scholarly journals An extension of the method of brackets. Part 1

2017 ◽  
Vol 15 (1) ◽  
pp. 1181-1211 ◽  
Author(s):  
Ivan Gonzalez ◽  
Karen Kohl ◽  
Lin Jiu ◽  
Victor H. Moll
Keyword(s):  
Efficient Method ◽  
Blow Up ◽  
Divergent Series ◽  
Half Line ◽  
Definite Integrals ◽  
Small Collection ◽  
Formal Representations

Abstract The method of brackets is an efficient method for the evaluation of alarge class of definite integrals on the half-line. It is based on a small collection of rules, some of which are heuristic. The extension discussed here is based on the concepts of null and divergent series. These are formal representations of functions, whose coefficients an have meromorphic representations for n ∈ ℂ, but might vanish or blow up when n ∈ ℕ. These ideas are illustrated with the evaluation of a variety of entries from the classical table of integrals by Gradshteyn and Ryzhik.

A representation for Jost solutions and an efficient method for solving the spectral problem on the half line

2019 ◽  
Vol 43 (16) ◽  
pp. 9304-9319 ◽  
Author(s):  
Briceyda B. Delgado ◽  
Kira V. Khmelnytskaya ◽  
Vladislav V. Kravchenko
Keyword(s):  
Spectral Problem ◽  
Efficient Method ◽  
Half Line ◽  
Jost Solutions

scholarly journals An extension of the method of brackets. Part 2

2020 ◽  
Vol 18 (1) ◽  
pp. 983-995
Author(s):  
Ivan Gonzalez ◽  
Lin Jiu ◽  
Victor H. Moll
Keyword(s):  
Mellin Transform ◽  
Feynman Diagrams ◽  
Half Line ◽  
Definite Integrals ◽  
Small Set

Abstract The method of brackets, developed in the context of evaluation of integrals coming from Feynman diagrams, is a procedure to evaluate definite integrals over the half-line. This method consists of a small number of operational rules devoted to convert the integral into a bracket series. A second small set of rules evaluates this bracket series and produces the result as a regular series. The work presented here combines this method with the classical Mellin transform to extend the class of integrands where the method of brackets can be applied. A selected number of examples are used to illustrate this procedure.

scholarly journals A nonlinear diffusion equation with reaction localized in the half-line

2021 ◽  
Vol 4 (3) ◽  
pp. 1-24
Author(s):  
Raúl Ferreira ◽  
◽  
Arturo de Pablo ◽  
Keyword(s):  
Global Existence ◽  
Diffusion Equation ◽  
Heat Equation ◽  
Nonlinear Diffusion ◽  
Blow Up ◽  
Nonlinear Diffusion Equation ◽  
Half Line ◽  
Blow Up Rate ◽  
Global Reaction

<abstract><p>We study the behaviour of the solutions to the quasilinear heat equation with a reaction restricted to a half-line</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ u_t = (u^m)_{xx}+a(x) u^p, $\end{document} </tex-math></disp-formula></p> <p>$ m, p &gt; 0 $ and $ a(x) = 1 $ for $ x &gt; 0 $, $ a(x) = 0 $ for $ x &lt; 0 $. We first characterize the global existence exponent $ p_0 = 1 $ and the Fujita exponent $ p_c = m+2 $. Then we pass to study the grow-up rate in the case $ p\le1 $ and the blow-up rate for $ p &gt; 1 $. In particular we show that the grow-up rate is different as for global reaction if $ p &gt; m $ or $ p = 1\neq m $.</p></abstract>

scholarly journals Numerical quadrature over smooth, closed surfaces

2016 ◽  
Vol 472 (2194) ◽  
pp. 20160401 ◽  
Author(s):  
J. A. Reeger ◽  
B. Fornberg ◽  
M. L. Watts
Keyword(s):  
Computational Complexity ◽  
Efficient Method ◽  
Basis Function ◽  
Numerical Approximation ◽  
One Dimension ◽  
Radial Basis Function Interpolation ◽  
Definite Integrals ◽  
Closed Surfaces ◽  
Appl Math ◽  
Subsequent Integration

The numerical approximation of definite integrals, or quadrature, often involves the construction of an interpolant of the integrand and its subsequent integration. In the case of one dimension it is natural to rely on polynomial interpolants. However, their extension to two or more dimensions can be costly and unstable. An efficient method for computing surface integrals on the sphere is detailed in the literature (Reeger & Fornberg 2016 Stud. Appl. Math. 137 , 174–188. ( doi:10.1111/sapm.12106 )). The method uses local radial basis function interpolation to reduce computational complexity when generating quadrature weights for any given node set. This article generalizes this method to arbitrary smooth closed surfaces.

Singularity formation in the one-dimensional supercooled Stefan problem

1996 ◽  
Vol 7 (2) ◽  
pp. 119-150 ◽  
Author(s):  
Miguel A. Herrero ◽  
Juan J. L. Velázquez
Keyword(s):  
Stefan Problem ◽  
Finite Time ◽  
Blow Up ◽  
Half Line ◽  
Singularity Formation ◽  
Asymptotics Of Solutions ◽  
One Dimensional ◽  
The One

It is well-known that solutions to the one-dimensional supercooled Stefan problem (SSP) may exhibit blow-up in finite time. If we consider (SSP) in a half-line with zero flux conditions at t = 0, blow-up occurs if there exists T < ∞ such that limt↑Ts(t) > 0 and lim inft↑T⋅(t) = – ∞,s(t) being the interface of the problem under consideration. In this paper, we derive the asymptotics of solutions and interfaces near blow-up. We shall also use these results to discuss the possible continuation of solutions beyond blow-up.

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