The present paper is devoted to constructing L2 type difference analog of the Caputo fractional derivative. The fundamental features of this difference operator are studied and it is used to construct difference schemes generating approximations of the second and fourth order in space and the $(3-\alpha )$th-order in time for the time fractional diffusion equation with variable coefficients. Difference schemes were also constructed for the variable-order diffusion equation and the generalized fractional-order diffusion equation of the Sobolev type. Stability of the schemes under consideration as well as their convergence with the rate equal to the order of the approximation error are proven. The received results are supported by the numerical computations performed for some test problems.

本文致力于构造Caputo分数阶导数的L2型差分模拟。研究了这种差分算子的基本特征，并用它来构造差分格式，在空间中生成二阶和四阶近似值，以及$(3-\alpha )$具有可变系数的时间分数扩散方程的时间阶数。还构建了变阶扩散方程和Sobolev型广义分数阶扩散方程的差分格式。证明了所考虑方案的稳定性以及它们以等于近似误差阶数的速率收敛。对一些测试问题进行的数值计算支持了接收到的结果。