On solution of integral equation of Abel-Volterra type
type: Item
instance of: scholarly article
title: On solution of integral equation of Abel-Volterra type
zbMATH Open document ID: 0823.45002
author: A. A. Kilbas
author: Megumi Saigo
published in: Differential and Integral Equations
publication date: 1995-11-05T00:00:00Z
review text: The authors study asymptotic expansions of the solution of the equation \[ \varphi (x) = a(x) \int_ 0^ x (x - t)^{\alpha - 1} \varphi (t) dt + f(x), \] as \(x \to 0 +\), where \(0 < \alpha < 1\). It is assumed that \(a\) and \(f\) have expansions of the form \(a(x) \sim \sum_{k = - 1}^ \infty a_ k x^{\alpha k}\), \(f(x) \sim \sum_{k = - 1}^ \infty f_ k x^{\alpha k}\), and it is shown that then the solution has a similar expansion \(\varphi (x) \sim \sum_{k = - 1}^ \infty \varphi_ k x^{\alpha k}\) when \(x \to 0 +\). Recursive formulas for calculating the coefficients \(\varphi_ k\) are given. The case where \(a(x) = ax^{\alpha m}\), \(m = - 1,0, \dots\), and where one can find an explicit expression for the solution is studied, too.
reviewed by: Gustaf Gripenberg
Mathematics Subject Classification ID: 45E10
Mathematics Subject Classification ID: 45D05
Mathematics Subject Classification ID: 41A60
zbMATH DE Number: 763452
zbMATH Keywords: integral equation of Abel-Volterra kind
zbMATH Keywords: asymptotic expansions
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 763452