Singular solution

Summary

A singular solution ys(x) of an ordinary differential equation is a solution that is singular or one for which the initial value problem (also called the Cauchy problem by some authors) fails to have a unique solution at some point on the solution. The set on which a solution is singular may be as small as a single point or as large as the full real line. Solutions which are singular in the sense that the initial value problem fails to have a unique solution need not be singular functions. In some cases, the term singular solution is used to mean a solution at which there is a failure of uniqueness to the initial value problem at every point on the curve. A singular solution in this stronger sense is often given as tangent to every solution from a family of solutions. By tangent we mean that there is a point x where ys(x) = yc(x) and y's(x) = y'c(x) where yc is a solution in a family of solutions parameterized by c. This means that the singular solution is the envelope of the family

Official source

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Stability of slow blow-up solutions for the critical focussing nonlinear wave equation on $\mathbb{R}^{3+1}$

Stefano Francesco Burzio

In this brief survey we outline the recent advances on the stability issues of certain finite time type II blow-up solutions for the energy critical focusing wave equation

\Box u=-u^{5}

\mathbb{R}^{3+1}

. Hereafter we use the convention

\Box=-\partial^{2}_{t}+\triangle

. The objective of this article is twofold: firstly we describe the construction of singular solutions contained in \cite{krieger2009slow} and \cite{krieger2014full}, and secondly we undertake a detailed analysis of its stability properties enclosed in \cite{krieger2017stability} and \cite{krieger2017optimal}.

2018

, ,

This work focuses on the development of a posteriori error estimates for fourth-order, elliptic, partial differential equations. In particular, we propose a novel algorithm to steer an adaptive simulation in the context of Kirchhoff plates and Kirchhoff-Love shells by exploiting the local refinement capabilities of hierarchical B-splines. The method is based on the solution of an auxiliary residual-like variational problem, formulated by means of a space of localized spline functions. This space is characterized by C1 continuous B-splines with compact support on each active element of the hierarchical mesh. We demonstrate the applicability of the proposed estimator to Kirchhoff plates and Kirchhoff-Love shells by studying several benchmark problems which exhibit both smooth and singular solutions. In all cases, we obtain optimal asymptotic rates of convergence for the error measured in the energy norm and an excellent approximation of the true error.

MATHICSE2019

A hierarchical approach to the a posteriori error estimation of isogeometric Kirchhoff plates and Kirchhoff–Love shells

Pablo Antolin Sanchez, Annalisa Buffa, Luca Coradello

This work focuses on the development of a posteriori error estimates for fourth-order, elliptic, partial differential equations. In particular, we propose a novel algorithm to steer an adaptive simulation in the context of Kirchhoff plates and Kirchhoff–Love shells by exploiting the local refinement capabilities of hierarchical B-splines. The method is based on the solution of an auxiliary residual-like variational problem, formulated by means of a space of localized spline functions. This space is characterized by c1 continuous B-splines with compact support on each active element of the hierarchical mesh. We demonstrate the applicability of the proposed estimator to Kirchhoff plates and Kirchhoff–Love shells by studying several benchmark problems which exhibit both smooth and singular solutions. In all cases, we obtain optimal asymptotic rates of convergence for the error measured in the energy norm and an excellent approximation of the true error.

2020

Stability of slow blow-up solutions for the critical focussing nonlinear wave equation on $\mathbb{R}^{3+1}$

Stefano Francesco Burzio

In this brief survey we outline the recent advances on the stability issues of certain finite time type II blow-up solutions for the energy critical focusing wave equation

\Box u=-u^{5}

\mathbb{R}^{3+1}

. Hereafter we use the convention

\Box=-\partial^{2}_{t}+\triangle

2018