Existence results and approximation methods for functional ordinary differential equations with singular diffusion-type differential operators
type: Item
instance of: scholarly article
title: Existence results and approximation methods for functional ordinary differential equations with singular diffusion-type differential operators
zbMATH Open document ID: 1009.34055
DOI: 10.1016/S0898-1221(02)00134-7
author: Alberto Cabada
author: J. Ángel Cid
author: Rodrigo L. Pouso
publication date: 2002-08-15T00:00:00Z
review text: In a previous paper, the first and the third author [Nonlinear Anal., theory Methods Appl. 42, No. 8, 1377-1796 (2000; Zbl 0964.34016)] proved the existence of extremal solutions \(u^*\) and \(u_*\) to an ordinary differential equation of the type \[ (k(u(x))u'(x))'=f(x,u(x))u'(x)+g(x,u(x))u(x), \quad \text{a.e. in } I=[0,\tau], \tag{1} \] with the initial data \[ u(0)=0,\quad k(u(0))u'(0)=0,\tag{2} \] where \(k\) is allowed to vanish for some values of \(u\), and therefore (1) is singular at these points. This paper continuous this research by considering the functional-differential equation \[ \frac{d}{dx}(k(u(x),u)u'(x))=f(x,u(x),u)u'(x)+g(x,u(x),u)u(x), \quad \text{a.e. in } I=[0,\tau],\tag{3} \] together with the initial condition \[ u(0)=0,\quad k(u(0),u)u'(0)=0.\tag{4} \] The authors give sufficient conditions for the existence of extremal solutions \(u^*\) and \(u_*\) to (3)--(4) and an approximate form of \(u^*\) and \(u_*\). The method which they use is the monotone iterative technique.
Mathematics Subject Classification ID: 34K05
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 34K07
zbMATH DE Number: 1781556
zbMATH Keywords: functional-differential equation
zbMATH Keywords: extremal solution
zbMATH Keywords: approximation
zbMATH Keywords: iterative method
zbMATH Keywords: singular diffusion-type differential operators
zbMATH Keywords: 964
reviewed by: Peixuan Weng
MaRDI profile type: Publication
cites work: Q5312819
cites work: Q5286659
cites work: Existence, uniqueness, and approximation of solutions to some nonlinear diffusion problems
cites work: Q4303531
cites work: Extremality results for discontinuous explicit and implicit diffusion problems
cites work: Approximate solutions to a new class of nonlinear diffusion problems
full work available at URL: s0898-1221(02)00134-7
OpenAlex ID: W1982154888
published in: Computers & Mathematics with Applications
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description: scientific article; zbMATH DE number 1781556