### Abstract:

An important element in rector safety and fuel performance assessments is the efficient calculation of the time dependent temperature profiles within reactor fuel pins. Because the fuel pins have temperature dependent thermal properties, the governing partial differential equations are nonlinear. Traditionally, finite difference or finite element methods have been used to solve this system. In this thesis an analytically based method is presented in which the transient temperature profiles are calculated using describing functions to characterize the system nonlinearities. While in the past the describing function approach has been used exclusively in the solution of ordinary differential equation systems, this study appears to be the first successful application of describing functions in the solution of partial differential equations. The solution requires a three step process. First, using the definition of the integral conductivity and defining in a similar manner the integral heat capacity, the governing equation is recast into a linear partial differential equation involving these thermal properties as the two dependent variables. Second, mathematical integral transforms are used to obtain an algebraic equation in the transformed time-space domain. Because the resulting algebraic equation is still in terms of the two dependent variables, the third step is to relate these variables to each other using the describing function approach. Using this technique a factor of 40 reduction in computer time was achieved over a similar finite element calculation. In addition, the approach is general and can be applied to other nonlinear partial differential equation systems.