學術上求拉氏反轉換(Inverse Laplace Transform)的數值方法很多，Gaver-Stehfest方法是較常用於力學問題分析的一種方法，原因在於此方法在執行上較為方便，而在應用拉氏方法及Gaver-Stehfest公式計算振動問題的常微分方程時，往往只在一段時間區間內得到滿意的結果，在長時間區間上則誤差很大。為了克服此問題可用時間軸平移的方式來分段計算，就可得到不錯的結果，本文則是把此改良的方法，具體地應用於結構動力學的問題上，考察其表現。

In academic world, there are many numerical methods to compute the Inverse Laplace Transform. Gaver-Stehfest formula is the most popular method to apply in dynamics theory analysis since it can perform conveniently. It usually obtains satisfied result in certain time while applying Laplace Transform and Gaver-Stehfest formula to calculate ODES of vibrations. While simulating the ODES, we can use some theories about Laplace transform and extend the effective region of the Gaver-Stehfest formula by some technique of local extension to reduce the error. This research is mainly focusing on applying the modified methods on Dynamics Of Structures to evaluate the result.

[1] Mario Paz and William E. Leigh, “Structural Dynamics:theory and computation.” New York : Van Nostrand Reinhold, c1980
[2] Madhujit Mukhopadhyay , “Vibrations, dynamics and Structural systems.” Rotterdam: Brookfield, VT: A. A. Balkema, 2000
[3] Anil K. Chopra, “Dynamics of Structures:Theory and applications to earthquake engineering.” Englewood Cliffs, N.J : Prentice Hall, c1995
[4] Erwin Kreyszig, “Advanced engineering mathematics.” Hoboken, NJ : John Wiley, c2006
[5] Dennis G. Zill,Michael R.Cullen, “Advanced engineering mathematics.” Sudbury, Mass : Jones and Bartlett, c2000
[6] Peter V. O’Neil, “Advanced engineering mathematics.” Boston : PWS-Kent Pub. Co., c1995
[7] H. Rutishauser, “Lectures on Numerical Mathmatics.” Translated by W. Gautschi, Birkhauser, Boston, 1990.
[8] M. J. Maron and R. J. Lopez, “Numerical Analysis: A Practical Approach.” Wadsworth Publishing Company, Belmont, California, 1990.
[9] J. Penny and G. Lindfield, “Numerical Methods Using Matlab.” Ellis Horwood, New York, 1995
[10] R. A. Schapery, ” approximate methods of transform inversion for viscoelast -ic stress analysis” Proc.4th.U.S. Nat.Congress.
[11] E. Detournay and A. H-D. Cheng, “poroelastic response of borehole in a non-Hydrostatic stress field,” Int. J. Rock. Meth. Min. Sci&Geomech. Abstr, 25 (1988) 171-182.
[12] H.S. Chohan, R.S. Sandhu and W.E. Wolfe, ”A semi-discrete procedure for dynamic response analysis of saturated soils.” Int. J. Numer. Analyt. Meth. Geomech., 15 (1991) 471-496.
[13] J.R. Booker. And J.C. Small, ＂A method of computing the consolidation behavior of layered soils using direct numerical inversion of laplace transform,” Int. J. Numer. Analyt. Meth. Geomech., 11 (1987) 363-380
[14] S.L. Chen, L.M. Zhang and L.Z Chen,“ Consolidation of a finite transverse -ly isdropic soil layer on a rough impervious base.” Journal of Engrg. Mech. ASCE, 131 (2005) 1279-1290.
[15] R.K.N.O. Rajapakse and T. Senjuntichai,” An direct boundary integral equation method for poroelasticity.＂Int. J. Numer. Analyt. Meth. Geomech., 19 (1995) 587-614
[16] D. P. Gaver, Jr., “Observing stochastic processes, and approximate transform inversion.” Operational Res., 14 (1966) 444-459
[17] H. Stehfest, Comm. Acm., 13 (1970) 47..
[18] E. L. Post, “Generalized differentiation.” Trans. Amer. Meth. Soc., 32 (1960) 723-781
[19] D. W. Widder, “The inversion of the Laplace integral and the related moment problem.” Amer. Meth. Soc. Trans 36 (1934) 107-200
[20] D. W. Widder, The Laplace Transform. Princeton University Press, Princetion, NJ. (1946)
[21] S. Sykore, V. Bortollotti and P. Fautazzini, “PERFIDI: parametrically enabled relaxation filters winth double and multiple inversion.” Magnetic Resonance Imaging 25 (2007) 529-532
[22] C. Montella, “ LSV modeling of electrochemical systems through numerical inversion of inversion of Laplace transform. I-The GS-LSV algorithm.” J. Electroamalytical chemistry, 614 (2008) 121-130.
[23] B.davies and B Martin, “Numerical inversion of the Laplace transform: a survey and comparision of methods.” J. Computational Plrys., 33 (1979) 1-32
[24] 陳正豪，「對Gaver-Stehfest公式之研究與探討」，碩士論文，國立中央大學