On an integral equation of the problem of heat conduction with domain boundary moving by law of t = x 2

Abstract

In the article it is shown that the homogeneous Volterra integral equation of the second kind, to which
the homogeneous boundary value problem of heat conduction in the degenerating domain is reduced, has
a nonzero solution. The boundary of the domain moves with a variable velocity. It is shown that the norm
of the integral operator acting in classes of continuous functions is equal to 1. Mellin transformation is
applied to the obtained integral equation. It is proved that for certain values of the spectral parameter the
eigenvalues of the integral equation will be simple.