of Computational Methods ((EACM,volume 6))

  • 575 Accesses

Abstract

The nonlinear initial-boundary-value problems (NIBVPs), i.e., the time-dependent NPDEs, provide a quantitative description for many nonlinear phenomena which play crucial roles in various scientific and engineering applications, including fluid mechanics, electromagnetic waves, nonlinear optics, population dynamics, solid-state physics, chemical kinetics, and plasma physics.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 89.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 119.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Hu HY (2006) Applied nonlinear dynamics. Aviation Industry Press, Beijing (in Chinese)

    Google Scholar 

  2. Baumann G (2005) Mathematica for theoretical physics: classical mechanics and nonlinear dynamics. Springer Science and Business Media Inc., New York

    Book  MATH  Google Scholar 

  3. Cole JD (1968) Perturbation method in applied mathematics. Blaisdell Publishing Company, Waltham

    MATH  Google Scholar 

  4. Mei C (1972) Finite element displacement method for larger amplitude free oscillations of beams and plates. Comput Struct 3:163–174

    Article  Google Scholar 

  5. Feng Y, Bert CW (1992) Application of the quadrature method to flexural vibration analysis of a geometrically nonlinear beam. Nonlinear Dyn 3:13–18

    Article  Google Scholar 

  6. Bhashyam GR, Prathap G (1980) Galerkin finite element method for non-linear beam vibrations. J Sound Vib 72:191–203

    Article  MATH  Google Scholar 

  7. Singh G, Sharma AK, Rao GV (1990) Large-amplitude free vibrations of beams-a discussion on various formulations and assumptions J. Sound Vib 142:77–85

    Article  Google Scholar 

  8. Guo Q (2004) Further Development of spline-based differential quadrature method and nonlinear vibration analysis of beam (in Chinese). Tsinghua University, Beijing

    Google Scholar 

  9. Chen SH (2007) Theorem analysis method of strongly nonlinear vibration system. Science Press, Beijing (in Chinese)

    Google Scholar 

  10. Wazwaz AM (2005) The tanh and the sine-cosine methods for compact and noncompact solutions of the nonlinear Klein-Gordon equation. Appl Math Comput 167:1179–1195

    MathSciNet  MATH  Google Scholar 

  11. Wazwaz AM (2005) The tanh and the sine-cosine methods for compact and noncompact solutions of the nonlinear Klein-Gordon equation. Appl Math Comput 167:1196–1210

    MathSciNet  MATH  Google Scholar 

  12. Kaya D, El-Sayed SM (2004) A numerical solution of the Klein-Gordon equation and convergence of the decomposition method. Appl Math Comput 156:341–353

    MathSciNet  MATH  Google Scholar 

  13. Basak KC, Ray PC, Bera RK (2009) Solution of non-linear Klein-Gordon equation with a quadratic non-linear term by Adomian decomposition method. Commun Nonlinear Sci Numer Simul 14:718–723

    Article  MathSciNet  MATH  Google Scholar 

  14. Chowdhury MSH, Hashim I (2009) Application of homotopy-perturbation method to Klein-Gordon and sine-Gordon equations. Chaos Solitons Fractals 39:1928–1935

    Article  MathSciNet  MATH  Google Scholar 

  15. Kanth A, Aruna K (2009) Differential transform method for solving the linear and nonlinear Klein-Gordon equation. Comput Phys Commun 180:708–711

    Article  MathSciNet  MATH  Google Scholar 

  16. Shakeri F, Dehghan M (2008) Numerical solution of the Klein-Gordon equation via He’s variational iteration method. Nonlinear Dyn 51:89–97

    Article  MathSciNet  MATH  Google Scholar 

  17. Yin F, Song J, Lu F (2014) A coupled method of Laplace trans-form and Legendre wavelets for nonlinear Klein-Gordon equations. Math Methods Appl Sci 37:781–792

    Article  MathSciNet  MATH  Google Scholar 

  18. Shao W, Wu X (2014) The numerical solution of the nonlinear Klein-Gordon and sine-Gordon equations using the Chebyshev tau meshless method. Comput Phys Commun 185:1399–1409

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. Bulbul B, Sezer M (2013) A new approach to numerical solution of nonlinear Klein-Gordon equation. Math Prob Eng 2013:869749.

