In this paper, we study the finite volume discretization method for balanced fractional diffusion equations where the fractional differential operators are comprised of both Riemann-Liouville and Caputo fractional derivatives. The main advantage of this approach is that new symmetric positive definite Toeplitz-like linear systems can be constructed for solving balanced fractional diffusion equations when diffusion functions are non-constant. It is different from non-symmetric Toeplitz-like linear systems usually obtained by existing numerical methods for fractional diffusion equations. The preconditioned conjugate gradient method with circulant and banded preconditioners can be applied to solve the proposed symmetric positive definite Toeplitz-like linear systems. Numerical examples, for both of one- and two- dimensional cases, are given to demonstrate the good accuracy of the finite volume discretization method and the fast convergence of the preconditioned conjugate gradient method.

在本文中，我们研究了分数微分算子同时由Riemann-Liouville和Caputo分数阶导数组成的平衡分数阶扩散方程的有限体积离散化方法。这种方法的主要优点是，可以构造新的对称正定Toeplitz类线性系统，用于在扩散函数非恒定时求解平衡分数阶扩散方程。它不同于通常通过分数扩散方程的现有数值方法获得的非对称Toeplitz线性系统。具有循环和带状预处理器的预处理共轭梯度方法可用于求解所提出的对称正定Toeplitz线性系统。一维和二维情况的数值示例，