On the integral equation of an adjoint boundary value problem of heat conduction

Abstract

An integral equation is considered, to which a nonhomogeneous first boundary value problem with an adjoint heat conduction operator is reduced. The problem is set in an infinite plane angle, that is, a boundary of the domain moves with a constant velocity, and the domain degenerates to a point at the initial moment of time. The incompressibility of the integral operator for the equation under study is shown. Using the relations for an independent variable, the equation under study is equivalently reduced to a certain simplified equation. With the help of replacements for independent variables, the equation is reduced to an integral equation with a difference kernel. By applying the Laplace transform, the obtained equation is reduced to an ordinary first - order differential equation (linear). Its solution is found. By using the inverse Laplace transform, a solution of the nonhomogeneous integral equation under study is obtained in the form of a convergent series in some domain.