Analysis of temperature fields is important for many engineering applications. The account of actual operating conditions of these structures frequently leads to mixed heating condition. The authors of this paper developed a new effective method of solutions derivation for mixed boundary-value unsteady heat conduction problems. This paper considers the cylinder with at the part of surface of which the temperature distribution is known. Outside this area the heat transfer by Newton's law is performed. To the heat conductivity problem it is applied the Laguerre integral transformation in time variables and integral Fourier transformation in spatial variable. As a result the triangular sequence of ordinary differential equations is obtained. The general solution of these sequences is obtained in the form of algebraic convolution. Taking into account the mixed boundary conditions leads to dual integral equations. For solution of this problem it is proposed the method of Neumann's series. By this method the problem is reduced to the infinite system of algebraic equations, for which the convergence of reduction procedure is proved. Finally, the unknown temperature is submitted as a series of Laguerre polynomials. The coefficient of these series is Fourier integrals.