A bibliography on numerical inversion of the Laplace transform and applications (original) (raw)
Related papers
Mathematics in Engineering, Science and Aerospace, 2014
In real situations, sometimes it is difficult or rather impossible to find Laplace transform inversion in classical way. Such situations are tackled by numerical evaluation of inverse Laplace transform. The numerical techniques for finding inverse of Laplace transforms were introduced in the sixties by Bellman et al. Since then enormous progress has taken place. This paper mainly discusses series methods for numerically inverting Laplace transforms such as by (1) Euler, Post-Widder and Crump, (2) Fast Fourier transform and (3) Laguerre Legendre and Chebyshev polynomials. Historical development and instances of certain engineering applications have been taken into consideration.
On the numerical inversion of Laplace transforms
ACM Transactions on Mathematical Software, 1993
Three frequently used methods for numerically inverting Laplace transforms are tested on complicated transforms taken from the literature. The first method is a straightforward application of the trapezoidal rule to Bromwich's integral. The second method, developed by Weeks [22], integrates Bromwich's integral by using Laguerre polynomials. The third method, devised by Talbot [18], deforms Bromwich's contour so that the integrand of Bromwich's integral is small at the beginning and end of the contour. These methods are also applied to joint Laplace-Fourier transform problems. All three methods give satisfactory results; Talbot's, however, has an accurate method for choosing required parameters.
Numerical inversion of the Laplace transform
2005
We give a short account on the methods for numerical inversion of the Laplace transform and also propose a new method. Our method is inspired and motivated from a problem of the evaluation of the Müntz polynomials (see [1]), as well as the construction of the generalized Gaussian quadrature rules for the Müntz systems (see [2]). As an illustration of our method we consider an example with 100 real poles distributed uniformly on (−1/2, 100). A numerical investigation shows the efficiency of the proposed method.
An accurate numerical inversionof Laplace transforms based on the location of their poles
Computers & Mathematics with Applications, 2004
we introduce an efficient and easily implemented numerical method for the inversion of Laplace transforms, using the analytic continuation of integrands of Bromwich's integrals. After deforming the Bromwich's contours so that it consists of the union of small circles around singular points, we evaluate the Bromwich's integrals by quadrature rules. We prove that the error bound of our method has spectral accuracy of type ~N -t-eps/z P, 0 < e < 1 and provide several numerical examples. O
On the Laguerre Method for Numerically Inverting Laplace Transforms
Informs Journal on Computing, 1996
The Laguerre method for numerically inverting Laplace transforms is an old established method based on the 1935 Tricomi-Widder theorem, which shows (under suitable regularity conditions) that the desired function can be represented as a weighted sum of Laguerre functions, where the weights are coefficients of a generating function constructed from the Laplace transform using a bilinear transformation. We present a new variant of the Laguerre method based on: (1) using our previously developed variant of the Fourier-series method to calculate the coefficients of the Laguerre generating function, (2) developing systematic methods for scaling, and (3) using Wynn's ǫ-algorithm to accelerate convergence of the Laguerre series when the Laguerre coefficients do not converge to zero geometrically fast. These contributions significantly expand the class of transforms that can be effectively inverted by the Laguerre method. We provide insight into the slow convergence of the Laguerre coefficients as well as propose a remedy. Before acceleration, the rate of convergence can often be determined from the Laplace transform by applying Darboux's theorem. Even when the Laguerre coefficients converge to zero geometrically fast, it can be difficult to calculate the desired functions for large arguments because of roundoff errors. We solve this problem by calculating very small Laguerre coefficients with low relative error through appropriate scaling. We also develop another acceleration technique for the case in which the Laguerre coefficients converge to zero geometrically fast. We illustrate the effectiveness of our algorithm through numerical examples.
An analysis of bilinear transform polynomial methods of inversion of Laplace transforms
Numerische Mathematik, 1995
Methods for the numerical inversion of a Laplace transform F(s) which use a special bilinear transformation of s are particularly e ective in many cases and are widely used. The main purpose of this paper is to analyze the convergence and conditioning properties of a special class of such methods, characterized by the use of Lagrange interpolation. The results derived apply both to complex and real inversion, and show that some known inversion methods are in fact in this class.
Numerical inversion of Laplace transform based on Bernstein operational matrix
Mathematical Methods in the Applied Sciences, Wiley & Sons, 2018
This paper provides a technique to investigate the inverse Laplace transform by using orthonormal Bernstein operational matrix of integration. The proposed method is based on replacing the unknown function through a truncated series of Bernstein basis polynomials and the coefficients of the expansion are obtained using the operational matrix of integration. This is an alternative procedure to find the inversion of Laplace transform with few terms of Bernstein polynomials. Numerical tests on various functions have been performed to check the applicability and efficiency of the technique. The root mean square error between exact and numerical results is computed, which shows that the method produces the satisfactory results. A rough upper bound for errors is also estimated.
On an approximate method of laplace inversion
USSR Computational Mathematics and Mathematical Physics, 1972
A=1 where F @) is the Laplace transform of the function f(t), are described. Estimates for the error of the approximation in a finite interval are obtained in the class of integral functions.