一阶常系数微分方程的积分因子研究
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摘要
在现代电力和工程计算实例应用中,很多数学模型均是以一阶常系数微分方程的形式构建起来的.高于一阶常系数微分方程的初值或是边值问题均可等效转换为一阶常微分方程组问题,研究一阶常系数微分方程组的数值求解具有特殊的基础性意义.引入积分因子的概念,提出一阶常系数微分方程求解的基本方法.推导出四种常见的一阶常系数微分方程的积分因子的一般形式,并给出不同形式积分因子存在的充要条件,使求解过程更简单,更清晰.该方法可用于求解当前常规方法无法求解的微分方程.
In the application of modern electric power and engineering calculation examples,many mathematical models are constructed in the form of first-order constant coefficient differential equations.The initial value or boundary value problems of higher than the first-order differential equations with constant coefficients can be converted into the first-order ordinary differential equations.It is of special fundamental significance to study the numerical solution of first order differential equations with constant coefficients.By introducing the concept of integral factor,this paper proposes a basic method for solving firstorder differential equations with constant coefficients.The general forms of integral factors for four common first-order differential equations with constant coefficients are deduced,and the necessary and sufficient conditions for the existence of different forms of integral factors are given.This method can be used to solve the differential equations which cannot be done by conventional methods.
In the application of modern electric power and engineering calculation examples,many mathematical models are constructed in the form of first-order constant coefficient differential equations.The initial value or boundary value problems of higher than the first-order differential equations with constant coefficients can be converted into the first-order ordinary differential equations.It is of special fundamental significance to study the numerical solution of first order differential equations with constant coefficients.By introducing the concept of integral factor,this paper proposes a basic method for solving firstorder differential equations with constant coefficients.The general forms of integral factors for four common first-order differential equations with constant coefficients are deduced,and the necessary and sufficient conditions for the existence of different forms of integral factors are given.This method can be used to solve the differential equations which cannot be done by conventional methods.
引文
[1]何文正,陈晨,文竞舟.预应力圆弧曲梁振动微分方程推导及求解[J].人民长江,2016,47(17):88-92.
[2]盛钰婷.三阶常微分方程的某些非线性特征值问题的正解[J].科学中国人,2016,23(5):513-519.
[3]林文贤.一类具阻尼项的偶阶中立型微分方程的振动性分析[J].吉林大学学报:理学版,2017,55(5):1073-1076.
[4]张滑,刘春凤.分数阶R-L微分方程初值问题解的存在唯一性[J].数学的实践与认识,2016,46(3):267-272.
[5]蔡钢.Banach空间上有限时滞退化微分方程的适定性[J].数学物理学报,2018,38(2):264-275.
[6]王信峰,陈玉花,张莉.Banach空间非线性二阶混合型脉冲积分-微分方程的解[J].数学的实践与认识,2017,47(4):266-272.
[7]姚慧丽,刘婷.一类随机泛函积分微分方程的p-期望伪概自守温和解[J].哈尔滨理工大学学报,2017,22(4):135-142.
[8]刘宏影,吕学琴.再生核结合配置法求解一类带有积分边值条件的四阶非线性微分方程[J].数学物理学报,2016,36(5):928-936.
[9]王玥.复高阶微分方程的解[J].数学学报(中文版),2017,60(4):651-660.
[10]曹玉林,马建萍.基于微分方程的移动自组网病毒传播模型研究[J].计算机工程,2017,43(1):172-177.
[11]姚慧丽,刘婷.一类随机泛函积分微分方程的p-期望伪概自守温和解[J].哈尔滨理工大学学报,2017,22(4):135-142.
[12]魏会贤,董文雷,郭彦平.一类时标上三阶微分方程的渐进性态[J].数学的实践与认识,2017,47(11):307-312.
[13]徐鸿鹏,尹社会,张勇.一个超混沌类Lorenz系统的非线性动力学行为及计算机仿真[J].电子设计工程,2016,24(10):38-41.
[14]王平,范磊,梁铃,等.基于改进微分进化算法的发电机调速系统参数辨识[J].自动化与仪器仪表,2018(1):47-50.
[15]朱鹏程,程浪,蔡栩.基于伪微分算法的电液稳定平台研究[J].自动化与仪器仪表,2017(10):29-32.
[2]盛钰婷.三阶常微分方程的某些非线性特征值问题的正解[J].科学中国人,2016,23(5):513-519.
[3]林文贤.一类具阻尼项的偶阶中立型微分方程的振动性分析[J].吉林大学学报:理学版,2017,55(5):1073-1076.
[4]张滑,刘春凤.分数阶R-L微分方程初值问题解的存在唯一性[J].数学的实践与认识,2016,46(3):267-272.
[5]蔡钢.Banach空间上有限时滞退化微分方程的适定性[J].数学物理学报,2018,38(2):264-275.
[6]王信峰,陈玉花,张莉.Banach空间非线性二阶混合型脉冲积分-微分方程的解[J].数学的实践与认识,2017,47(4):266-272.
[7]姚慧丽,刘婷.一类随机泛函积分微分方程的p-期望伪概自守温和解[J].哈尔滨理工大学学报,2017,22(4):135-142.
[8]刘宏影,吕学琴.再生核结合配置法求解一类带有积分边值条件的四阶非线性微分方程[J].数学物理学报,2016,36(5):928-936.
[9]王玥.复高阶微分方程的解[J].数学学报(中文版),2017,60(4):651-660.
[10]曹玉林,马建萍.基于微分方程的移动自组网病毒传播模型研究[J].计算机工程,2017,43(1):172-177.
[11]姚慧丽,刘婷.一类随机泛函积分微分方程的p-期望伪概自守温和解[J].哈尔滨理工大学学报,2017,22(4):135-142.
[12]魏会贤,董文雷,郭彦平.一类时标上三阶微分方程的渐进性态[J].数学的实践与认识,2017,47(11):307-312.
[13]徐鸿鹏,尹社会,张勇.一个超混沌类Lorenz系统的非线性动力学行为及计算机仿真[J].电子设计工程,2016,24(10):38-41.
[14]王平,范磊,梁铃,等.基于改进微分进化算法的发电机调速系统参数辨识[J].自动化与仪器仪表,2018(1):47-50.
[15]朱鹏程,程浪,蔡栩.基于伪微分算法的电液稳定平台研究[J].自动化与仪器仪表,2017(10):29-32.