A new perturbative technique for solving integro-partial differential equations

Integro-partial differential equations occur in many contexts in mathematic al physics. Typical examples include time-dependent diffusion equations con taining a parameter (e.g., the temperature) that depends on integrals of th e unknown distribution function. The standard approach to solving the resul ting nonlinear partial differential equation involves the use of predictor- corrector algorithms, which often require many iterations to achieve an acc eptable level of convergence. In this paper we present an alternative proce dure that allows us to separate a family of integro-partial differential eq uations into two related problems, namely (i) a perturbation equation for t he temperature, and (ii) a linear partial differential equation for the dis tribution function. We demonstrate that the variation of the temperature ca n 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 t he approach is confirmed by comparing the (input) continued-fraction temper ature profile with the (output) temperature computed by integrating the res ulting distribution function. (C) 1999 American Institute of Physics. [S002 2-2488(99)03410-6].