A technique for increasing the accuracy of the FFT-based method of numerical inversion of Laplace transforms

Hwang, Chyi‐Ching; Wu, Rong-Yuang; Lu, Ming-Jeng

doi:10.1016/0898-1221(94)90146-5

Cited by 6 publications

(3 citation statements)

References 4 publications

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“…For example, Cunha and Viloche [5] have presented an iterative method for the Numerical inversion of Laplace transform, and Dong [6] has introduced a regularization method for this purpose. In [4], Crump has used Fourier series approximation (see also [8] in this regard) while Miller and Guy in [15] have used Jacobi polynomials, and Sidi [23] has applied a window function for Laplace transform inversion. Finally, Piessens' works [19,20] are a good bibliography that one can refer to it for the Laplace transform numerical inversion.…”

Section: Evaluation Of Inverse Laplace Transform Using Rational Classmentioning

confidence: 99%

On rational classical orthogonal polynomials and their application for explicit computation of inverse Laplace transforms

Masjed‐Jamei

Dehghan

2005

Mathematical Problems in Engineering

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From the main equation(ax2+bx+c)y″n(x)+(dx+e)y′n(x)−n((n−1)a+d)yn(x)=0,n∈ℤ+, six finite and infinite classes of orthogonal polynomials can be extracted. In this work, first we have a survey on these classes, particularly on finite classes, and their corresponding rational orthogonal polynomials, which are generated by Mobius transformx=pz−1+q,p≠0,q∈ℝ. Some new integral relations are also given in this section for the Jacobi, Laguerre, and Bessel orthogonal polynomials. Then we show that the rational orthogonal polynomials can be a very suitable tool to compute the inverse Laplace transform directly, with no additional calculation for finding their roots. In this way, by applying infinite and finite rational classical orthogonal polynomials, we give three basic expansions of six ones as a sample for computation of inverse Laplace transform.

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Section: Evaluation Of Inverse Laplace Transform Using Rational Classmentioning

confidence: 99%

On rational classical orthogonal polynomials and their application for explicit computation of inverse Laplace transforms

Masjed‐Jamei

Dehghan

2005

Mathematical Problems in Engineering

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“…Recently, numerous different methods have been adopted for analytically and numerically inverting the Laplace transform, which is well suited to different types of functions. Lee and Sheen developed analytic continuation of Bromwich integral and determined the integral by using quadrature rule, Dubner and Abate expressed the function in Fourier cosine series, further it was improved by Durbin that the trapezoidal rule can be applied and the function can be expressed in trigonometric series, Iqbal studied the regularization method, Cuomo et al have described a numerical technique based on collocation method and Laguerre series exapansion of inverse function, Abate et al investigated the Laguerre method, Post‐Widder formula has been adopted in the work of Frolov and Kitaev, Massouros and Genin presented a Taylor series approach, Laguerre matrix polynomial series to the numerical inversion of Laplace transforms of matrix functions is used in the work of Sastre et al and other references therein . An extensive survey and comparison of methods were discussed in the work of Davis and Martin …”

Section: Introductionmentioning

confidence: 99%

Numerical inversion of Laplace transform based on Bernstein operational matrix

Rani

Mishra

Cattani

2018

Math Methods in App Sciences

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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. KEYWORDSLaplace transform, numerical inverse Laplace transform, operational matrices, orthonormal Bernstein polynomialswhere c is a positive real number and all the poles of F(s) lie at the left side of the line s = c. Sometimes, inverse Laplace transform becomes complicated when the analytic inverse cannot be found. Hence, to calculate the inverse Laplace transform, numerical techniques are adopted.Recently, numerous different methods have been adopted for analytically and numerically inverting the Laplace transform, which is well suited to different types of functions. Lee and Sheen 5 developed analytic continuation of Bromwich integral and determined the integral by using quadrature rule, Dubner and Abate 6 expressed the function in Fourier cosine series, further it was improved by Durbin 7 that the trapezoidal rule can be applied and the function can be expressed in trigonometric series, Iqbal 8 studied the regularization method, Cuomo et al 9 have described a numerical technique based Math Meth Appl Sci. 2018;41:9231-9243.wileyonlinelibrary.com/journal/mma

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“…For the finite series approximation, a system of orthogonal polynomials [5 -7] or splines [8] is often chosen as the expansion basis. As for the use of numerical Laplace inversion methods for time domain analysis of time-delay systems, the FFT-based algorithms [9] are computationally most efficient and the algorithm based on numerical integration of Bromwich's integral can guarantee solution accuracy [10].…”

Section: Introductionmentioning

confidence: 99%

Use of the Lambert W function for time-domain analysis of feedback fractional delay systems

Yan

Hwang

2006

IEE Proceedings - Control Theory and Applications

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The Lambert W function is defined as the multivalued inverse of the function w ! we w ¼ z. It has been applied to stability analysis of a class of fractional delay systems whose transcendental characteristic equation (TCE) can be remodelled in the form (as þ b)e cs þ d ¼ 0. The approach of using the Lambert W function to time-domain analysis of a class of feedback fractional-order time-delay systems is extended. It should be noted that, owing to the multivaluedness of a transfer function of fractional order, the approach has two pitfalls that must be circumvented with care. Because remodelling the TCE of a feedback fractional delay system to allow for the Lambert W function representation of roots introduces superfluous poles to the original TCE, a clarification of the relationship between the roots of the remodelled TCEs and the poles of the system is provided. As a result, the time response function of the system can be approximated by a finite series of eigenmodes written in terms of Lambert W functions. As the singularities of a fractional-order system include both the poles and the branch cut(s) of the transfer function, the neglect of the response portion contributed by the branch cut(s) incurs a significant transient response error. In order to compensate for such a transient response error, three schemes of optimal approximation with specified poles are developed. Simulation results show that the proposed approaches to time-domain analysis of feedback fractional delay systems can indeed enlarge the application scope of the emerging Lambert W function.

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A technique for increasing the accuracy of the FFT-based method of numerical inversion of Laplace transforms

Cited by 6 publications

References 4 publications

On rational classical orthogonal polynomials and their application for explicit computation of inverse Laplace transforms

On rational classical orthogonal polynomials and their application for explicit computation of inverse Laplace transforms

Numerical inversion of Laplace transform based on Bernstein operational matrix

Use of the Lambert W function for time-domain analysis of feedback fractional delay systems

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