A New Perturbative Technique for Solving Integro-Partial Differential Equations

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.