This work addresses the problem of thermal shock in a light water cooled nuclear reactor undergoing emergency cooling following a hypothetical loss of coolant accident. Portions of this work provide novel approaches to heat conduction problems. Moreover, the scope of this analysis is broader than that which is currently sought in industry. In the first solution to the problem, the general heat conduction equation in one dimension is solved analytically by the method of variation of parameters. The domain of the solution is the radius of the bimetallic, upper cylindrical region of the pressure vessel, The solution considers spatially and time-varying internal heat generation. The two-dimensional analysis of the problem is begun next. Assuming axisymmetric heat conduction, the steady-state profile, which exists in the pressure vessel at the time the loss of coolant accident occurs, is determined by separation of variables. This analysis is innovative because of the following: the solution is composed of four analytical solutions of the two-dimensional Poisson equation, two in cylindrical coordinates and two in the spherical coordinate system, each pair spanning media having different thermal properties. The inhomogeneous heat generation term in each of the four regions is a function of an independent variable. The eigensolutions derived for the cylindrical section are joined along the mutual boundary to the eigensolutions of the spherical regions by first imposing continuity constraints on the dependent variables and their first derivatives followed by an exploitation of the orthogonal nature of, the eigensolutions. Contrary to one-dimensional results, a two-dimensional analysis indicates that maximum temperatures do not necessarily occur at the outer insulated boundary of the pressure vessel. The results of this analysis are verified by numerical techniques and are used as the initial conditions for the two-dimensional, transient analysis that follows. The transient analysis is formulated by both finite difference and finite element techniques for the purposes of method comparison and verification of results. Each technique results in a set of linear, first order, ordinary differential equations which are solved exactly in time by matrix methods instead of the usual time stepping, numerical procedures. The merits and demerits of matrix methods used in conjunction with each numerical technique for handling the spatial variables of the problem are enumerated. The analysis considers boiling heat transfer, and results indicate axisymmetric heat flow into the lower hemispherical region has a mitigating effect on radial temperature gradients in only the lower 20% of the cylindrical region. The effect of different clad thicknesses is also discussed. / Ph. D.