A priori bounds of solutions of the nonlinear integral convolution type equation and their applications
type: Item
instance of: scholarly article
title: A priori bounds of solutions of the nonlinear integral convolution type equation and their applications
zbMATH Open document ID: 0824.45006
DOI: 10.1007/BF01208385
author: S. N. Askhabov
author: M. A. Betilgiriev
published in: Mathematical Notes
publication date: 1995-09-12T00:00:00Z
review text: In connection with applications in the theories of filtration, shock waves, heat exchange, and others an equation of the form \[ u^ \alpha(x)= \int^ x_ 0 k(x- t) u(t)+ f(x),\qquad \alpha> 1,\;x> 0,\tag{1} \] has been considered in the literature in the class of non- negative \(\Gamma\)-functions continuous on \([0, \infty)\) and under the rather restrictive condition \(f(0)= 0\). Since equations of the form (1) are exactly solvable only in a few particular cases, we see that both for the theory and applications the development of approximate methods for equations of this type as well as the stability of solutions under changes of given functions \(k\), \(f\) are of great importance. In the present paper we obtain new a priori bounds for solutions of equation (1) that make the corresponding bounds of the authors more precise. Using these bounds enables us to prove the existence and uniqueness of a solution of equation (1) for \(f\) not necessarily vanishing at \(x= 0\) and to define the precise boundaries of the solution. For \(\alpha\geq 2\) we show that the solution may be obtained by the method of sequential approximations for which the error bound and that of the degree of their convergence to the precise solution are given, as well as the stability of the solution relative to changes of \(k\) and \(f\) in the same metric is also proved.
Mathematics Subject Classification ID: 45G10
Mathematics Subject Classification ID: 45L05
Mathematics Subject Classification ID: 45M10
zbMATH DE Number: 795594
zbMATH Keywords: nonlinear integral convolution type equation
zbMATH Keywords: filtration
zbMATH Keywords: shock waves
zbMATH Keywords: heat exchange
zbMATH Keywords: stability
zbMATH Keywords: a priori bounds
zbMATH Keywords: existence
zbMATH Keywords: uniqueness
zbMATH Keywords: method of sequential approximations
zbMATH Keywords: error bound
zbMATH Keywords: convergence
MaRDI profile type: Publication
cites work: On a nonlinear convolution equation occurring in the theory of water percolation
cites work: On a nonlinear volterra equation
cites work: Integral equations of convolution type with power nonlinearity
cites work: Q3496750
cites work: On the existence and uniqueness of non-negative solutions of a certain non-linear convolution equation
cites work: Time Dependent Temperature Distribution in Radiating Solids
cites work: Q5583334
full work available at URL: https://doi.org/10.1007/bf01208385
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description: scientific article; zbMATH DE number 795594