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

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Journal of Vibration and Shock ›› 2021, Vol. 40 ›› Issue (18) : 119-123.

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

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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.

Key words

single parameter / Bernstein series / variable fractional order / nonlinear boundary value problem(NBVP)

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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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