    Google Scholar 

  21. Jang TS (2014) An integral equation formalism for solving the nonlinear Klein-Gordon equation. Appl Math Comput 243:322–338

    MathSciNet  MATH  Google Scholar 

  22. Lai H, Ma C (2011) Lattice Boltzmann model for generalized nonlinear wave equations. Phys Rev E 84:046708

    Article  Google Scholar 

  23. Kuo PY, Luis V (1984) Numerical solution of a nonlinear wave equation in polar coordinates. Appl Math Comput 14:313–329

    MathSciNet  MATH  Google Scholar 

  24. Argyris J, Haase M (1987) An engineer’s guide to soliton phenomena: application of the finite element method. Comput Methods Appl Mech Eng 61:71–122

    Article  MathSciNet  MATH  Google Scholar 

  25. Dehghan M, Shokri A (2009) Numerical solution of the nonlinear Klein-Gordon equation using radial basis functions. J Comput Appl Math 230:400–410

    Article  MathSciNet  MATH  Google Scholar 

  26. Rashidinia J, Mohammadi R (2010) Tension spline approach for the numerical solution of nonlinear Klein-Gordon equation. Comput Phys Commun 181:78–91

    Article  MathSciNet  MATH  Google Scholar 

  27. Lakestani M, Dehghan M (2010) Collocation and finite difference-collocation methods for the solution of nonlinear Klein-Gordon equation. Comput Phys Commun 181:1392–1401

    Article  MathSciNet  MATH  Google Scholar 

  28. Khuri SA, Sayfy A (2010) A spline collocation approach for the numerical solution of ageneralized nonlinear Klein-Gordon equation. Appl Math Comput 216:1047–1056

    MathSciNet  MATH  Google Scholar 

  29. Rashidinia J, Ghasemi M, Jalilian R (2010) Numerical solution of the nonlinear Klein-Gordon equation. J Comput Appl Math 233:1866–1878

    Article  MathSciNet  MATH  Google Scholar 

  30. Yin F, Tian T, Song J, Zhu M (2015) Spectral methods using Legendre wavelets for nonlinear Klein\Sine-Gordon equations. J Comput Appl Math 275:321–334

    Article  MathSciNet  MATH  Google Scholar 

  31. Bratsos AG (2009) On the numerical solution of the Klein-Gordon equation. Numer Methods Part Diff Eq 25:939–951

    Article  MathSciNet  MATH  Google Scholar 

  32. Dehghan M, Mohebbi A, Asgari Z (2009) Fourth-order compact solution of the nonlinear Klein-Gordon equation. Numer Alg 52:523–540

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  34. Wei GW (2000) Discrete singular convolution for the sine-Gordon equation. Physica D 137:247–259

    Article  MathSciNet  MATH  Google Scholar 

  35. Dehghan M, Shokri A (2008) A numerical method for one-dimensional nonlinear sine-gordon equation using collocation and radial basis functions. Numer Methods Part Diff Eq 24:687–698

    Article  MathSciNet  MATH  Google Scholar 

  36. Sari M, Gurarslan G (2011) A sixth-order compact finite difference method for the one-dimensional sine-Gordon equation. Int J Numer Methods Biomed Eng. 27:1126–1138

    Article  MathSciNet  MATH  Google Scholar 

  37. Bratsos AG (2008) A numerical method for the one-dimensional sine-Gordon equation. Numer Methods Part Diff Eq 24:833–844

    Article  MathSciNet  MATH  Google Scholar 

  38. Dehghan M, Ghesmati A (2010) Application of the dual reciprocity boundary integral equation technique to solve the nonlinear Klein-Gordon equation. Comput Phys Commun 181:1410–1418

    Article  MathSciNet  MATH  Google Scholar 

  39. Cui M (2009) Fourth-order compact scheme for the one-dimensional sine-Gordon equation. Numer Methods Part Diff Eq 25:685–711

    Article  MathSciNet  MATH  Google Scholar 

  40. Chen JB (2006) Symplectic and multisymplectic Fourier pseudo spectral discretizations for the Klein-Gordon equation. Lett Math Phys 75:293–305

    Article  MathSciNet  MATH  Google Scholar 

  41. Wang Q, Cheng D (2005) Numerical solution of damped non-linear Klein-Gordon equations using variational method and finite element approach. Appl Math Comput 162:381–401

