一类非线性二阶常微分方程边值问题的解析解

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ｄ^２ｙ／ｄｔ^２＝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≈ｅ^ｙ／２，故求解Ｐｏｉｓｓｏｎ－Ｂｏｌｔｚｍａｎｎ方程近似解问题转化为求ｄ^２ｙ／ｄｔ^２＝ｅ^ｙ／２解析解。因此，寻找常微分方程解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ｄ^２ｙ／ｄｔ^２＝ｅ^ｙ／ａ在等边值条件下的解析解表达式。

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

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.