A new perturbative technique for solving integro-partial differential equations

Abstract

Integro-partial differential equations occur in many contexts in mathematical physics. Typical examples include time-dependent diffusion equations containing a parameter ͑e.g., the temperature͒ 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 ͑i͒ a perturbation equation for the temperature, and ͑ii͒ 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 ͑input͒ continued-fraction temperature profile with the ͑output͒ temperature computed by integrating the resulting distribution function.

Exact solution for the Green's function describing time-dependent thermal Comptonization

“…We therefore conclude that the variation of the electron temperature in the corona must be included in models for the spectral evolution occurring during the flare. If the transient is very intense, the gas is expected to be radiation‐dominated, and the electron temperature should therefore track the inverse‐Compton temperature of the radiation (Becker & Begelman 1986; Becker 1999). This issue has been explored by Böttcher (2001), who performed Monte Carlo simulations of disc/corona models with temperatures that responded self‐consistently to the inverse‐Compton cooling.…”