Fredholm-Volterra integral equation in contact problem.
type: Item
instance of: scholarly article
title: Fredholm-Volterra integral equation in contact problem.
zbMATH Open document ID: 1034.45005
DOI: 10.1016/S0096-3003(02)00101-7
author: Osama L. Moustafa
author: M. A. Abdou
published in: Applied Mathematics and Computation
publication date: 2003-09-09T00:00:00Z
review text: The author considers the Fredholm-Volterra integral equation \[ kP(x,y,t)+q\int\limits_0^\infty\int\limits_0^\infty \frac{P(\xi,\eta,t)\,d\xi\,d\eta}{\sqrt{(x-\xi)^2+(y-\eta)^2}} +q\int\limits_0^t F(t,\tau)P(x,y,\tau) \,d\tau=f(x,y,t) \tag{1} \] in the space \(L_2(\Omega)\times C(0,T)\), under the condition \[ \int\limits_0^\infty\int\limits_0^\infty P(x,y,t)\,dx\,dy=P(t). \] Using the partial Fourier transform \[ G_\alpha(x,t)=\int\limits_{-\infty}^\infty G(x,y,t)e^{i\alpha y}\,dy \] the authors claim that equation (1) reduces to the equation \[ kP_\alpha(x,y,t)+2q\int\limits_0^\infty\int\limits_0^\infty K_0(\alpha(x-\xi))P_\alpha(\xi,t)\,d\xi +q\int\limits_0^t F(t,\tau)P_\alpha(x,\tau) \,d\tau=F_\alpha(x,t). \tag{2} \] Equation (2) is treated with series expansion and reduction to an algebraic system. Reviewer's remark: In equation (1) integration is meant, perhaps, over the half plane \(\Omega=(-\infty,\infty)\times [0,\infty)\) and not over the quarter plane \(\Omega= [0,\infty)\times [0,\infty)\) because in the latter case, after the Fourier transform, we should deal with analytic functions. The main auxiliary result, Theorem 1 on page 204, is incorrect: a convolution operator with a kernel \(k_0\in L_1(-\infty,\infty)\) is compact in \(L_2(0,\infty)\) if and only if \(k_0=0\).
reviewed by: Roland Duduchava
Mathematics Subject Classification ID: 45E10
Mathematics Subject Classification ID: 74M15
zbMATH DE Number: 1977997
zbMATH Keywords: Fredholm-Volterra integral equation
zbMATH Keywords: convolution
zbMATH Keywords: contact problem
MaRDI profile type: Publication
cites work: Q5570459
cites work: Some approximate methods of solving integral equations of mixed problems
cites work: On the Eigenvalues of a Class of Integral Equations Arising in Laser Theory
cites work: Q4124250
cites work: Q4040933
cites work: Solution methods for integral equations. Theory and applications
cites work: Numerical solution of integral equations
cites work: Fredholm integral equation with potential kernel and its structure resolvent
cites work: Fredholm integral equation of the second kind with potential kernel
cites work: Fredholm-Volterra integral equation of the first kind and contact problem
cites work: Q5822969
cites work: Q5820804
cites work: On a method of obtaining spectral relationships for integral operators of mixed problems of mechanics of continuous media
cites work: Q3945677
cites work: Q3716532
cites work: A special relationship in spheroidal wave functions and its application to contact problems
cites work: Q3681889
full work available at URL: s0096-3003(02)00101-7
OpenAlex ID: W2000663703
description: scientific article; zbMATH DE number 1977997