﻿ 复合Bessel函数零点数值计算方法及分布规律
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 山东大学学报(工学版)  2018, Vol. 48 Issue (1): 71-77  DOI: 10.6040/j.issn.1672-3961.0.2017.085 0

### 引用本文

JI Anzhao, WANG Yufeng, LIU Xuefen. Numerical calculation method and distribution law of zero points of the compound Bessel function[J]. Journal of Shandong University (Engineering Science), 2018, 48(1): 71-77. DOI: 10.6040/j.issn.1672-3961.0.2017.085.

### 文章历史

Numerical calculation method and distribution law of zero points of the compound Bessel function
JI Anzhao, WANG Yufeng, LIU Xuefen
College of Energy Engineering, Longdong University, Qingyang 745000, Gansu, China
Abstract: In order to solve the problem of computing the zeros of compound Bessel functions, a modified optimization algorithm of the particle swarm and the quantum-behaved particle swarm were proposed. Most of the zero points of the compound Bessel function could be calculated by modified algorithm in the finite interval. In order to improve the searching ability of the zero points, the quantum-behaved particle swarm optimization algorithm with crossover operator was modified by using cross operator operation in combination with the characteristics of the two former algorithms. All the zero points of the compound Bessel function in the finite interval were found with the modified version of algorithm. The modified version of algorithm was faster with convergence rate and higher with zero points calculation accuracy. The calculation results showed that, except for the former three zero points, the following zero points showed linear relationship with their sequence on double logarithmic coordinate axis under the same parameters of the compound Bessel function. The straight-line fitting of the zero points and their sequence from different parameters was calculated. The results showed that the correlation coefficient was 99.99% and the relative error of zero point fitting was less than 0.5%, which could full fit the requirement of engineering calculation.
Key words: compound Bessel function    finite reservoirs    multimodal function    quantum particle swarm    crossover operator
0 引言

1 基本模型及解析

 $p\left( {r,t} \right) = {p_{\rm i}} - \frac{{q\mu }}{{2\pi kh}}\left\{ \frac{{2\eta t}}{{(r^2_{\rm De} - 1)r^2_{\rm w}}} + \frac{{r^2_{\rm De}}}{{r^2_{\rm De} - 1}}\left(\ln \frac{{{r_{\rm De}}}}{{{r_D}}} + \frac{{r^2_D}}{{2r^2_{\rm De}}} \right)- \frac{{3r^4_{\rm De} - 4r^2_{\rm De}\ln {r_{\rm De}} - 2r^2_{\rm De} - 1}}{{4{{(r^2_{\rm De} - 1)}^2}}} - \right.\\ \left. \quad \pi \sum\limits_{n = 1}^\infty {\frac{{\exp( - \alpha ^2_n\frac{{\eta t}}{{r^2_{\rm w}}}){J_1}({r_{\rm De}}{\alpha _n})[{Y_1}({\alpha _n}){J_0}({r_D}{\alpha _n}) - {J_1}({\alpha _n}){Y_0}({r_D}{\alpha _n})]}}{{{\alpha _n}[J^2_1({\alpha _n}{r_{\rm De}}) - J^2_1({\alpha _n})]}}} \right\},$ (1)

 $q\left( t \right) = \frac{{4\pi hk({p_{\rm i}} - {p_{\rm wf}}({r_{\rm w}},t))}}{u}\sum\limits_{n = 1}^\infty {\frac{{\exp( - \frac{{\alpha ^2_nkt}}{{\phi \mu {c_t}r^2_w}})[{Y_1}({r_{\rm De}}{\alpha _n}){J_1}({\alpha _n}) - {Y_1}({\alpha _n}){J_1}({\alpha _n}{r_{\rm De}})]}}{{{r_{\rm De}}[{Y_0}({r_{\rm De}}{\alpha _n}){J_0}({\alpha _n}) - {J_0}({r_{\rm De}}{\alpha _n}){Y_0}({\alpha _n})] + {Y_1}({\alpha _n}){J_1}({r_{\rm De}}{\alpha _n}) - {Y_1}({r_{\rm De}}{\alpha _n}){J_1}({\alpha _n})}}} ,$ (2)

