Efficient Algorithms for Solving Linear Parabolic Partial Differential Equations Using Green's Functions This thesis describes two related techniques for solving several types of parabolic partial differential equations (PDE's) using Green's functions. Both techniques involve using the Green's function for the linear part of the PDE and making certain approximations to derive computational algorithms. One algorithm uses the Green's function directly to form a computational solution (the Computational Green's Function or CGF algorithm) and the other algorithm uses an approximate solution obtained using standard finite element basis functions with product rule integration (the Product Rule Computational Green's Function or Product Rule CGF algorithm). Central Processing Unit (CPU) time required to execute the algorithms on scalar, vector, and parallel processors is compared with the time required to execute various classical numerical algorithms on the same CPU's Both homogeneous and inhomogeneous boundary conditions are examined. special consideration is given to two-dimensional problems using techniques that are also suitable for higher-dimensional problems. Methods developed to streamline the algorithms are examined in detail. Results show that these techniques can execute faster with less error than classical techniques under appropriate circumstances. Error analysis and identification of the circumstances are two of the main topics of this thesis.