Padé Approximants, Their Properties, and Applications to Hydrodynamic Problems

This paper is devoted to an overview of the basic properties of the Padé transformation and its generalizations. The merits and limitations of the described approaches are discussed. Particular attention is paid to the application of Padé approximants in the mechanics of liquids and gases. One of the disadvantages of asymptotic methods is that the standard ansatz in the form of a power series in some parameter usually does not reflect the symmetry of the original problem. The search for asymptotic ansatzes that adequately take into account this symmetry has become one of the most important problems of asymptotic analysis. The most developed technique from this point of view is the Padé approximation.

A Critical Evaluation and Modification of the Padé–Laplace Method for Deconvolution of Viscoelastic Spectra

This paper shows that using the Padé–Laplace (PL) method for deconvolution of multi-exponential functions (stress relaxation of polymers) can produce ill-conditioned systems of equations. Analysis of different sets of generated data points from known multi-exponential functions indicates that by increasing the level of Padé approximants, the condition number of a matrix whose entries are coefficients of a Taylor series in the Laplace space grows rapidly. When higher levels of Padé approximants need to be computed to achieve stable modes for separation of exponentials, the problem of generating matrices with large condition numbers becomes more pronounced. The analysis in this paper discusses the origin of ill-posedness of the PL method and it was shown that ill-posedness may be regularized by reconstructing the system of equations and using singular value decomposition (SVD) for computation of the Padé table. Moreover, it is shown that after regularization, the PL method can deconvolute the exponential decays even when the input parameter of the method is out of its optimal range.

Virtual corrections to gg → ZH in the high-energy and large-mt limits

We compute the next-to-leading order virtual corrections to the partonic cross-section of the processgg → ZH, in the high-energy and large-mtlimits. We use Padé approximants to increase the radius of convergence of the high-energy expansion in$$ {m}_t^2/s $$mt2/s,$$ {m}_t^2/t $$mt2/tand$$ {m}_t^2/u $$mt2/uand show that precise results can be obtained down to energies which are fairly close to the top quark pair threshold. We present results both for the form factors and the next-to-leading order virtual cross-section.

The effects of the dark energy on the static Schrödinger–Newton system — An Adomian Decomposition Method and Padé approximants based approach

The Schrödinger–Newton system is a nonlinear system obtained by coupling together the linear Schrödinger equation of quantum mechanics with the Poisson equation of Newtonian mechanics. In this work, we will investigate the effects of a cosmological constant (dark energy or vacuum fluctuation) on the Schrödinger–Newton system, by modifying the Poisson equation through the addition of a new term. The corresponding Schrödinger–Newton-[Formula: see text] system cannot be solved exactly, and therefore for its study one must resort to either numerical or semianalytical methods. In order to obtain a semianalytical solution of the system we apply the Adomian Decomposition Method, a very powerful method used for solving a large class of nonlinear ordinary and partial differential equations. Moreover, the Adomian series are transformed into rational functions by using the Padé approximants. The semianalytical approximation is compared with the exact numerical solution, and the effects of the dark energy on the structure of the Newtonian quantum system are fully investigated.