IngentaConnect Inversion of Noise-Free Laplace Transforms: Towards a Standardize...

Abstract:

The numerical inversion of Laplace transform arises in many applications of science and engineering whenever ordinary and partial differential equations or integral equations are solved. The increasing number of available numerical methods and computer codes has generated a need for well-documented sets of test problems. Using such sets, algorithm developers can evaluate the relative merits and drawbacks of their suggested new methods, and end-users can make judgments on the applicability of an individual method for a specific problem. Many areas in science and engineering, lead to problems that share three important properties: (i) the image function can be evaluated for real arguments, but not necessarily for complex ones; (ii) the original is known to be infinitely differentiable for times t > 0, (iii) the values of the image function can be obtained with any prescribed accuracy. The published test sets do not properly cover these applications, as many included problems are beyond of the specific class, while the remaining ones fail to address some of the potential difficulties arising in practice. The goal of this paper is to establish a common ground for problem classification, to list the requirements for the above class of problems, and to provide a carefully selected test set by addressing the deficiencies of the ones currently available. The findings of the paper are used to solve a problem of practical importance in modeling underground flow. Accompanying the paper, WEB links are provided to a list of more than 800 relevant publications (going back to 1795), to a Mathematica program to generate and solve the suggested set of test problems, and to a user friendly Java program to solve inversion problems for a restricted class of image functions.

Keywords: Laplace transform; Numerical inversion algorithm; Multi-precision calculation; Test problems

Document Type: Research article

Affiliations: 1: Harold Vance Department of Petroleum Engineering, Texas A&M University, 3116 TAMU, College Station, TX 77843-3116 2: Department of Biomedical Engineering, Boston University, 44 Cummington Street, Boston, MA 02215

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Inversion of Noise-Free Laplace Transforms: Towards a Standardized Set of Test Problems