Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 2009, Article ID 898195, 38 pages doi:10.1155/2009/898195 Research Article Inversion of the Laplace Transform from the Real Axis Using an Adaptive Iterative Method Sapto W. Indratno and Alexander G. Ramm Department of Mathematics, Kansas State University, Manhattan, KS 66506-2602, USA Correspondence should be addressed to Alexander G. Ramm, ramm@math.ksu.edu Received 24 July 2009; Accepted 19 October 2009 Recommended by Thomas Witelski A new method for inverting the Laplace transform from the real axis is formulated. This method is based on a quadrature formula. We assume that the unknown function f t is continuous with known compact support. An adaptive iterative method and an adaptive stopping rule, which yield the convergence of the approximate solution to f t, are proposed in this paper. Copyright q 2009 S. W. Indratno and A. G. Ramm. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Consider the Laplace transform Lf ( p ) :  ∞ 0 e -pt f tdt  F ( p ) , Re p> 0, 1.1 where L : X 0,b → L 2 0, ∞, X 0,b :  f ∈ L 2 0, ∞ | supp f ⊂ 0,b  , b> 0. 1.2 We assume in 1.2 that f has compact support. This is not a restriction practically. Indeed, if lim t →∞ f t 0, then |f t| <δ for t>t δ , where δ> 0 is an arbitrary small number. Therefore, one may assume that suppf ⊂ 0,t δ , and treat the values of f for t>t δ as noise. One may also note that if f ∈ L 1 0, ∞, then F ( p ) :  ∞ 0 f te -pt dt   b 0 f te -pt dt   ∞ b f te -pt dt : F 1 ( p )  F 2 ( p ) , 1.3