应用Bernstein多项式求解一类分数阶微分方程

给出了基于 Bernstein多项式求解分数阶微分方程的配置方法。首先，在 Bernstein级数的截断式中用tα（0〈α〈1）代替t得到分数阶Bernstein级数截断式，采用Caputo分数阶导数构建分数阶Bernstein级数截断式的矩阵形式。其次，把方程中的每一项用分数阶Bernstein级数截断式转换成矩阵形式，选取配置点，得到相应于非线性代数方程的基本矩阵方程。最后得到由条件矩阵形式和基本矩阵方程构成的新方程组，其解给出了截断项为N的近似解，同时给出了基于残余函数的误差分析。举例说明了这种方法的有效性和可行性。

The Numerical Solutions of a Class of Fractional Differential Equations by Means of the Bernstein Polynomials

A collocation method based on the Bernstein polynomials is presented for a class of fractional differential equations. By replacing t with tα（0〈α〈1） in the truncated Bernstein series, the truncated fractional Bernstein series is obtained and then it is transformed into the matrix form. By using Caputo fractional derivative, the matrix forms of the fractional derivatives are constructed for the truncated fractional Bernstein series. We convert each term of the problem to the matrix form by means of the truncated fractional Bernstein series. By using the collocation points, we have the basic matrix equation which corresponds to a system of nonlinear algebraic equations. Lastly, a new system of nonlinear algebraic equations is obtained by using the matrix forms of the conditions and the basic matrix equation. The solution of this system gives the approximate solution for the truncated limited N. An error analysis technique based on residual function is developed and applied to an example to demonstrate the validity and applicability of the proposed method.