A wavelet operational method for solving fractional partial differential equations numerically
A wavelet operational method for solving fractional partial differential equations numerically
复制标题
DOI：
10.1016/j.amc.2009.03.066
复制
发表时间：
2009-08
期刊：
影响因子：
--
通讯作者：
Jiunn-Lin Wu
中科院分区：
文献类型：
--
作者：
Jiunn-Lin Wu
分数阶微积分是导数和积分在非整数阶的推广，涉及分数阶微积分算子的偏微分方程称为分数阶偏微分方程。它们在科学和工程中有许多应用。然而，不仅解析解存在的情况有限，而且数值方法是非常复杂和困难的。本文建立了基于正交函数运算矩阵的模拟方法。我们在一个统一的框架内制定了一体化的操作矩阵。利用积分运算矩阵，提出了求解线性分数阶偏微分方程的一种新的数值方法。在该方法中，我们（1）使用Haar小波;（2）建立一个Lyapunov型矩阵方程;（3）得到适合计算机编程的代数方程。通过两个算例验证了该方法的简单性、清晰性和有效性。
Fractional calculus is an extension of derivatives and integrals to non-integer orders, and a partial differential equation involving the fractional calculus operators is called the fractional PDE. They have many applications in science and engineering. However not only the analytical solution existed for a limited number of cases, but also the numerical methods are very complicated and difficult. In this paper, we newly establish the simulation method based on the operational matrices of the orthogonal functions. We formulate the operational matrix of integration in a unified framework. By using the operational matrix of integration, we propose a new numerical method for linear fractional partial differential equation solving. In the method, we (1) use the Haar wavelet; (2) establish a Lyapunov-type matrix equation; and (3) obtain the algebraic equations suitable for computer programming. Two examples are given to demonstrate the simplicity, clarity and powerfulness of the new method.