    MathSciNet  MATH  Google Scholar 

  42. Guo PF, Liew KM, Zhu P (2015) Numerical solution of nonlinear Klein-Gordon equation using the element-free kp-Ritz method. Appl Math Model 39:2917–2928

    Article  MathSciNet  MATH  Google Scholar 

  43. Pontaza JP, Reddy JN (2004) Space-time coupled spectral/hp least-squares finite element formulation for the incompressible Navier-Stokes equations. J Comput Phys 197:418–459

    Article  MathSciNet  MATH  Google Scholar 

  44. Dorao CA, Jakobsen HA (2007) A parallel time–space least-squares spectral element solver for incompressible flow problems. Appl Math Comput 185:45–58

    Google Scholar 

  45. Fletcher CAJ (1984) Computational Galerkin methods. Springer, Berlin

    Book  MATH  Google Scholar 

  46. Malomed BA (1985) Inelastic interactions of solitons in nearly integrable systems II. Physica D 15:385–401

    Article  MathSciNet  MATH  Google Scholar 

  47. Kivshar YS, Fei Z, Vazquez L (1991) Resonant soliton-impurity interactions. Phys Rev Lett 67:1177–1180

    Article  Google Scholar 

  48. Fei Z, Kivshar YS, Vazquez L (1992) Resonant kink-impurity interactions in the sine-Gordon model. Phys Rev A 45:6019–6030

    Article  Google Scholar 

  49. Zhang F (1998) Breather scattering by impurities in the sine-Gordon model. Phys Rev E 58:2558–2563

    Article  Google Scholar 

  50. Liu XJ, Wang JZ, Zhou YH (2017) A space-time fully decoupled wavelet Galerkin method for solving a class of nonlinear wave problems. Nonlinear Dyn 90:599–616

    Article  MathSciNet  Google Scholar 

  51. Tzou HS, Anderson GL (eds) (1992) Intelligent structural systems. Kluwer Academic Publishers, Boston

    Google Scholar 

  52. Lee CK (1992) Piezoelectric laminates: theory and experiments for distributed sensors and actuators. In: Tzou HS, Anderson GL (eds) Intelligent structural systems. Academic Publishers and Kluwer, Boston

    Google Scholar 

  53. Zhou YH, Wang JZ (1998) A dynamic control model of piezoelectric cantilevered beam-plate based on wavelet theory. Acta Mechanica Sinica 30(6):719–727 (in Chinese)

    Google Scholar 

  54. Zhou YH, Wang JZ, Zheng XJ, Jiang Q (2000) Vibration control of variable thickness plates with piezoelectric sensors and actuators by wavelet theory. J Sound Vib 237(3):395–410

    Article  Google Scholar 

  55. Guo G, Chai TY (2002) Predictive control strategy for a cantilever. Proc CSEE (12):81–85 (in Chinese)

    Google Scholar 

  56. Tzou HS, Zhou YH (1995) Dynamics and control of piezoelectric circular plates with geometrical nonlinearity. J Sound Vib 188(2):189–207

    Article  Google Scholar 

  57. William JR, Amaratunga K (1994) Introduction to wavelet in engineering. Int J Numer Meth Eng 37:2365–2388

    Article  MathSciNet  MATH  Google Scholar 

  58. Motard RL, Joseph B (1994) Wavelet applications in chemical engineering. Kluwer Academic Publishers, Boston

    Book  Google Scholar 

  59. Kivshar YS, Malomed BA (1989) Dynamics of solitons in nearly integrable systems. Rev Mod Phys 61(4):763–915

    Article  Google Scholar 

  60. Dalfovo F, Giorgini S, Pitaevskii LP et al (1999) Theory of Bose-Einstein condensation in trapped gases. Rev Mod Phys 71(3):463

    Article  Google Scholar 

  61. Agrawal GP (2007) Nonlinear fiber optics, 4th edn. Academic Press, San Diego

    Google Scholar 

  62. Solli DR, Ropers C, Koonath P et al (2007) Optical rogue waves. Nature 450(7172):1054–1057

    Article  Google Scholar 

  63. Ye Y (2009) Self-similar solutions for nonlinear Schrödinger equations. Math Prob Eng

    Google Scholar 

  64. Subaşi M (2002) On the finite differences schemes for the numerical solution of two dimensional Schrödinger equation. Numer Methods Part Diff Eq: Int J 18(6):752–758

