Geometric interpretation of the Euler-Knopp summation method in the problem of inverting the Laplace transform

In this paper, we consider a method for inverting the Laplace transform F(s) = \(\int\limits_0^\infty {e^{ - st} f(t)dt} \), which consists in representing the original function by the Laguerre series

$f(t) = \sum\limits_{k = 0}^\infty {a_k L_k (bt).} $

(1)

First, we perform a conformal mapping of the plane (s), which depends on parameter ξ. The value of the parameter is determined by the location of the singular points of the given representation. Under this mapping, series (1) takes the form

$f(t) = \frac{{b - \xi }}{b}\exp (\xi t)\sum\limits_{k = 0}^\infty {c_k L_k ((b - \xi )t).} $

It is demonstrated that such inverting scheme is equivalent to applying the Picone-Tricomi method with further acceleration of the rate of convergence of series (1) using the Euler-Knopp nonlinear procedure

$\sum\limits_{k = 0}^\infty {a_k z^k } = \sum\limits_{k = 0}^\infty {A_k (p)\frac{{z^k }}{{(1 - pz)^{k + 1} }},} A_k (p) = \sum\limits_{j = 0}^k {\left( \begin{gathered} k \hfill \\ j \hfill \\ \end{gathered} \right)( - p)^{k - j} a_j } .$

Under this approach, the original function is represented by the series

$f(t) = \exp \left( {\frac{{bpt}}{{p - 1}}} \right)\sum\limits_{k = 0}^\infty {\frac{{A_k (p)}}{{(1 - p)^{k + 1} }}L_k } \left( {\frac{{bpt}}{{1 - p}}} \right),$

where parameters ξ and p are related by the formula p = x/(ξ ? b). Unlike many other methods for summation of series, in the scheme suggested, there is no need to investigate the regularity conditions.

     

Acceleration of the convergence of the Laguerre series in the problem of inverting the Laplace transform

When the Laplace transform is inverted numerically, the original function is sought in the form of a series in the Laguerre polynomials. To accelerate the convergence of this series, the Euler-Knopp method is used. The techniques for selecting the optimal value of the parameter of the transform on the real axis and in the complex plane are proposed.     

Inversion of the multi-dimensional Laplace transform-expansion by Laguerre series

In this paper an algorithm to numerically invert two-dimensional Laplace transform known in closed form as an analytic function is presented. The method is based on expanding the inverse function in a series of products of (generalized) Laguerre polynomials. It is based on the method by Weeks (1966) and the generalized version presented by Piessens and Branders (1971) for the one-dimensional case.     

The numerical performance of Tricomi's formula for inverting the Laplace transform

Summary A method for inverting the Laplace transform which uses an expansion into Laguerre polynomials is considered. By means of a recently established generalization of the Euler-Knopp transformation the rate of convergence of the series of Laguerre polynomials is accelerated. For computing the transformed series a recursive algorithm is given. Results of theoretical and practical nature make the usefulness of the new transformation evident.     

Solution of Coulomb problem by Laplace transform

The bound state energies and scattering phase shifts for the Coulomb potential are obtained from both the Schrodinger and Dirac equations by taking a Laplace transform. Inversion of transforms is not required, and the nonrelativistic eigenvalue problem is solved without even obtaining the transforms explicitly. The nonrelativistic scattering amplitude appears after solving a first-order differential equation.     

Laguerre polynomials and the inverse Laplace transform using discrete data

We consider the problem of finding a function defined on (0,∞) from a countable set of values of its Laplace transform. The problem is severely ill-posed. We shall use the expansion of the function in a series of Laguerre polynomials to convert the problem in an analytic interpolation problem. Then, using the coefficients of Lagrange polynomials we shall construct a stable approximation solution. Error estimate is given. Numerical results are produced.     

Error analysis of a Collocation method for numerically inverting a Laplace transform in case of real samples

In [S. Cuomo, L. D’Amore, A. Murli, M.R. Rizzardi, Computation of the inverse Laplace transform based on a collocation method which uses only real values, J. Comput. Appl. Math., 198 (1) (2007) 98–115] the authors proposed a Collocation method (C-method) for real inversion of Laplace transforms (Lt), based on the truncated Laguerre expansion of the inverse function:

where σ, b are parameters and c_k, kN, are the MacLaurin coefficients of a function depending on the Lt. The computational kernel of a C-method is the solution of a Vandermonde linear system, where the right hand side is obtained evaluating the Lt on the real axis. The Bjorck Pereira algorithm has been used for solving the Vandermonde linear system, providing a computable componentwise error bound on the solution.

For an inversion problem on discrete data F is known on a pre-assigned set of points (we refer to these points as samples of F) only and the major challenge is to deal with a significative loss of information. A natural approach to overcome this intrinsic difficulty is to construct a suitable fitting model that approximates the given data. In this case, we show that such approach leads to a C-method with perturbed right hand side, and then we use again the Bjorck Pereira algorithm.

Starting from the error introduced by the fitting model, we study its propagation in order to determine the maximum attainable accuracy on f_N. Moreover we derive a computable error bound that allows to get the suitable value of the parameter N that gives the maximum attainable accuracy.     

The Lebesgue constants for Borel summation of Laplace series

The unboundedness of the set of Lebesgue constants (norms) arising from the application of a summation method to an appropriate series implies the existence of a continuous function whose associated series fails to be summable by that method. Here it is established that the Lebesgue constantsL (t, B) arising from thet-th Borel exponential mean of Laplace series equal 2^{3/4 -3/2} (1/4)t ^1/4 +O(1), ast , showing that this phenomenon (known as the Du Bois-Reymond singularity) occurs for Borel summation of Laplace series.     

On analytic continuation of a hypergeometric series using the Euler-Knopp transformation

The Euler-Knopp transformation is considered in terms of the problems of regularity and acceleration of the rate of convergence. The object of study is the hypergeometric series

On absolute summation of Fourier series by subsequences

В работе изучается сл едующая задача. Пусть заданы числа 0<α≦1 и β<α. При каки х условиях на строго во зрастающую последов ательность натуральных чисел {n _k } _k ^t8 =1 для всех 2π-периодических функ ций \(f(x) \sim \sum\limits_{v = - \infty }^\infty {c_v e^{ivx} } \) , принадлежащих к лассу Lip α, равномерно пох будет выполнено неравенство $$\sum\limits_{k = 1}^\infty {|\sum\limits_{n_k \leqq |v|< n_{k + 1} } {c_v e^{ivx} } |n_k^\beta< \infty ?} $$ .     

On the inversion of the n-dimensional Laplace transform

In this paper we study the inversion of the multidimensionalLaplace transform by a combination of a general partial-fractionexpansion formula and the theory of residues. The ideas maybe applied to nonlinear systems defined by Volterra series.     

A method of summation of the Fourier-Jacobi series

A summation method is constructed for the Fourier-Jacobi series, which has properties similar to the properties of the de la Vallée-Poussin methods of summation of the Fourier series by the trigonometric system.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 5, pp. 676–680, May, 1993.     

Laguerre series direct method for variational problems

A direct method for solving variational problems via Laguerre series is presented. First, an operational matrix for the integration of Laguerre polynomials is introduced. The variational problems are reduced to the solution of algebraic equations. An illustrative example is given.