In this paper, based on semi-discrete difference scheme, a new direct algorithm on heat conduction equations is presented. For the Dirichlet and Neumann boundary conditions, the ordinary differential equations are derived from spatial difference scheme, and then the semi-analytical solution of original equation is obtained by the sine or cosine transforms. The corresponding convergence is also analyzed. The precision analysis on central difference and compact difference schemes is also presented by the corresponding eigenvalue theory. In addition, this direct algorithm may be directly developed into double sine or cosine transforms for two-dimensional heat conduction equations by the matrix tensor product. Since there does not exist the time discrete such that it has very high computational efficiency.