Study On The Periodic Solution Of The Boundary Value Problem Of The Nonlinear Differential Equation

The nonlinear differential equation, especially, the second-order one is paid closed attention because they relate wide field. The existence and unique of the periodic solution of these problems are always one of the central issue of the research. People start to seek numerical solution recently. The standard solution way is simple, accurate and fast, so the numerical continual method become the first choose. In second-order nonlinear differential equation, the Lienard and Duffing type equations are applied widely in atmosphere. But people always get special type equation when studying actual problem. They usually use approximate ways to have solution. This makes the deviation of solution in some degree. In this paper, we will discuss the existing condition of the periodic solution of Duffing type equation with damping term. We use homeomorphism and fixed point method to prove the periodic solution's existence. We try to loose its condition, analysis and then get the existing condition of periodic solution according to defining atmospheric dynamics equation. Then, we use numerical continual method to get the periodic numerical solution. There is deep physical background in the boundary value problem of semi-linear elliptic equation. The problems are widely applied in springiness mechanics, electromagnetism, filuid mechanics. If we can get the accurate numerical solution, this has important value in theory and number. We can use the numerical continual method to make overall soften algorithms on the base of expanding soften domain in order to get the periodic numerical solution.