01拉普拉斯变换定义

1.变换公式

2. 常见函数的Laplace变换

02 Laplace数值逆变换

根据拉普拉斯逆变换的公式，可以看到，f(t)可以变成如下的公式。

#!/usr/local/bin/python
# -*- coding: gbk -*-
#============================================================
# TEST1.PY                     -- by Dr. ZhuoQing 2020-07-10
#
# Note:
#============================================================from headm import *#################################################################
#   Function InvLap(t,omega,sigma,nint), numerically inverts a  #
#   Laplace transform F(s) into f(t) using the Fast Fourier     #
#   Transform (FFT) algorithm for a specific time "t", an       #
#   upper frequency limit "omega", a real parameter "sigma"     #
#   and the number of integration intervals "nint" .            #
#                                                               #
#   Function F(s) is defined in separate as Fs(s) (see code     #
#   below). Fs(s) has to be changed accordingly everytime the   #
#   user wants to invert a different function.                  #
#                                                               #
#   I suggest to use omega>100 and nint=50*omega. The higher    #
#   the values for omega, the more accurate the results will be #
#   in general, but at the expense of longer processing times.  #
#                                                               #
#   Sigma is a real number which must be a little bigger than   #
#   the real part of rightmost pole of the function F(s). For   #
#   example, F(s) = 1/s + 1/(s-2) + 1/(s+1) has poles for s=0,  #
#   s=2 and s=-1. Hence, sigma must be made equal to, say,      #
#   2.05 so as to keep all poles at the left of this value.     #
#   The analytical inverse for this simple function is          #
#   f(t) = 1 + exp(-t) + exp(2t). For t=1.25, omega=200,        #
#   nint=10000 and sigma=2.05, the numerical inversion yields   #
#   f(1.25) ~= 13.456844516, or -0.09% away from the actual     #
#   analytical result, 13.468998757 (results truncated to 9     #
#   decimal places). This is the example used in this code.     #
#                                                               #
#   Creator: Fausto Arinos de Almeida Barbuto (Calgary, Canada) #
#   Date: May 18, 2002                                          #
#   E-mail: fausto_barbuto@yahoo.ca                             #
#                                                               #
#   Reference:                                                  #
#   Huddleston, T. and Byrne, P: "Numerical Inversion of        #
#   Laplace Transforms", University of South Alabama, April     #
#   1999 (found at http://www.eng.usouthal.edu/huddleston/      #
#   SoftwareSupport/Download/Inversion99.doc)                   #
#                                                               #
#   Usage: invoke InvLap(t,omega,sigma,nint), for t>0.          #
#                                                               #
#################################################################
#   We need cmath because F(s) is a function operating on the
#   complex argument s = a + bj
from math import ceil
from cmath import *#   *** Driver InvLap function  ***
def InvLap(t,omega,sigma,nint):
#   Sanity check on some parameters.omega = ceil(omega)nint = ceil(nint)if omega <= 0:omega = 200if nint <= 0:nint = 10000return (trapezoid(t,omega,sigma,nint))#   *** Function trapezoid computes the numerical inversion. ***
def trapezoid(t,omega,sigma,nint):sum = 0.0delta = float(omega)/nintwi = 0.0#   The for-loop below computes the FFT Inversion Algorithm.
#   It is in fact the trapezoidal rule for numerical integration.for i in range(1,(nint+1)):witi = complex(0,wi*t)wf = wi + deltawfti = complex(0,wf*t)fi = (exp(witi)*Fs(complex(sigma,wi))).realff = (exp(wfti)*Fs(complex(sigma,wf))).realsum = sum + 0.5*(wf-wi)*(fi+ff)wi = wfreturn ((sum*exp(sigma*t)/pi).real)#   *** The Laplace function F(s) is defined here.  ***
def Fs(s):
#    return (1.0/s + 1.0/(s+1.0) + 1.0/(s-2.0))return 1.0/s#   Function InvLap(t,omega,sigma,nint) is invoked.
printf(InvLap(1.25, 200, 3.05, 10000))#------------------------------------------------------------
#        END OF FILE : TEST1.PY
#============================================================

03一些常见函数数值逆变换结果

f(t)=sin(t)

def Fs(s):return 1/(s*s+1)

^{▲ sin(t)数值逆变换结果}

f(t)=t

^{▲ 函数t的Laplace数值逆变换结果}
f(t)=exp(-t)

^{▲ exp(-t)函数数值逆变换结果}

※ 结论

本文测试了一个简易版本的数值Laplace逆变换的程序。

这个算法需要人工给出积分的上下限，积分限的实部等。如果参数给定不合适，计算出来的数值误差较大。

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逆Laplace数值逆变换

01拉普拉斯变换定义

1.变换公式

2. 常见函数的Laplace变换

02 Laplace数值逆变换

03一些常见函数数值逆变换结果

※ 结论

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