Abstract
This paper is concerned with the real invserion of a Laplace tranform function F(s) equals L[f(t)]. The real inversion problem is that of reconstructing f(t) from known values of F(s), given only at real points. Numerical inversion of a Laplace transform is a very difficult problem to be solved in general, as shown in the survey written by Davies and Martin in 1979. Actually, this is still true. It is well known that the numerical solution of the real inversion problem is much more difficult than that of the complex one. Briefly, this is due to an intrinsic ill-posedness of the real inversion problem in the sense that small changes in data can cause arbitrary large changes in the solutionl. This is reflected in ill-conditioning of the discrete model. We propose a numerical method for the real inversion problem based on a Fourier series expansion of f(t). Introducing some kind of regularization we show how it is possible to approximate the Fourier coefficients of f(t), and then to compute the inverse Laplace function, only using the knowledge of the restriction of F(s) on the real axes.
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Luisa D'Amore, Luisa D'Amore, Almerico Murli, Almerico Murli, } "Real inversion of a Laplace transform", Proc. SPIE 2570, Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, (9 October 1995); doi: 10.1117/12.224163; https://doi.org/10.1117/12.224163
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