2 复合Bessel函数的特性及零点求解策略

 $f(x,{r_{\rm De}}) = {Y_1}({r_{\rm De}}x){J_0}\left( x \right) - {Y_0}\left( x \right){J_1}({r_{\rm De}}x),$ (3)

 图 1 不同参数rDe情况下的复合Bessel函数示意图 Figure 1 Sketch map of compound Bessel function in different parameters rDe

 $F(x,{r_{\rm De}}) = |f(x,{r_{\rm De}})|。$ (4)
3 复合Bessel函数零点求解算法

3.1 修正普通粒子群算法

 $x^j_{i,t + 1} = x^j_{i,t} + \omega v^j_{i,t} + {c_1}r^j_1\left( {p^j_{i,t} - x^j_{i,t}} \right),$ (5)

3.2 修正量子粒子群算法

 $x^j_{i,t + 1} = \beta ^j_{i,t}p^j_{i,t} + \left( {1 - \beta ^j_{i,t}} \right)p^j_{g,t} \pm a|c^j_t - x^j_{i,t}|\ln\left( { - \frac{1}{{u^j_{i,t}}}} \right),$ (6)

 $x^j_{i,t + 1} = \beta ^j_{i,t}p^j_{i,t} + \left( {1 - \beta ^j_{i,t}} \right)x^j_{i,t} \pm a|p^j_{i,t} - x^j_{i,t}|\ln\left( { - \frac{1}{{u^j_{i,t}}}} \right)。$ (7)
3.3 修正带交叉算子量子粒子群算法

 $x^j_{i,t} = {\beta _{i,t}}p^j_{i,t} + \left( {1 - {\beta _{i,t}}} \right)x^j_{i,t} + \frac{{p^j_{b,t} - p^j_{c,t}}}{2} + a\left| {\frac{{p^j_{b,t} - p^j_{c,t}}}{2}} \right|\ln\left( { - \frac{1}{{u^j_{i,t}}}} \right),$ (8)

4 复合Bessel函数零点分布规律 4.1 复合Bessel函数零点计算结果

 $a(t)=\frac{1.58}{1+\exp(0.02t)},$ (9)

 图 2 修正PSO/QPSO/CQPSO算法迭代次数与绝对误差关系 Figure 2 The relationships between number iterations andabsolute error of PSO/QPSO/CQPSO modified algorithm
4.2 复合Bessel函数零点分布规律

 图 3 不同参数rDe情况下复合Bessel函数零点分布 Figure 3 Distribution of zeros of compound Bessel function indifferent parameters rDe

 $\lg\left( {{\alpha _i}} \right) = k\lg\left( i \right) + m,$ (10)

5 结论及建议

(1) 复合Bessel函数f(x, rDe)是以rDe为参数的多峰函数, 函数零点有无穷多个, 当参数rDe取值较小时, 零点值较大, 且分布相对较稀疏; 而当参数rDe取值较大时, 零点值较小, 且分布密集。在零点求取过程中, 若参数rDe较小, 可适当放大搜索范围; 反之, 则要缩小搜索范围。

(2) 对PSO算法和QPSO算法进行个体历史最优位置和群体最优位置加权为吸引点的计算方法进行修正, 同时对QPSO算法进行分析基础之上对CQPSO算法进行修正, 将这3种修正算法求解的结果与二分法逐步搜索算法进行对比, 说明3种算法在求解复合Bessel函数零点时是可行的。

(3) 通过修正CQPSO算法与其他两种算对比, 修正CQPSO算法收敛速度快, 且计算精度高, 说明修正CQPSO算法中的交叉算子操作在优化多峰函数时能够取得良好的效果。

(4) 复合Bessel函数除去前3个零点, 后续零点与零点次序在双对数坐标系中满足直线关系, 对同一参数rDe拟合结果表明:零点与零点次序的线性相关程度高, 拟合零点与求解零点相对误差不超过0.5%, 因此可通过给出3种优化算法获取任意参数rDe前面有限个零点, 然后通过直线方程拟合可以获取函数其他任意零点, 在工程计算避免复杂的Bessel函数的计算。

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