    Article  MathSciNet  MATH  Google Scholar 

  65. Bao W, Jaksch D (2003) An explicit unconditionally stable numerical method for solving damped nonlinear Schrödinger equations with a focusing nonlinearity. SIAM J Numer Anal 41(4):1406–1426

    Article  MathSciNet  MATH  Google Scholar 

  66. Bao W, Jaksch D, Markowich PA (2003) Numerical solution of the Gross-Pitaevskii equation for Bose-Einstein condensation. J Comput Phys 187(1):318–342

    Article  MathSciNet  MATH  Google Scholar 

  67. Wang H (2005) Numerical studies on the split-step finite difference method for nonlinear Schrödinger equations. Appl Math Comput 170(1):17–35

    Article  MathSciNet  MATH  Google Scholar 

  68. Kalita JC, Chhabra P, Kumar S (2006) A semi-discrete higher order compact scheme for the unsteady two-dimensional Schrödinger equation. J Comput Appl Math 197(1):141–149

    Article  MathSciNet  MATH  Google Scholar 

  69. Dehghan M, Shokri A (2007) A numerical method for two-dimensional Schrödinger equation using collocation and radial basis functions. Comput Math Appl 54(1):136–146

    Article  MathSciNet  MATH  Google Scholar 

  70. Sweilam NH (2007) Variational iteration method for solving cubic nonlinear Schrödinger equation. J Comput Appl Math 207(1):155–163

    Article  MathSciNet  MATH  Google Scholar 

  71. Dehghan M, Mirzaei D (2008) Numerical solution to the unsteady two-dimensional Schrödinger equation using meshless local boundary integral equation method. Int J Numer Meth Eng 76(4):501–520

    Article  MATH  Google Scholar 

  72. Alomari AK, Noorani MSM, Nazar R (2009) Explicit series solutions of some linear and nonlinear Schrodinger equations via the homotopy analysis method. Commun Nonlinear Sci Numer Simul 14(4):1196–1207

    Article  MathSciNet  MATH  Google Scholar 

  73. Liao H, Sun Z, Shi H (2010) Error estimate of fourth-order compact scheme for linear Schrödinger equations. SIAM J Numer Anal 47(6):4381–4401

    Article  MathSciNet  MATH  Google Scholar 

  74. Mertens FG, Quintero NR, Bishop AR (2010) Nonlinear Schrödinger equation with spatiotemporal perturbations. Phys Rev E 81(1):016608

    Article  Google Scholar 

  75. Tian ZF, Yu PX (2010) High-order compact ADI (HOC-ADI) method for solving unsteady 2D Schrödinger equation. Comput Phys Commun 181(5):861–868

    Article  MathSciNet  MATH  Google Scholar 

  76. Gao Z, Xie S (2011) Fourth-order alternating direction implicit compact finite difference schemes for two-dimensional Schrödinger equations. Appl Numer Math 61(4):593–614

    Article  MathSciNet  MATH  Google Scholar 

  77. He D, Pan K (2017) An unconditionally stable linearized CCD–ADI method for generalized nonlinear Schrödinger equations with variable coefficients in two and three dimensions. Comput Math Appl 73(11):2360–2374

    Article  MathSciNet  MATH  Google Scholar 

  78. Wang S, Zhang L (2011) Split-step orthogonal spline collocation methods for nonlinear Schrödinger equations in one, two, and three dimensions. Appl Math Comput 218(5):1903–1916

    MathSciNet  MATH  Google Scholar 

  79. Zhu H, Chen Y, Song S et al (2011) Symplectic and multi-symplectic wavelet collocation methods for two-dimensional Schrödinger equations. Appl Numer Math 61(3):308–321

    Article  MathSciNet  MATH  Google Scholar 

  80. Kong L, Duan Y, Wang L, Yin X, Ma Y (2012) Spectral-like resolution compact ADI finite difference method for the multi-dimensional Schrödinger equations. Math Comput Model 55(5–6):1798–1812

    Article  MATH  Google Scholar 

  81. Xu Y, Zhang L (2012) Alternating direction implicit method for solving two-dimensional cubic nonlinear Schrödinger equation. Comput Phys Commun 183(5):1082–1093

