Skip to main content

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

  • 571 Accesses

Abstract

Since Isaac Newton published his famous book of the Mathematical Principles of Natural Philosophy 300 years ago, the Newton classical mechanics has been recognized as an open of modern science through rigorous logical reasoning, precise mathematical tools, and accurate calculation results [1–3].

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. Newton I (1802) Mathematical principles of natural philosophy. A. Strahan

    Google Scholar 

  2. Hartog JP (1948) Mechanics. General Publishing Company, Ontario

    MATH  Google Scholar 

  3. Wei N (2004) Basis and application of nonlinear science. Science Press, Beijing

    Google Scholar 

  4. Nicolis C (1995) Introduction to nonlinear science. Cambridge University Press, Cambridge

    Book  Google Scholar 

  5. Lui L (1997) Introduction to nonlinear physics. Springer-Verlag, New York

    MATH  Google Scholar 

  6. Wilson JR (1981) Nonlinear system theory. The Johns Hopkins University Press, Maryland

    MATH  Google Scholar 

  7. Scott AC (2007) The Nonlinear universe: Chaos, emergence, life. Springer, Berlin Heidelberg, New York

    Google Scholar 

  8. Hu HY (2006) Applied nonlinear dynamics. Aviation Industry Press, Beijing

    Google Scholar 

  9. Thompson JMT, Stewart HB (2002) Nonlinear dynamics and chaos. Wiley, Chichester

    MATH  Google Scholar 

  10. Lin JZ, Ruan XD, Chen BG et al (2005) Fluid mechanics. Tsinghua University Press, Beijing

    Google Scholar 

  11. Zheng XJ, Liu XE (2005) A nonlinear constitutive model for Terfenol-D rods. J Appl Phys 97:053901

    Article  Google Scholar 

  12. Zhu ZX, Huang Y (1998) Science in the 21st century-nonlinear science. Chin Sci Bull 7:1–2

    Google Scholar 

  13. Tadmor E (2012) A review of numerical methods for nonlinear partial differential equations. Bull Am Math Soc 49:507–554

    Article  MathSciNet  MATH  Google Scholar 

  14. Neilan M, Glowinski R, Feng X (2013) Recent developments in numerical methods for fully nonlinear second order partial differential equations. Soc Indus Appl Math Rev 55:205–267

    MathSciNet  MATH  Google Scholar 

  15. Acton JR, Squire PT (1985) Solving equations with physical understanding. Adam Hilger Ltd., Boston

    Google Scholar 

  16. Elwakil SA, El-labany SK, Zahran MA (2002) Modified extended tanh-function method for solving nonlinear partial differential equations. Phys Lett A 299:179–188

    Google Scholar 

  17. Sedov LI (1982) Similarity methods and dimensional analysis in mechanics. Science Press, Beijing

    MATH  Google Scholar 

  18. Yan XL (2004) The exact solution of a nonlinear equation. Economic Science Press, Beijing

    Google Scholar 

  19. Lu WR (2003) Variational methods in differential equations. Science Press, Beijing

    Google Scholar 

  20. Struwe M (2008) Variational methods, applications to nonlinear partial differential equations and Hamiltonian systems. Springer-Verlag, Berlin

    MATH  Google Scholar 

  21. Li JC, Zhou XC (1998) Asymptotic methods in mathematical physics. Science Press, Beijing

    Google Scholar 

  22. Yang BJ, Zhao YF (2003) Methods of advanced mathematical physics. Beijing University of Posts and Telecommunications Press, Beijing

    Google Scholar 

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

    MATH  Google Scholar 

  24. Nayfeh AH (2000) Perturbation method. Wiley, New York

    Book  MATH  Google Scholar 

  25. Von Dyke M (1975) Perturbation methods in fluid mechanics. The Parabolic Press, Stanford

    MATH  Google Scholar 

  26. Zheng XJ, Zhou YH (1989) On convergence of interpolated iterative method of geometrically nonlinear equations of circular plates. Sci China A 32:316–327

