Fourier theory may be derived from analytical considerations of the heat transfer process in a substance and neglecting of higher order terms in Fourier analysis becomes important at high input power fluxes. It is apparent from the previous work that the kinetic theory equations governing the heat conduction process are more general than the Fourier equation. The special case may exist where the kinetic theory solutions converge to solutions extracted from the Fourier analysis. Consequently, research into the convergence of kinetic theory approach to Fourier analysis becomes necessary. To achieve this, the special case in which local thermal equilibrium is assumed to exist between the electrons and atoms at any section on the surface region of the solid substance is studied in the present work, providing that the study does not allow the existence of phase changes. For this purpose, an analytical approach is introduced and a computer solution of the reduced integral equations is obtained. It is found that the solutions of electron kinetic theory equations based on the assumption of existence of thermal equilibrium between electrons and atoms are identical with those obtained from Fourier theory.