论文标题
直线差分方程的一般精确闭合溶液
A General Exact Closed-Form Solution for Nonlinear Differential Equation of Pendulum
论文作者
论文摘要
在本文中，研究了摆的非线性微分方程，以找到精确的封闭形式解决方案，满足控制方程以及初始条件。引入了建议方法中使用的新概念。关于任何任意系统的控制方程表示其固有的属性，这表明非线性项会导致系统具有变量标识。因此，原始函数作为变量在解决方案中的变量包括在内，可以考虑到管理方程的变化。为了找到精确的封闭形式解决方案，非线性微分方程的变化趋于零，在这种情况下，具有局部线性微分方程的系统具有确定的身份，具有确定的局部答案。结果表明，一般答案是表面上的任意曲线，一种新开发的概念称为超级功能，不同的初始条件将不同的曲线作为特定的解决方案。将结果与有限差的结果进行比较显示了与任何任意振幅和初始条件的确切一致性。
In the present paper, the nonlinear differential equation of pendulum is investigated to find an exact closed form solution, satisfying governing equation as well as initial conditions. The new concepts used in the suggested method are introduced. Regarding the fact that the governing equation for any arbitrary system represents its inherent properties, it is shown that the nonlinear term causes that the system to have a variable identity. Hence, the original function is included as a variable in the solution to can take into account the variation of governing equation. To find the exact closed form solution, the variation of the nonlinear differential equation tends to zero, where in this case the system with a local linear differential equation has a definite identity with a definite local answer. It is shown that the general answer is an arbitrary curve on a surface, a newly developed concept known as super function, and different initial conditions give different curves as particular solutions. The comparison of the results with those of finite difference shows an exact agreement for any arbitrary amplitude and initial conditions.