    MathSciNet  MATH  Google Scholar 

  27. Von Kármán T (1910) Festigkeits problem in maschinenbau. Encyklopädie Der Mathematischen Wissenschaften 4:311–385

    Google Scholar 

  28. Way S (1934) Bending of circular plate with large deflection. J Appl Mech 56:627–636

    Google Scholar 

  29. Vincent JJ (1931) The bending of a thin circular plate. Phil Mag 12:185–196

    Article  MATH  Google Scholar 

  30. Chien W (1947) Large deflection of a circular clamped plate under uniform pressure. Chin J Phys 7:102–113

    MathSciNet  Google Scholar 

  31. Chien W, Yeh K (1954) On the large deflection of circular plates. China Sci 3:405–436

    MathSciNet  Google Scholar 

  32. Qian WC (1989) A selection of scientific papers by Qian Weichang. Fujian Education Press, Fuzhou

    Google Scholar 

  33. Qian WC, Lin HS, Hu HC, Yeh KY (1954) Large deflection of elastic circular thin plate. Science Press, Beijing

    Google Scholar 

  34. Zheng XJ (1990) Theory and application of large deflection of circular thin plate. Jilin Science & Technology Publishing House, Changchun

    Google Scholar 

  35. Yeh KY, Zheng XJ, Zhou YH (1989) An analytical formula of the exact solution to von Kármán’s equations of a circular plate under a concentrated load. Int J Non-Linear Mech 24:551–560

    Article  MATH  Google Scholar 

  36. Zheng XJ, Lee J (1995) On the convergence of the Chien’s perturbation method for von Kármán plate equations. Int J Eng Sci 33:1085–1094

    Article  MATH  Google Scholar 

  37. Zheng XJ, Zhou YH (1990) Analytical formulas of solutions of geometrically nonlinear equations of axisymmetric plates and shallow shells. Acta Mech Sin 6:69–80

    Article  MATH  Google Scholar 

  38. Zhou YH, Zheng XJ (1989) On the range of applicability of von Karman plate equations. J Appl Mech 56:724–726

    Article  Google Scholar 

  39. Yeh KY, Zheng XJ, Wang XZ (1990) On some properties and calculation of the exact solution to von Kármán’s equations of circular plates under a concentrated load. Int J Non-Linear Mech 25:17–26

    Article  MATH  Google Scholar 

  40. Zheng XJ, Zhou YH (1988) The exact solution of large deflection problem of circular thin plate of elastic foundation under concentrated load. Chin J Theor Appl Mech 20:161–171

    Google Scholar 

  41. Zheng XJ, Zhou YH (1987) On exact solution of Karman’s equations of rigid clamped circular plate and shallow spherical shell under a concentrated load. Appl Math Mech 8:1057–1068

    Article  MATH  Google Scholar 

  42. He JH (2006) Some asymptotic methods for strongly nonlinear equations. Int J Mod Phys B 20:1141–1199

    Article  MathSciNet  MATH  Google Scholar 

  43. Chapra SC, Canale RP (2010) Numerical methods for engineers. McGraw-Hill, New York

    Google Scholar 

  44. Atkinson KA (1989) An Introduction to numerical analysis. Wiley, New York

    MATH  Google Scholar 

  45. Lambert JD (1991) Numerical methods for ordinary differential systems: the Initial Value Problem. Wiley, New York

    MATH  Google Scholar 

  46. Khiyal MSH (2005) Implementation of Newmark’s method for second order initial value problems. J Appl Sci 5:402–410

    Article  Google Scholar 

  47. Iserles A (1996) A first course in the numerical analysis of differential equations. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  48. Causon DM, Mingham CG (2010) Introductory finite difference method for PDEs. Ventus Publishing, London

    Google Scholar 

  49. Delfour M, Fortin M, Payr G (1981) Finite-difference solutions of a non-linear Schrödinger equation. J Comput Phys 44:277–288

    Article  MathSciNet  MATH  Google Scholar 

  50. Chen YM (1975) Numerical computation of dynamic stress intensity factors by a Lagrangian finite-difference method. Eng Fract Mech 7:653–660

    Article  Google Scholar 

  51. Brian PLT (1961) A finite-difference method of high-order accuracy for the solution of three-dimensional transient heat conduction problems. AIChE J 7:367–370

    Article  Google Scholar 

  52. Beam RM, Warming RF (1978) An implicit factored scheme for the compressible Navier-Sstokes equations. Am Inst Aeronaut Astronaut J 16:393–402

    Article  MATH  Google Scholar 

  53. Narasimhan TN, Witherspoon PA (1976) An integrated finite difference method for analyzing fluid flow in porous media. Water Resour Res 12:57–64

    Article  Google Scholar 

  54. Yuste SB, Quintana-Murillo J (2012) A finite difference method with non-uniform timesteps for fractional diffusion equations. Comput Phys Commun 183:2594–2600

    Article  MathSciNet  MATH  Google Scholar 

  55. Buckmire R (2004) Application of a Mickens finite-difference scheme to the cylindrical Bratu-Gelfand problem. Numer Methods Partial Differen Equations Int J 20:327–337

