Classroom Note: Fourier Method for Laplace Transform Inversion

A method is described for inverting the Laplace transform. The performance of the Fourier method is illustrated by the inversion of the test functions available in the literature. Results are shown in the tables.


Introduction
During the past few decades, methods based on integral transforms, in particular, the Laplace transforms, are being increasingly employed in mathematics, physics, mechanics and other engineering sciences.Laplace transforms have a wide variety of applications in the solution of differential, integral and difference equations.To solve such equations by Laplace transform, one applies the Laplace transform to the equation, obtaining an equation for the transform of the required function.The latter equation is usually considerably simpler than the initial equation and its solution is often a function of quite simple structure.One must then derive the solution of the original equation from its Laplace transform, that is invert the Laplace transform.
In the terminology of ill-posed problems, the Laplace transform is a severely ill-posed problem.Unfortunately many problems of physical interest lead to Laplace transforms whose inverses are not readily expressed in terms of tabulated functions, although there exist extensive tables of transforms and their inverses.It is highly desirable, therefore, to have methods for appropriate numerical inversion.
The notion of ill-posedness is usually attributed to Hadamard [9].A modern treatment of the concept appears in Tikhonov and Arsenin [22].In an ill-posed inverse problem, a classical least squares, minimum distance or cross-validation solution may not be uniquely defined.Moreover the sensitivity of such solutions to slight perturbations in the data is often unacceptably large.
The basic principle common to all such methods is as follows: seek a solution that is consistent both with the observed data and prior notions about the physical behavior of the phenomenon under study.Different practical problems have led to unique strategies for implementation of this principle such as the method of regularization (Tikhonov and Arsenin [22]), (Varah [23]), maximum entropy (Jaynes [10], Mead [15]), quasi-reversibility (Lattes and Lions [12]) and cross-validation (Wahba [25]).
Regularization methods have also been discussed by (Varah [23], Essah and Delves [6]) and by (Bertero [3])); other methods are also available in the literature for the numerical inversion of Laplace transform which have been described by (Norden [16]) and (Salzer [19]).However no single method gives optimum results for all purposes and for all occasions.For a detailed bibliography, the reader is referred to (Piessens and Pissens and Branders, [17,18]).Several methods and a comparison is given by (Davis [4]) and (Talbot [21]).

Laplace Transform an Incorrectly Posed Problem
The problem of the recovery of a real function f (t), t ≥ 0, given its Laplace transform for real values of s, is an ill-posed problem and, therefore, affected by numerical instability.

McWhirter and Pike's Method for Laplace Transform Inversion
Under the assumptions that McWhirter and Pike [13,14] show that the solution f (t) of equation (1.1) may be represented in terms of a continuous eigen-expansion as follows: 1) where ψ ± ω (s) are the real and imaginary parts of and the eigenvalues λ ω are real: Here Γ(z) is the complex Gamma function (see, e.g.[1,5]).
In order to approximate where and The ill-posedness of the problem reflected by the very rapid decay of λ ωn with increasing n.Thus the inclusion of too many terms in the expansion (2.4) leads to large oscillations in f N (t), whereas too few terms do not give a sufficiently accurate solution.McWhirter and Pike [14] evaluate the coefficients a ± n in (2.5) by quadrature and determine N in (2.4) by trial and error.

Our Method
We are interested in finding where κ n is complex as defined earlier, ω n is real and a n are the complex coefficients to be determined.We use the notations as ∼ represents Mellin transform, ∧ denotes Fourier transform.Consider which is the Mellin transform of g(s), λ being complex.From (3.1) and (3.2) we obtain which is a well-known relationship between MTs and FTs, obtained by substituting for s = e −t in equation (3.2).
where the coefficients c m are given to 7 decimal places in Table 1.Thus and Having written a n in the form (3.7) it is sometimes possible, when g(t) is given analytically, to evaluate a n exactly from tables of Fourier transforms (see [1,5]).This has the advantage of removing quadrature errors from the coefficients in the expansion (2.4) which are amplified by small eigenvalues.

Numerical Examples
Example 1. McWhirter and Pike [14] We have For reasons of comparison with McWhirter and Pike [14] we choose H = 0.136 and we tabulate the error in the numerical solution (2.4) versus N in Table 2.The optimal N is clearly 24.
we have For H = 0.136 and N = 20, the numerical solution obtained is the best giving the least error norm and is exceedingly better than Varah's solution.

Conclusion
Our method worked very well over both the test problems and the results obtained are shown in Tables 2 and 3.The method is easy to understand as compared with other more technical methods and yields equally good results.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning (2.1) numerically, McWhirter and Pike replace the semi-infinite interval [0, ∞) by the finite interval [L 1 , L 2 ], where 0 < L 1 << 1 and ∞ > L 2 >> 1.By introducing a spacing H = 2π T , where T = log L 2 − log L 1 , and a discrete spectrum ω n = nH, they replace the integral (2.1) by the finite sum

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation