Asymptotics of the solutions to singularly perturbed multidimensional integral equations (Q1892232)

The object of the study in the paper is the singularly perturbed integral equation of the form \[ \varepsilon h_ \varepsilon(x)+ \int_ T R(x- y) h_ \varepsilon(y) dy= f(x),\tag{1} \] \(x\in T\), where \(\varepsilon> 0\) is a parameter, \(T\) is a bounded domain in \(\mathbb{R}^ n\) with a smooth boundary and \(f(x)\) is a given smooth function. Moreover, \(R(x)= P(D) G(x)\), where \(P(D)\) is a differential operator and \(G(x)\) is a fundamental solution. Extending some methods developed earlier an asymptotic solution of the equation (1) is constructed. The estimate of the error of that solution is given. Some examples of application are also provided.