For solution of one integral equation with one variable integration limit and the other infinite one

We consider the integral equation of the second kind of Volterra type with a variable lower limit of integration. Exception of the equation is, firstly, that the interval of integration is infinite, and secondly, in a convergence of lower limit to the upper one the integral tends to unity. Integral equations of this kind arise in the solution of boundary value problems of the theory of heat conduction in non-cylindrical domains. Also the value problems for spectrally loaded parabolic equations are reduced to them.