Asymptotics of integrodifferential models with integrable kernels
The paper deals with the equation \[ \epsilon y^{\prime }(t)=g(t)+{\frac{1}{\Gamma (1-\beta)}} \int^{t}_{0}{\frac{k(t,s)}{(t-s)^{\beta}}}y(s)ds,\;0<t\leq T,\;y(0)=y_{0}(\epsilon),\eqno(1) \] where \(0<\epsilon \leq 1\) and \(0<\beta <1\), with continuous functions \(g(t)\) and \(k(t,s)\) such that \(k(t,t)<0\) for \(0\leq t\leq T\). When \(\epsilon =0\), (1) coincides with the classical Abel integral equation; for example, see the book by \textit{R. Gorenflo} and \textit{S. Vessella} [Abel integral equations. Analysis and applications (1991; Zbl 0717.45002)].The main results states that if \(y(t,\epsilon)\) is the exact solution of (1), then this equation has the asymptotic solution \(y_0(t,\epsilon)\in C[0,T]\) such that \(| y(t,\epsilon)-y_0(t,\epsilon)|\) is uniformly of order \(0(\epsilon^{\alpha})\), as \(\epsilon \to 0\), on \([0,T]\). The proof of this results is based on the additive decomposition technique and on the theory of the Riemann-Liouville fractional integrals; for example, see Section 2.3 in the book by \textit{S. G. Samko}, the reviewer and \textit{O. I. Marichev} [Fractional integrals and derivatives: theory and applications (1993; Zbl 0818.26003)]. The approach presented is demonstrated to find an approximate solution of the integral equation connected with the boundary value problem for the heat equation \(u_{xx}-u_{t}=0\), which models the diffusion of a gas through a liquid. Application of the equation (1) to one problem of viscoelasticity is indicated.
scientific article; zbMATH DE number 1930048