    Article  MathSciNet  MATH  Google Scholar 

  82. Abbasbandy S, Ghehsareh HR, Hashim I (2013) A meshfree method for the solution of two-dimensional cubic nonlinear Schrödinger equation. Eng Anal Boundary Elem 37(6):885–898

    Article  MathSciNet  MATH  Google Scholar 

  83. Antoine X, Bao W, Besse C (2013) Computational methods for the dynamics of the nonlinear Schrödinger/Gross-Pitaevskii equations. Comput Phys Commun 184(12):2621–2633

    Article  MathSciNet  MATH  Google Scholar 

  84. Dehghan M, Emami-Naeini F (2013) The Sinc-collocation and Sinc-Galerkin methods for solving the two-dimensional Schrödinger equation with nonhomogeneous boundary conditions. Appl Math Model 37(22):9379–9397

    Article  MathSciNet  MATH  Google Scholar 

  85. Wang T, Guo B, Xu Q (2013) Fourth-order compact and energy conservative difference schemes for the nonlinear Schrödinger equation in two dimensions. J Comput Phys 243:382–399

    Article  MathSciNet  MATH  Google Scholar 

  86. Taleei A, Dehghan M (2014) Time-splitting pseudo-spectral domain decomposition method for the soliton solutions of the one-and multi-dimensional nonlinear Schrödinger equations. Comput Phys Commun 185(6):1515–1528

    Article  MathSciNet  MATH  Google Scholar 

  87. Tay WC, Tan EL (2014) Pentadiagonal alternating-direction-implicit finite-difference time-domain method for two-dimensional Schrödinger equation. Comput Phys Commun 185(7):1886–1892

    Article  MathSciNet  MATH  Google Scholar 

  88. Zhang LW, Deng YJ, Liew KM et al (2014) The improved complex variable element-free Galerkin method for two-dimensional Schrödinger equation. Comput Math Appl 68(10):1093–1106

    Article  MathSciNet  MATH  Google Scholar 

  89. Zhang LW, Liew KM (2014) An element-free based solution for nonlinear Schrödinger equations using the ICVMLS-Ritz method. Appl Math Comput 249:333–345

    MathSciNet  MATH  Google Scholar 

  90. Bhrawy AH, Abdelkawy MA (2015) A fully spectral collocation approximation for multi-dimensional fractional Schrödinger equations. J Comput Phys 294:462–483

    Article  MathSciNet  MATH  Google Scholar 

  91. Li LZ, Sun HW, Tam SC (2015) A spatial sixth-order alternating direction implicit method for two-dimensional cubic nonlinear Schrödinger equations. Comput Phys Commun 187:38–48

    Article  MathSciNet  MATH  Google Scholar 

  92. Mocz P, Succi S (2015) Numerical solution of the nonlinear Schrödinger equation using smoothed-particle hydrodynamics. Phys Rev E 91(5):053304

    Article  MathSciNet  Google Scholar 

  93. Quintero NR, Mertens FG, Bishop AR (2015) Soliton stability criterion for generalized nonlinear Schrödinger equations. Phys Rev E 91(1):012905

    Article  MathSciNet  Google Scholar 

  94. Sheu TWH, Lin L (2015) Dispersion relation equation preserving FDTD method for nonlinear cubic Schrödinger equation. J Comput Phys 299:1–21

    Article  MathSciNet  MATH  Google Scholar 

  95. Liu XJ, Wang JZ, Zhou YH (2016) A space–time fully decoupled wavelet Galerkin method for solving two-dimensional Burgers’ equations. Comput Math Appl 72(12):2908–2919

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2021 The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Zhou, YH. (2021). Space–Time Fully Decoupled Wavelet-Based Solution to Nonlinear Problems. In: Wavelet Numerical Method and Its Applications in Nonlinear Problems. Engineering Applications of Computational Methods, vol 6. Springer, Singapore. https://doi.org/10.1007/978-981-33-6643-5_8

Download citation

  • DOI: https://doi.org/10.1007/978-981-33-6643-5_8

  • Published:

  • Publisher Name: Springer, Singapore

  • Print ISBN: 978-981-33-6642-8

  • Online ISBN: 978-981-33-6643-5

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics