Padé approximants, density of rational functions in A∞(Ω) and smoothness of the integration operator

We consider the problem of finding approximate analytical solutions for nonlinear equations typical of physics applications. The emphasis is on the modification of the method of Padé approximants that are known to provide the best approximation for the class of rational functions, but do not provide sufficient accuracy or cannot be applied at all for those nonlinear problems, whose solutions exhibit behavior characterized by irrational functions. In order to improve the accuracy, we suggest a method of self-similarly corrected Padé approximants taking into account the irrational functional behavior. The idea of the method is in representing the sought solution as a product of two factors, one of which is given by a self-similar root approximant, responsible for irrational functional behavior, and the other being a Padé approximant corresponding to a rational function. The efficiency of the method is illustrated by constructing very accurate solutions for nonlinear differential equations. A thorough investigation is given proving that the suggested method is more accurate than the method of standard Padé approximants.

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On the Maximum and Minimum Modulus of Rational Functions

Canadian Journal of Mathematics ◽

10.4153/cjm-2000-035-3 ◽

2000 ◽

Vol 52 (4) ◽

pp. 815-832 ◽

Cited By ~ 1

Author(s):

D. S. Lubinsky

Keyword(s):

Unit Ball ◽

Rational Function ◽

Rational Functions ◽

Padé Approximants ◽

Convergence Result ◽

Linear Measure ◽

Minimum Modulus ◽

Pade Approximants

AbstractWe show that if m, n ≥ 0, λ > 1, and R is a rational function with numerator, denominator of degree ≤ m, n, respectively, then there exists a set ⊂ [0, 1] of linear measure such that for r ∈ ,Here, one may not replace , for any ε > 0. As our motivating application, we prove a convergence result for diagonal Padé approximants for functions meromorphic in the unit ball.

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

Modern Physics Letters A ◽

10.1142/s0217732321500383 ◽

2021 ◽

pp. 2150038

Author(s):

Man Kwong Mak ◽

Chun Sing Leung ◽

Tiberiu Harko

Keyword(s):

Dark Energy ◽

Poisson Equation ◽

Decomposition Method ◽

Adomian Decomposition Method ◽

Rational Functions ◽

Padé Approximants ◽

Pade Approximants ◽

The Poisson Equation ◽

Adomian Decomposition ◽

Constant Dark

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.

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