Computation of the Laplace inverse transform by application of the wavelet theory

Abstract

SUMMARYAn e cient and robust method of solving Laplace inverse transform is proposed based on the wavelet theory. The inverse function is expressed as a wavelet expansion with rapid convergence. Several examples are provided to demonstrate the methodology. As an example of application, the proposed inversion method is applied to the dynamic analysis of a single-degree-of-freedom spring-mass-damper system whose damping is described by a stress-strain relation containing fractional derivatives. The results are compared with previous studies.

On the inversion of multicomponent NMR relaxation and diffusion decays in heterogeneous systems

“…[9], according to Eqs. [6] and [7], we have h͑T͒ ϭ P͑␣͒, [10] which means that the distribution of scaled relaxation times T is proportional to the distribution of pore sizes ␣. In this particular case, the relaxation time, and not the relaxation rate, is proportional to the physical heterogeneity parameter.…”

A simplified formula of Laplace inversion based on wavelet theory

“…For simplicity, we specify [1], it can be shown that = 7 in the case of N = 6. Using this approximation of , Equation (1) can be simpliÿed as…”

“…By applying wavelet theory, we have previously suggested an inversion formula of Laplace transform as [1] …”

“…Recently, we have proposed [1] a numerical method of Laplace inversion via wavelet expansion of functions in the transform domain. The e ciency and robustness of this method have been illustrated by both theoretical analysis and numerical examples.…”

“…The method of Laplace transform arises in many ÿelds of science and engineering [1][2][3][4][5][6][7][8][9]. One of the challenges for using this method is that analytical inversion is sometimes formidable or even impossible, in which case numerical techniques become necessary.…”