离散分数阶微积分的矩阵方法II：部分分数阶微分方程
Matrix approach to discrete fractional calculus II: partial fractional differential equations
摘要：
提出了一种新的离散化方法，该方法能方便地离散具有任意实阶导数（所谓分数阶导数）和时滞的偏微分方程，并以各种类型的分数阶扩散方程的数值解为例加以说明.所建议的方法是Podlubny矩阵方法的发展（Fractional Calculus and Applied Analysis，vol.3，no.4，2000，359- 386）。四个例子的数值解的分数扩散方程的各种组合的时间/空间分数导数（整数/整数，分数/整数，整数/分数，分数/分数）相对于时间和空间变量，以说明如何简单和一般的建议的方法。第五个例子说明了该方法同样可以简单地用于含时滞的分数阶微分方程。一组MATLAB例程的方法，以及用于解决的例子的示例代码的实施已经开发。
英文摘要：
A new method that enables easy and convenient discretization of partial differential equations with derivatives of arbitrary real order (so-called fractional derivatives) and delays is presented and illustrated on numerical solution of various types of fractional diffusion equation. The suggested method is the development of Podlubny's matrix approach (Fractional Calculus and Applied Analysis, vol. 3, no. 4, 2000, 359--386). Four examples of numerical solution of fractional diffusion equation with various combinations of time/space fractional derivatives (integer/integer, fractional/integer, integer/fractional, and fractional/fractional) with respect to time and to the spatial variable are provided in order to illustrate how simple and general is the suggested approach. The fifth example illustrates that the method can be equally simply used for fractional differential equations with delays. A set of MATLAB routines for the implementation of the method as well as sample code used to solve the examples have been developed.
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