  Integro-partial differential equations occur in many contexts in mathematical
physics. Typical examples include time-dependent diffusion equations containing
a parameter (e.g., the temperature) that depends on integrals of the unknown
distribution function. The standard approach to solving the resulting nonlinear
partial differential equation involves the use of predictor-corrector
algorithms, which often require many iterations to achieve an acceptable level
of convergence. In this paper we present an alternative procedure that allows
us to separate a family of integro-partial differential equations into two
related problems, namely (i) a perturbation equation for the temperature, and
(ii) a linear partial differential equation for the distribution function. We
demonstrate that the variation of the temperature can be determined by solving
the perturbation equation before solving for the distribution function.
Convergent results for the temperature are obtained by recasting the divergent
perturbation expansion as a continued fraction. Once the temperature variation
is determined, the self-consistent solution for the distribution function is
obtained by solving the remaining, linear partial differential equation using
standard techniques. The validity of the approach is confirmed by comparing the
(input) continued-fraction temperature profile with the (output) temperature
computed by integrating the resulting distribution function.
