Solution of a Boundary Value Problem for an Elliptic Equation with a Small Noninteger Order Degeneracy

Abstract

We consider the Dirichlet boundary value problem for an elliptic type equation with irregular noninteger-order degeneration in a rectangle. The coefficients of the differential operator are supposed to be analytic. We construct a formal solution by using the method of spectral separation of singularities in the form of a series; the character of the nonanalytic dependence of the solution on the variable \(y\) in a neighborhood of \(y=0 \) is written out explicitly. We prove the convergence of the series to the classical solution using the Green’s function method.