可变阶分数微分方程的数值解法
Numerical solution of variable order fractional differential equations
AI总结：
本文提出了一种求解可变阶分数微分方程的数值方法，适用于线性和非线性方程，通过近似可变阶分数导数建立简单表达式，并通过示例验证了方法的有效性和准确性。
摘要：
本文提出了一种求解可变阶（VO）分数微分方程（FDE）的数值方法。该方法适用于线性和非线性VO-FDEs。采用Caputo型可变阶分数导数。首先建立了一个近似可变阶分数导数的简单表达式，然后基于此近似建立了解决VO-FDEs线性和非线性（包括显式和隐式）的程序。还处理了具有可变系数的VO-FDEs。通过求解描述可变阶分数振子响应的二阶VO-FDE（线性和非线性（Duffing））来说明该方法。然而，该方法可以简单地扩展到更高阶的VO-FDEs。除了有效性外，该方法还易于实现和编程。获得的结果验证了所开发方法的有效性和准确性。
英文摘要：
A method for the numerical solution of variable order (VO) fractional differential equations (FDE) is presented. The method applies to linear as well as to nonlinear VO-FDEs. The Caputo type VO fractional derivative is employed. First, an simple expression, which approximates the VO fractional derivative, is established and then a procedure based on this approximation is developed to solve VO-FDEs linear and nonlinear, both explicit and implicit. VO-FDEs with variable coefficients are also treated. The method is illustrated by solving the second order VO-FDE describing the response of the VO fractional oscillator, linear and nonlinear (Duffing). However, it can be straightforwardly extended to higher order VO-FDEs. The presented method, in addition to its effectiveness, is simple to implement and program on a computer. The obtained results validate the efficiency and accuracy of the developed method
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