一种单参变量Bernstein序列及其在含变分数阶非线性边界值问题中的应用

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振动与冲击 ›› 2021, Vol. 40 ›› Issue (18) : 119-123.

论文

王春秀1 ，周星德1 ，方立雪2，金奕潼1，石贤增1

作者信息 +

Single parameter bernstein series and its application to nonlinear boundary value problems with variable fractional order

WANG Chunxiu1，ZHOU Xingde1，FANG Lixue2，JIN Yitong1，SHI Xianzeng1

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摘要

针对含变分数阶的非线性边界值问题(NBVP)，通过每项增加一个参变量的方式，该研究提出了一种含单参变量Bernstein序列(SBS)，把其用于问题的求解，具体过程如下：引入单参变量构造SBS，进而把变分数阶项转换为以SBS为基的多项式表示；对于非线性边界值问题，目标函数中的积分采用高斯勒让德积分方法近似表示；考虑到非线性优化时存在多解现象，引入遗传算法以期同时获得所有次优解，进而以次优解作为初始值，采用软件MATLAB优化模块获得最优解；给出二个仿真实例。从计算结果来看，该方法求解的精度与Hassani等的研究结果一致，且均比采用同样项数的Bernstein多项式精度高。

Abstract

Aiming at solving nonlinear boundary value problems(NBVPs) with variable fractional order, a novel single Bernstein sequence (SBS) where a parameter was attached to each item was proposed.The specific process for solving the problems was as follows: firstly, the SBS was constructed with the single parameter, and the variable fraction order term was transformed into a polynomial expression based on SBS.The integral in the objective function of the nonlinear boundary value problems was approximated into an analytical expression by the Gauss-Legendre integral method.Then, considering the phenomenon of multiple solutions existing in nonlinear optimization procedure, a genetic algorithm was introduced to obtain all the sub-optimal solutions at the same time.Finally, the sub-optimal solutions were taken as the initial values and the optimal solutions were obtained by using the MATLAB optimization module.Two simulation examples were provided.The results indicate that the accuracy of the method is consistent with those of Hassani, and is higher than that by using a Bernstein polynomial with the same number of terms.

导出引用

王春秀,周星德，方立雪，金奕潼，石贤增. 一种单参变量Bernstein序列及其在含变分数阶非线性边界值问题中的应用[J]. 振动与冲击, 2021, 40(18): 119-123

WANG Chunxiu，ZHOU Xingde，FANG Lixue，JIN Yitong，SHI Xianzeng. Single parameter bernstein series and its application to nonlinear boundary value problems with variable fractional order[J]. Journal of Vibration and Shock, 2021, 40(18): 119-123

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