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.     

Two regularization methods for a Cauchy problem for the Laplace equation

A Cauchy problem for the Laplace equation in a rectangle is considered. Cauchy data are given for y=0, and boundary data are for x=0 and x=π. The solution for 0<y?1 is sought. We propose two different regularization methods on the ill-posed problem based on separation of variables. Both methods are applied to formulate regularized solutions which are stably convergent to the exact one with explicit error estimates.     

A characterization of inverse Radon transform on the Laguerre hypergroup

Let K=[0,∞)×R be the Laguerre hypergroup which is the fundamental manifold of the radial function space for the Heisenberg group. In this note we give another characterization for a subspace of S(K) (Schwartz space) such that the Radon transform Rα on K is a bijection. We show that this characterization is equivalent to that in [M.M. Nessibi, K. Trimèche, Inversion of the Radon transform on the Laguerre hypergroup by using generalized wavelets, J. Math. Anal. Appl. 208 (1997) 337-363]. In addition, we establish an inversion formula of the Radon transform Rα in the weak sense.     

The a posteriori Fourier method for solving the Cauchy problem for the Laplace equation with nonhomogeneous Neumann data

In the present paper, the Cauchy problem for the Laplace equation with nonhomogeneous Neumann data in an infinite “strip” domain is considered. This problem is severely ill-posed, i.e., the solution does not depend continuously on the data. A conditional stability result is given. A new a posteriori Fourier method for solving this problem is proposed. The corresponding error estimate between the exact solution and its regularization approximate solution is also proved. Numerical examples show the effectiveness of the method and the comparison of numerical effect between the a posteriori and the a priori Fourier method are also taken into account.     

Determining surface heat flux in the steady state for the Cauchy problem for the Laplace equation

In this paper, we consider the Cauchy problem for the Laplace equation, in a strip where the Cauchy data is given at x = 0 and the flux is sought in the interval 0<x?1. This problem is typical ill-posed: the solution (if it exists) does not depend continuously on the data. We study a modification of the equation, where a fourth-order mixed derivative term is added. Some error stability estimates for the flux are given, which show that the solution of the modified equation is approximate to the solution of the Cauchy problem for the Laplace equation. Furthermore, numerical examples show that the modified method works effectively.     

Inverse two-sided Laplace transform for probability density functions

We present a method for the numerical inversion of two-sided Laplace transform of a probability density function. The method assumes the knowledge of the first M derivatives at the origin of the function to be antitransformed. The approximate analytical form is obtained by resorting to maximum entropy principle. Both entropy and L₁-norm convergence are proved. Some numerical examples are illustrated.     

Orthogonal polynomials on the unit circle associated with the Laguerre polynomials

Using the well-known fact that the Fourier transform is unitary, we obtain a class of orthogonal polynomials on the unit circle from the Fourier transform of the Laguerre polynomials (with suitable weights attached). Some related extremal problems which arise naturally in this setting are investigated.

     

Wavelet regularization for an inverse heat conduction problem

In this paper we consider an inverse heat conduction problem which appears in some applied subjects. This problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the data. The Meyer wavelets are applied to formulate a regularized solution which is convergent to exact one on an acceptable interval when data error tends to zero.     

An auxiliary parameter method using Adomian polynomials and Laplace transformation for nonlinear differential equations

In this article, we proposed an auxiliary parameter method using Adomian polynomials and Laplace transformation for nonlinear differential equations. This method is called the Auxiliary Laplace Parameter Method (ALPM). The nonlinear terms can be easily handled by the use of Adomian polynomials. Comparison of the present solution is made with the existing solutions and excellent agreement is noted. The fact that the proposed technique solves nonlinear problems without any discretization or restrictive assumptions can be considered as a clear advantage of this algorithm over the numerical methods.     

A modified method for a non-standard inverse heat conduction problem

A non-standard inverse heat conduction problem is considered. Data are given along the line x = 1 and the solution at x = 0 is sought. The problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the data. In order to solve the problem numerically it is necessary to employ some regularization method. In this paper, we study a modification of the equation, where a fourth-order mixed derivative term is added. Error estimates for this equation are given, which show that the solution of the modified equation is an approximation of the heat equation. A numerical implementation is considered and a simple example is given. Some numerical results show the usefulness of the modified method.     

