A Regularization Method for Approximating the Inverse Laplace Transform

A new method for approximating the inerse Laplace transform is presented. We first change our Laplace transform equation into a convolution type integral equation, where Tikhonov regularization techniques and the Fourier transformation are easily applied. We finally obtain a regularized approximation to the inverse Laplace transform as finite sum

JACOBI POLYNOMIALS USED TO INVERT THE LAPLACE TRANSFORM

Given the Laplace transform F(s) of a function f(t), we develop a new algorithm to find on approximation to f(t) by the use of the dassical Jacobi polynomials. The main contribution of our work is the development of a new and very effective method to determine the coefficients in the finite series ex-pansion that approximation f(t) in terms of Jacobi polynomials. Some numerical examples are illustrated.

The New Inversion of Laplace Transform

The new inversion formula of the Laplace transform is considered. In the formula we use only the positive values ofx SiCoLT（x） = c S（x）, L（S（x）） = T（x）, c = const., x 〉 O,from the real axis. Si is the sinus transform, Co is the cosines transform of Fourier and L is the Laplace transform.

SOLVING COUPLED PSEUDO-PARABOLIC EQUATION USING A MODIFIED DOUBLE LAPLACE DECOMPOSITION METHOD

In this paper, the modification of double Laplace decomposition method is pro- posed for the analytical approximation solution of a coupled system of pseudo-parabolic equation with initial conditions. Some examples are given to support our presented method. In addition, we prove the convergence of double Laplace transform decomposition method applied to our problems.

Calculating the transient behavior of grounding systems using inverse Laplace transform

This paper deals with a unified and novel approach for analyzing the frequency and time domain performance of grounding systems.The proposed procedure is based on solving the full set of Maxwell's equations in the frequency domain,and enables the exact computation of very near fields at the surface of the grounding grid,as well as far fields,by simple and accurate closed-form expressions for solving Sommerfeld integrals.In addition,the soil ionization is easily considered in the proposed method.The frequency domain responses are converted to the time domain by fast inverse Laplace transform.The results are validated and have shown acceptable accuracy.

Data inversion of multi-dimensional magnetic resonance in porous media

Since its inception in the 1970s,multi-dimensional magnetic resonance(MR)has emerged as a powerful tool for non-invasive investigations of structures and molecular interactions.MR spectroscopy beyond one dimension allows the study of the correlation,exchange processes,and separation of overlapping spectral information.The multi-dimensional concept has been re-implemented over the last two decades to explore molecular motion and spin dynamics in porous media.Apart from Fourier transform,methods have been developed for processing the multi-dimensional time-domain data,identifying the fluid components,and estimating pore surface permeability via joint relaxation and diffusion spectra.Through the resolution of spectroscopic signals with spatial encoding gradients,multi-dimensional MR imaging has been widely used to investigate the microscopic environment of living tissues and distinguish diseases.Signals in each voxel are usually expressed as multi-exponential decay,representing microstructures or environments along multiple pore scales.The separation of contributions from different environments is a common ill-posed problem,which can be resolved numerically.Moreover,the inversion methods and experimental parameters determine the resolution of multi-dimensional spectra.This paper reviews the algorithms that have been proposed to process multidimensional MR datasets in different scenarios.Detailed information at the microscopic level,such as tissue components,fluid types and food structures in multi-disciplinary sciences,could be revealed through multi-dimensional MR.

Long-term settlement prediction of high-speed railway bridge pile foundation

The process and characteristics of loading on high-speed railway bridge pile foundation were firstly obtained by means of field research and analysis,and the corresponding loading function was presented.One-dimensional consolidation equation of elastic multilayered soils was then established with single drainage or double drainages under multilevel loading.Moreover,the formulas for calculating effective stress and settlement were derived from the Laplace numerical inversion transform.The three-dimensional composite analysis method of bridge pile group was improved,where the actual load conditions of pile foundation could be simulated,and the consolidation characteristics of soil layers beneath pile were also taken into account.Eventually,a corresponding program named LTPGS was developed to improve the calculation efficiency.The comparison between long-term settlement obtained from the proposed method and the in-situ measurements of pile foundation was illustrated,and a close agreement is obtained.The error between computed and measured results is less than 1 mm,and it gradually reduces with time.It is shown that the proposed method can effectively simulate the long-term settlement of pile foundation and program LTPGS can provide a reliable estimation.

A Numerical Algorithm for Arbitrary Real-Order Hankel Transform

The Hankel transform is widely used to solve various engineering and physics problems,such as the representation of electromagnetic field components in the medium,the representation of dynamic stress intensity factors,vibration of axisymmetric infinite membrane and displacement intensity factors which all involve this type of integration.However,traditional numerical integration algorithms cannot be used due to the high oscillation characteristics of the Bessel function,so it is particularly important to propose a high precision and efficient numerical algorithm for calculating the integral of high oscillation.In this paper,the improved Gaver-Stehfest(G-S)inverse Laplace transform method for arbitrary real-order Bessel function integration is presented by using the asymptotic characteristics of the Bessel function and the accumulation of integration,and the optimized G-S coefficients are given.The effectiveness of the algorithm is verified by numerical examples.Compared with the linear transformation accelerated convergence algorithm,it shows that the G-S inverse Laplace transform method is suitable for arbitrary real order Hankel transform,and the time consumption is relatively stable and short,which provides a reliable calculation method for the study of electromagnetic mechanics,wave propagation,and fracture dynamics.

Analytical solutions of citrate-phosphate coupled model of rice(Oryza sativa L.)roots

The citrate secreted by the rice(Oryza sativa L.)roots will promote the absorption of phosphate,and this process is described by the Kirk model.In our work,the Kirk model is divided into citrate sub-model and phosphate sub-model.In the citrate sub-model,we obtain the analytical solution of citrate with the Laplace transform,inverse Laplace transform and convolution theorem.The citrate solution is substituted into the phosphate sub-model,and the analytical solution of phosphate is obtained by the separation variable method.The existence of the solutions can be proved by the comparison test,the Weierstrass M-test and the Abel discriminating method.

Efficient Numerical Valuation of Continuous Installment Options

In this work we investigate the novel Kryzhnyi method for the numerical inverse Laplace transformation and apply it to the pricing problem of continuous installment options.We compare the results with the one obtained using other classical methods for the inverse Laplace transformation,like the Euler summation method or the Gaver-Stehfest method.