一类非线性二阶常微分方程边值问题的解析解  

Analytical Solution of Boundary Value Problems for a Class of Second Order Nonlinear Ordinary Differential Eqations

作　　者：刘国志[1] 刘倩囡 

机构地区：[1]辽宁石油化工大学理学院,辽宁抚顺113001 [2]华北电力大学(北京)能源动力与机械工程学院,北京102206

出　　处：《辽宁石油化工大学学报》2012年第3期88-90,共3页Journal of Liaoning Petrochemical University

基　　金：国家自然科学基金资助项目(50771052)

摘　　要：在计算胶粒间双电层相互作用能时,需要计算胶粒间的电位分布,而两个胶粒间的电位分布满足Poisson-Boltzmann方程d2 y/dt2=sinhy,这是一个二阶非线性常微分方程,无法求出其解析解。但是当y≥1时,sinhy≈ey/2,故求解Poisson-Boltzmann方程近似解问题转化为求d2 y/dt2=ey/2解析解。因此,寻找常微分方程解析解的问题是工程实际的需要。通过对二阶非线性常微分方程边值问题的研究,给出了一类二阶非线性常微分方程d2 y/dt2=ey/a在等边值条件下的解析解表达式。As calculating the double layer interaction between colloidal particles,it is necessary to calculate potential distribution between colloidal particles.Then potential distribution between two colloidal particles meet the Poisson-Boltzmann equation d2y/dt2=sinhy.Due to its nonlinear nature,no analytical solution of the equation has been found,however when y≥1,sinhy≈ey/2,the approximate solution of Poisson-Boltzmann equation is transformed into solving analytical solution of d2y/dt2=ey/2.Hence,finding analytical solution of nonlinear ordinary differential equation is the need of engineering.Therefore through researching boundary value problem of second order nonlinear ordinary differential equation,analytical solution of a class of second order nonlinear ordinary differential equation d2y/dt2=ey/a was obtained under the conditions of equal boundary value.

关 键 词：微分方程 POISSON-BOLTZMANN方程 解析解 

分 类 号：O175.8[理学—数学]