This dissertation presents a novel developed semi-analytic mathematic scheme, the finite difference distributed transfer function method (FDDTFM), succeeds both static/steady-state and dynamic/unsteady-state analysis on the two-dimensional elasticity, heat conduction, and plate problems in the Cartesian and the polar coordinate systems. The scheme uses the finite difference method (FDM) along the y/r direction and the distributed transfer function method (DTFM) along the x/theta direction. The FDDTFM doesn't need to re-derive equations because of changing boundary conditions; use series function for complicated geometry.; The Navier's displacement equations by the FDDTFM examine the static and dynamic aspects of the linear two-dimensional elastic plane problems with arbitrary boundary conditions in the two different coordinate systems. The static results of square, annular sector, and L-shaped regions with arbitrary boundary conditions show up to 384 times faster convergence than those of the FEM. The free vibration results of square and annular sector regions show the similar phenomenon.; The dissertation applies the FDDTFM to solve the steady-state and unsteady-state heat conduction problems on the two-dimensional region with arbitrary boundary conditions in the two different coordinate systems. The steady-state results of square, annular sector and L-shaped regions with arbitrary boundary conditions examples present up to 10 times faster convergence by comparing with those of the FEM. Exact general time response solutions show the one-dimensional heat conduction problems by the DTFM with the residue theorem work very well. The general time response results of the two-dimensional heat conduction on square and composite regions also present the comparable fact.; The proposed method analyzes the static and dynamic parts of the Kirchhoff plate problems with arbitrary boundary conditions in the two coordinate systems. The static results of rectangular and annular sector plates with arbitrary boundary conditions are up to 126 times faster convergence than those of the FEM. The free vibration resuls of square and annualar sector regions show the similar trend.; The FDDTFM guarantees teh convergences of those two-dimensional elastic continua problems. The appendices contain over 20,000 possibilities of the boundary conditions on the simple geometry.