求解积分-偏微分方程的一种新的摄动方法
A New Perturbative Technique for Solving Integro-Partial Differential Equations
摘要：
积分-偏微分方程出现在数学物理的许多背景中。典型的例子包括包含参数（例如，温度）取决于未知分布函数的积分。求解所得非线性偏微分方程的标准方法涉及使用预测-校正算法，其通常需要多次迭代以实现可接受的收敛水平。在本文中，我们提出了一个替代的程序，使我们能够分离成两个相关的问题，即（i）的扰动方程的温度，和（ii）的分布函数的线性偏微分方程的积分-偏微分方程的家庭。我们证明了温度的变化可以通过求解微扰方程来确定，然后再求解分布函数。收敛的结果，得到的温度通过重铸的发散扰动扩展作为一个连续的分数。一旦确定了温度变化，则通过使用标准技术求解剩余的线性偏微分方程来获得分布函数的自洽解。该方法的有效性确认通过比较（输入）连续分数温度分布与（输出）温度计算所得的分布函数进行整合。
英文摘要：
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.
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