The method based on block pulse functions (BPFs) has been proposed to solve different kinds of fractional differential equations (FDEs). However, high accuracy requires considerable BPFs because they are piecewise constant and not so smooth. As a result, it increases the dimension of operational matrix and computational burden. To overcome this deficiency, a novel numerical method is developed to solve fractional differential equations. The method is based upon hybrid of BPFs and Bernstein polynomials (HBBPs), which are piecewise smooth. The HBBPs operational matrix of fractional‐order integral is derived to reduce the FDEs to a system of algebraic equations. Then the numerical solution of the FDEs is obtained through solving the system of algebraic equations. The convergence analysis is conducted for the suggested scheme, and the upper bound of error of the solution is given. Finally, illustrative examples are presented to demonstrate the validity, applicability, and efficiency of the proposed technique in contrast with other approaches.

中文翻译：

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使用块脉冲函数和Bernstein多项式求解分数阶微分方程

提出了一种基于块脉冲函数（BPF）的方法来求解不同种类的分数阶微分方程（FDE）。但是，高精度需要相当大的BPF，因为它们是分段恒定的，并且不是那么平滑。结果，它增加了运算矩阵的尺寸和计算负担。为了克服这一缺陷，开发了一种新颖的数值方法来求解分数阶微分方程。该方法基于分段平滑的BPF和Bernstein多项式（HBBP）的混合。导出分数阶积分的HBBP运算矩阵，以将FDE简化为代数方程组。然后通过求解代数方程组来获得FDE的数值解。对建议的方案进行了收敛性分析，给出了解的误差上限。最后，给出了说明性示例，以证明与其他方法相比，所提出技术的有效性，适用性和效率。