Skip to main content
SpringerLink
Log in

Wavelet-Based Laplace Transformation for Initial- and Boundary-Value 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))

  • 570 Accesses

Abstract

If the function \(f(t)\) is defined on \(t \ge 0\), and the integral \(\int_{0}^{ + \infty } {f(t)e^{ - st} } dt\) (s is a complex parameter) converges in a domain of s, such an integral is referred to as the Laplace transform of \(f(t)\), denoted by.

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

Access this chapter

Log in via an institution
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. Al-Shuaibi A (2001) Inversion of the Laplace transform via post-Widder formula. Integr Transform Spec Funct 11(3):225–232

    Article  MathSciNet  Google Scholar 

  2. Cohen AM (2007) Numerical methods for Laplace transform inversion. Springer, New York

    MATH  Google Scholar 

  3. Daubechies I (1988) Orthonormal bases of compactly supported wavelets. Commun Pure Appl Math 41(7):909–996

    Article  MathSciNet  Google Scholar 

  4. Daubechies I (1993) Orthonormal bases of compactly supported wavelets II: Variations on a theme. SIAM J Math Anal 24(2):499–519

    Article  MathSciNet  Google Scholar 

  5. Beylkin G, Coifman R, Rokhlin V (1991) Fast wavelet transforms and numerical algorithms. Commun Pure Appl Math 44(2):141–183

    Article  MathSciNet  Google Scholar 

  6. Rossikhin YA, Shitikova MV (1997) Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids. Appl Mech Rev 50(1):15–67

    Article  Google Scholar 

  7. Ingman D, Suzdalnitsky J (2001) Iteration method for equation of viscoelastic motion with fractional differential operator of damping. Comput Methods Appl Mech Eng 190(37–38):5027–5036

    Article  Google Scholar 

  8. Miller KS (1993) The Mittag-Leffler and Related Functions. Integral transforms and special functions. Integr Transform Spec Funct 1(1):41–49

    Google Scholar 

  9. Agrawal OP (2001) Stochastic analysis of dynamic system containing fractional derivatives. J Sound Vib 247(5):927–938

    Article  Google Scholar 

  10. Jones DIG (2001) Handbook of viscoelastic vibration damping. Wiley, Arizona

    Google Scholar 

  11. Enelund M, Mähler L, Runesson K, Josefson BL (1999) Formulation and integration of the standard linear viscoelastic solid with fractional order rate laws. Int J Solid Struct 36(16):2417–2442

    Article  Google Scholar 

  12. Oldham KB, Spanier J (1974) The Fractional calculus: integrations and differentiations of arbitrary order. Academic Press, New York

    MATH  Google Scholar 

  13. Samko SG, Kilbas AA, Marichev OI (1993) Fractional integrals and derivatives. Gordon and Breach Science Publisher, Yverdon, Yverdon-les-Bains, Switzerland

    MATH  Google Scholar 

  14. Suarez LE, Shokooh A (1997) An eigenvector expansion method for the solution of motion containing derivatives. J Appl Mech 64(3):629–635

    Article  MathSciNet  Google Scholar 

  15. Enelund M, Olsson P (1999) Damping described by fading memory-analysis and application to fractional derivative models. Int J Solids Struct 36(7):939–970

    Article  MathSciNet  Google Scholar 

  16. Caputo M, Mainardi F (1971) Linear models of dissipation in anelastic solids. La Rivista Del Nuovo Cimento 1(2):161–198

    Article  Google Scholar 

  17. Yuanlu LI (2010) Solving a nonlinear fractional differential equation using Chebyshev wavelets. Commun Nonlinear Sci Numer Simul 15(9):2284–2292

    Article  MathSciNet  Google Scholar 

  18. Wang JZ, Zhou YH, Gao HJ (2003) Computation of the Laplace inverse transform by application of the wavelet theory. Commun Numer Methods Eng 19:959–975

    Article  MathSciNet  Google Scholar 

  19. Liu XJ, Wang JZ, Wang XM, Zhou YH (2014) Exact solutions of multi-term fractional diffusion-wave equations with Robin type boundary conditions. Appl Math Mech 35(001):49–62

    Article  MathSciNet  Google Scholar 

  20. Ray SS, Chaudhuri KS, Bera RK (2006) Analytical approximate solution of nonlinear dynamic system containing fractional derivative by modified decomposition method. Appl Math Comput 182(1):544–552

    MathSciNet  MATH  Google Scholar 

  21. Atanackovic TM, Stankovic B (2008) On a numerical scheme for solving differential equations of fractional order. Mech Res Commun 35(7):429–438

    Article  MathSciNet  Google Scholar 

  22. Huang F, Guo B (2010) General solutions to a class of time fractional partial differential equations. Appl Math Mech (English Edition) 31(7):815–826

    Article  MathSciNet  Google Scholar 

  23. Mainardi F (1996) Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos Solitons Fractals 7(9):1461–1477

    Article  MathSciNet  Google Scholar 

  24. Caputo M (1967) Linear models of dissipation whose Q is almost frequency independent-II. Geophys J Int 13(5):529–539

    Article  Google Scholar 

  25. Nigmatullin RR (1986) The realization of the generalized transfer equation in a medium with fractal geometry. Physica Status Solidi (B). 133(1):425–430

    Article  Google Scholar 

  26. Nigmatullin RR (1984) To the theoretical explanation of the universal response. Physica Status Solidi (B). 123(2):739–745

    Article  Google Scholar 

  27. Daftardar-Gejji V, Bhalekar S (2008) Boundary value problems for multi-term fractional differential equations. J Math Anal Appl 345(2):754–765

    Article  MathSciNet  Google Scholar 

  28. Wang JZ (2011) Fractional stochastic description of hinge motions in single protein molecules. Chin Sci Bull 56:495–501

    Article  Google Scholar 

  29. Metzler R, Klafter J (2000) The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys Rep 339:1–77

    Article  MathSciNet  Google Scholar 

  30. Tarasov VE (2013) Review of some promising fractional physical models. Int J Mod Phys B 27(09):1330005

    Article  MathSciNet  Google Scholar 

  31. Welch SWJ, Ropper RAL, Duren RG (1999) Application of time-based fractional calculus methods to viscoelastic creep and stress relaxation of materials. Mech Time-Depend Mater 3:279–303

    Article  Google Scholar 

  32. Metzler R, Klafter J (2000) Boundary value problems for fractional diffusion equations. Physica A 278(1–2):107–125

    Article  MathSciNet  Google Scholar 

  33. Zhou YH, Wang XM, Wang JZ, Liu XJ (2011) A wavelet numerical method for solving nonlinear fractional vibration, diffusion and wave equations. Comput Model Eng Sci 7(2):137–160

    MathSciNet  MATH  Google Scholar 

  34. Zhou YH, Wang JZ, Zheng XJ (2001) A numerical inversion of the Laplace transform by use of the scaling function transform of wavelet theory. Acta Mathematica Scientia 21A(1):86–93 (in Chinese)

    MathSciNet  MATH  Google Scholar 

  35. Rashidinia J, Ghasemi M, Jalilian R (2010) Numerical solution of the nonlinear Klein-Gordon equation. J Comput Appl Math 233(8):1866–1878

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanics and Engineering Science, College of Civil Engineering and Mechanics, Lanzhou University, Lanzhou, Gansu, China

    Prof. You-He Zhou

Authors
  1. Prof. You-He Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

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). Wavelet-Based Laplace Transformation for Initial- and Boundary-Value 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_6

Download citation

  • DOI: https://doi.org/10.1007/978-981-33-6643-5_6

  • 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

Search

Navigation