### Abstract:

Describing functions have traditionally been used to obtain the
solutions of systems of ordinary differential equations. In this
work the describing function concept has been extended to include
nonlinear, distributed parameter partial differential equations. A
three-stage solution algorithm is presented which can be applied to
any nonlinear partial differential equation. Two generalized
integral transforms were developed as the T-transform for the time
domain and the B-transform for the spatial domain. As specific
applications of the solution technique two cases are considered: the
heat conduction equation and the neutron diffusion equation.
The thermal diffusion describing function (TDDF) is developed
for conduction of heat in solids and a general iterative solution
along with convergence criteria is presented. The proposed solution
method is used to solve the problem of heat transfer in nuclear fuel
rods with annular fuel pellets. As a special instance the solid
cylindrical fuel pellet is examined. A computer program is written
which uses the describing function concept for computing fuel pin
temperatures in the radial direction during reactor transients. It was found that the quasi-linear method used in the describing
function method is as accurate as nonlinear treatments and as fast
as true linearization methods.
The second problem investigated was the neutron diffusion
equation which is intrinsically different from the first case.
Although, for most situations, it can be treated as a linear
differential equation, the describing function method is still
applicable. A describing function solution is derived for two
possible cases: constant diffusion coefficient and variable
diffusion coefficient. Two classes of describing functions are
defined for each case which portray the leakage and absorption
phenomena. A study of the convergence criteria is also included.
For the specific case of a slab reactor criticality problem the
comparison between analytical and describing function solutions
revealed an excellent agreement.