An efficient algorithm for regularization of Laplace transform inversion in real case

We address design of a numerical algorithm for solving the linear system arising in numerical inversion of Laplace transforms in real case [L. D’Amore, A. Murli, Regularization of a Fourier series method for the Laplace transform inversion with real data, Inverse Problems 18 (2002) 1185–1205]. The matrix has a condition number that grows almost exponentially and the singular values decay gradually towards zero. In such a case, because of this intrinsic strong instability, the main difficulty of any numerical computation is the ability of discovering at run time, only using data, what is the maximum attainable accuracy on the solution.

In this paper, we use GMRES with the aim of relating the current residuals to the maximum attainable accuracy of the approximate solution by using a suitable stopping rule. We prove that GMRES stops after, at most, as many iterations as the number of the largest eigenvalues (compared to the machine epsilon). We use a split preconditioner that symmetrically precondition the initial system. By this way, the largest eigenvalue dynamically provides the estimate of the condition number of the matrix.     

A note on the coefficients of inverse polynomials

We describe some relations on the coefficients of a polynomial in terms of the map that induces and use them to characterize the coefficients of the inverse polynomials of some special classes of permutation polynomials.     

Limit relations for the complex zeros of Laguerre and q-Laguerre polynomials

For each the nth Laguerre polynomial has an m-fold zero at the origin when α=−m. As the real variable α→−m, it has m simple complex zeros which approach 0 in a symmetric way. This symmetry leads to a finite value for the limit of the sum of the reciprocals of these zeros. There is a similar property for the zeros of the q-Laguerre polynomials and of the Jacobi polynomials and similar results hold for sums of other negative integer powers.     

Truncation functions and Laplace transform

We note the increase in the parameter c of the symmetrized Laplace transform of a random variable truncated at level c. We illustrate this remark for both the exponential and the uniform variables.     

Error Bounds for Asymptotic Expansion of the Conditional Variance of the Scale Mixtures of the Multivariate Normal Distribution

Let X= A ^1/2 G be a scale mixture of a multivariate normal distribution with X, G ⁿ , Gis a multivariate normal vector, and A is a positive random variable independent of the multivariate random vector G. This study presents asymptotic results of the conditional variance-covariance, Cov(X ₂|X ₁), X ₁ ^m , m < n, under some moment expressions. A new representation form is also presented for conditional expectation of the scale variable on the random vector X ₁ ^m , m < n. Both the asymptotic expression and the representation are manageable and in computable form. Finally, an example is presented to illustrate how the computations are carried out.     

A stability estimate of an inverse problem in financial prospection

In many option pricing models, there is a need to derive a stability estimate of their solutions. This paper is concerned with the stability estimate of the regularized solution arising in the inverse problem of option pricing. This kind of inverse problems, where one looks for causes for observed or desired effects, are usually “ill-posed”, i.e., their solution is not unique or unstable with respect to data perturbations. Stability estimate is needed for stable solution of these ill-posed problems. If one cannot guarantee the stability of a solution to these problems, then the problem does not make sense and there is no hope of handling them numerically. It is shown here that one can achieve stability by using so called stabilizing functional suggested by Tikhonov [Linear Integral Equation, Applied Mathematical Sciences, Springer, New york, 1989].     

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 an approximation to f(t) by the use of the classical 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 expansion that approximation f(t) in terms of Jacobi polynomials. Some numerical examples are illustrated.     

Wavelets and regularization of the Cauchy problem for the Laplace equation

In this paper, a Cauchy problem for two-dimensional Laplace equation in the strip 0<x?1 is considered again. This is a classical severely ill-posed problem, i.e., the solution (if it exists) does not depend continuously on the data, a small perturbation in the data can cause a dramatically large error in the solution for 0<x?1. The stability of the solution is restored by using a wavelet regularization method. Moreover, some sharp stable estimates between the exact solution and its approximation in Hr(R)-norm is also provided.