A WEB page maintained by Peter Valkó

A WEB page maintained by:

Peter Valkó

[official WEB page]

[personal WEB page]

[email]

Last edited:

March 30, 2007

Numerical Inversion of Laplace Transform

A challenge for developers of numerical methods

"Shall I refuse my dinner because I do not fully understand ... digestion?"

Heaviside

The numerical inversion of Laplace transform arises in many applications of science and engineering whenever ordinary and partial differential equations or integral equations are to be solved. The basic idea of any integral transform method is that in the image domain equations are usually simpler than the original ones. Therefore, it is relatively easy to obtain the image of the solution. Once the image is known, one has to invert it to go back to the original domain, that is to obtain the solution of the original equation.

In the case of Laplace transform, for this purpose one can use detailed Tables of Laplace transform pairs and some tricks known as the basic properties of Laplace transform. A more recent alternative is to use Computer Algebra Systems (CAS) - to do the table lookup (and the tricks.)

Unfortunately, it is not always easy to find the inverse (that is the original of the image) using the tables or CAS. One possible reason is that the inverse is not a named function or can not be represented by any simple “formula”. In general, one would need a numerical algorithm for the inversion problem.

Many such algorithms have been suggested and this WEB page is devoted to them.

Go to the page: Sets of Test Problems

where you can find supporting material to the paper (published in the journal Inverse Problems in Engineering) written together with S. Vajda of Boston University

If you have already installed Java Web Start, just click

BigNumber-Stehfest Java Application

Go to The List [PDF] of more than 1500 publications maintained together with Vladimír Vojta

Valkó, P.P. and Abate, J. : Comparison of Sequence Accelerators for the Gaver Method of Numerical Laplace Transform Inversion, Computers and Mathematics with Application, Vol. 48 (Iss.3-40) 2004 pp. 629-636. [PDF]

Mathematica source code of the GWR algorithm: http://library.wolfram.com/infocenter/MathSource/4738/

Abate, J. and Valkó, P.P.: Multi-precision Laplace transform inversion, International Journal for Numerical Methods in Engineering, Vol. 60 (Iss. 5-7) 2004 pp 979–993. [PDF]

Mathematica source code of the FT algorithm:

Numerical Inversion of Laplace Transform with Multiple Precision Using the Complex Domain

http://library.wolfram.com/infocenter/MathSource/5026/

What is the BigNumber-Stehfest algorithm?

It is a modification of the original (Gaver-Stehfest) algorithm. The number of terms in the Stehfest summation (N) is successively doubled

The precision of the internal arithmetic calculations and the F(z) evaluations is increased simultaneously with the number of terms in the algorithm (for N terms N decimal digit precision is required).

A simple convergence criterion is applied: the digits being the same in the approximants calculated with the number of terms N and 2N are considered already "final" e.g. significant.

A “real-form” of the Laplace transform is used.

The Stehfest coefficients (being rational fractions) are stored in an integer representation, resulting in the required precision. (Currently the coefficients are available for maximum 4096 terms.)

What is Java Web Start?

With Java Web Start technology, you launch applications simply by clicking on a Web page link. If the application is not present on your computer, Java Web Start automatically downloads all necessary files.

A Java program written according to the rules of Java Web Start (JNLP protocol) should run on virtually any computer (Windows, or Unix or whatever.) Of course (?) Java Web Start itself has to be installed as a plug-in to your WEB browser. If it has not been installed on your computer yet, please go to the Java WEB-Start site of SUN Microsystems (TM) and download it first. (It is free!) You will find versions for various platforms.

What is the Gaver-Stehfest algorithm?

It is a relatively simple numerical method to invert the Lapalace transform. If you want to read the original paper, you will find its bibliographic info in The List. (You may easily search for Stehfest.) If you want to find all the publications dealing with the algorithm, you may want to check out letter “S” in the third column.

What is the Gaver-Wynn Rho algorithm?

The well-known Stehfest algorithm uses a certain linear convergence acceleration scheme to a series (originally introduced by Gaver) that approximates the inverse.

The Gaver Wynn Rho algorithm uses another (nonlinear) acceleration scheme to the same series. The acceleration scheme itself was first suggested by Wynn. Its use in numerical Laplace transform inversion is suggested in the paper above.

In what language is the BigNumber-Stehfest algorithm available?

The professional version is written in Mathematica.

A light-weight version is available in Java.

If you have Mathematica, you might want to download some programs from the

Sets of Test Problems

page.

What is Mathematica?

(That is the funniest question I have ever heard.)

Some links of interest:

Luisa D’Amore

Whitt

Abate and Whitt