    Article  MathSciNet  MATH  Google Scholar 

  56. Qiu JB (1991) Theory and application of weighted residual value method. Aerospace Press, Beijing

    Google Scholar 

  57. Wang XZ (2006) Computational mechanics. Lanzhou University Press, Lanzhou

    Google Scholar 

  58. Bramble JH, Schatz AH (1971) Least squares methods for 2mth order elliptic boundary-value problems. Math Comput 25:1–32

    MATH  Google Scholar 

  59. Xiu D, Hesthaven JS (2006) High-order collocation methods for differential equations with random inputs. Soc Indus Appl Math J Sci Comput 27:1118–1139

    MathSciNet  MATH  Google Scholar 

  60. Cockburn B, Shu CW (2006) The local discontinuous Galerkin method for time-dependent convection-diffusion systems. Soc Indus Appl Math J Numer Anal 35:2440–2463

    MathSciNet  MATH  Google Scholar 

  61. Wang P, Yao Y, Tulin MP (1995) An efficient numerical tank for non-linear water waves, based on the multi-subdomain approach with BEM. Int J Numer Meth Fluids 20:1315–1336

    Article  MATH  Google Scholar 

  62. Feng X, Neilan M (2009) Vanishing moment method and moment solutions for fully nonlinear second order partial differential equations. J Sci Comput 38:74–98

    Article  MathSciNet  MATH  Google Scholar 

  63. Atluri SN, Zhu T (1998) A new Meshless Local Petrov-Galerkin approach in computational mechanics. Comput Mech 22:117–127

    Article  MathSciNet  MATH  Google Scholar 

  64. Zhu T, Zhang J, Atluri SN (1998) A meshless local boundary integral equation method for solving nonlinear problems. Comput Mech 22:174–186

    Article  MathSciNet  MATH  Google Scholar 

  65. Zhang X, Liu Y (2004) Meshless method. Tsinghua University Press, Beijing

    Google Scholar 

  66. Sauter SA, Schwab C (2011) Boundary element methods. Springer-Verlag, Berlin

    Book  MATH  Google Scholar 

  67. Sutradhar A, Paulino GH, Gray LJ (2008) Symmetric Galerkin boundary element method. Springer-Verlag, Berlin

    MATH  Google Scholar 

  68. Caglar H, Caglar N, Özer M (2008) Fifth-degree B-spline solution for nonlinear fourth-order problems with separated boundary conditions. J Phys Conf Ser 96:012031

    Article  Google Scholar 

  69. Wang XC (2004) Finite element method. Tsinghua University Press, Beijing

    Google Scholar 

  70. Desai YM, Eldho TI, Shah AH (2011) Finite element method with applications in engineering. Dorling Kindersley Pvt. Ltd, Noida

    Google Scholar 

  71. Wang XC, Shao M (1997) Basic principles and numerical methods of finite element method. Tsinghua University Press, Beijing

    Google Scholar 

  72. Chen Y, Lee J, Eskandarian A (2006) Meshless methods in solid mechanics. Springer Science & Business Media, Inc., New York

    MATH  Google Scholar 

  73. Gu ZQ, Kou GM (1997) Active vibration control. National Defense Industry Press, Beijing

    Google Scholar 

  74. Ja YH (1983) Using the shooting method to solve boundary-value problems involving nonlinear coupled-wave equations. Opt Quant Electron 15:529–538

    Article  Google Scholar 

  75. Bert CW, Malik M (1996) Differential quadrature method in computational mechanics: a review. Appl Mech Rev 49:1–28

    Article  Google Scholar 

  76. Ali AH, Al-Saif ASJ (2008) Adomian decomposition method for solving some models of nonlinear partial differential equations. Basrah J Sci A 26:1–11

    Google Scholar 

  77. Liao S (2012) Homotopy analysis method in nonlinear differential equations. Springer-Verlag, Berlin

    Book  MATH  Google Scholar 

  78. Meyer Y (1992) Wavelets and operators. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  79. Cui JT (1995) Introduction to wavelet analysis. Xi ‘an: Xi ‘an Jiaotong University Press

    Google Scholar 

  80. Daubechies I (1988) Orthonormal bases of compactly supported wavelets. Commun Pure Appl Math 41:909–996

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to You-He Zhou .

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). Introduction . 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_1

Download citation

  • DOI: https://doi.org/10.1007/978-981-33-6643-5_1

  • 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