### Highlights from Gaver-Stehfest algorithm for inverse Laplace transform

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# Gaver-Stehfest algorithm for inverse Laplace transform

### Wahyu Srigutomo (view profile)

15 Feb 2006 (Updated )

Gaver-Stehfest algorithm for inverse Laplace transform

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%
% ilt=gavsteh(funname,t,L)
%
% funname The name of the function to be transformed.
% t The transform argument (usually a snapshot of time).
% ilt The value of the inverse transform
% L number of coefficient ---> depends on computer word length used
% (examples: L=8, 10, 12, 14, 16, so on..)
%
% Numerical Inverse Laplace Transform using Gaver-Stehfest method
%
%Refferences:
% 1. Villinger, H., 1985, Solving cylindrical geothermal problems using
% Gaver-Stehfest inverse Laplace transform, Geophysics, vol. 50 no. 10 p.
% 1581-1587
% 2. Stehfest, H., 1970, Algorithm 368: Numerical inversion of Laplace transform,
% Communication of the ACM, vol. 13 no. 1 p. 47-49
%
% Simple (and yet rush) examples included in functions fun1 and fun2 with
% their comparisons to the exact value (use testgs.m to run the examples)

MATLAB release MATLAB 6.0 (R12)

I am trying to solve the heat equation with both conduction and advective terms:

alpha*(d^2)T/(d^2)z-v*dT/dz-dT/dt (1)

where v=u*Cw/Cv, alpha is a constant. According to the literature the temperature frequency cariation solution of equation 1 is:

That=T0*exp(gamma*z)*exp(i*omega*t) (2)

where T0 and omega are constants. To get the actual temperature (T) value I am trying to use the Gaver-Stehfest numerical inversion. However, the values I get are not realistic, while my input parameters are fine. What am I doing wrong?

My input function is:

function f=fun_s(p)

%Define constants u=3.08*10^-8; %ms^-1 Cw=4.18*10^6; %Jm^-3K^-1 Cv=0.7*10^6; %Jm^-3K^-1 k=0.3; %Wm^-1K^-1 L=1; %m alpha=0.4*10^-6; %m^2 s^-1 omega=(2*pi())/(365.25*86400); theta0=11.5112; z=1;

v=u*Cw/Cv; gamma=(v-sqrt(v^2+4*i*omega*alpha))/(2*alpha);

f=theta0*(exp(gamma*z))*(exp(i*omega*p));

which I then run with the following to get a daily temperature value:

t=1:2*pi/365:2*pi; l=length(t);

for i=1:l T(i)=gavsteh('fun_s',t(i),18); end

Any help on this will be greatly appreciated!

Comment only
28 May 2010 Peng

### Peng (view profile)

this one totally helps a lot

Comment only
09 Jan 2010 Alireza Keramat

### Alireza Keramat (view profile)

Thank you. It was interesting because of its simplicity.

26 Aug 2009 Mark

### Mark (view profile)

Worked great for me. Nice job.

26 Feb 2009 Montaru Maxime

### Montaru Maxime (view profile)

You need to use multi-precision toolbox to enhance results. Very good results for one staircase simulation with a big L. Beware when you simulate multiple staircase.

20 Oct 2008 Binbin Qi

Comment only
18 Sep 2007 Andor Bariska

This is a solid implementation of the Stehfest Laplace Inversion algorithm. It computes the Stehfest-vector for arbitrary even L. Two examples are supplied, the inversion of 1/s^2 and 1/(s^2+1).

14 Aug 2007 Heinrich Schuchardt

The main advantage of the algorithm is that it only uses real numbers. Hence it can easily be ported to other programming languages.

The algorithm is accurate for overdamped and slightly underdamped systems. But it is not accurate for systems with prolonged oscillations.

16 May 2006 Quentin de Waziers

useles function

25 Apr 2006 Sylvain Chupin

This function is not able to fit a simple cosinus!