Studies On Several Classes Of Strongly Nonlinear Differential Systems

There are many nonlinear problems in science and engineering.Usually,these nonlinear problems can be represented by nonlinear ordinary differential equations or nonlinear partial differential equations.In the end,a lot of problems need to solve the differential equation(including ordinary differential equation,partial differential equation,integral-differential equation and algebraic-differential equation).There are experimental methods and theoretical methods for studying nonlinear systems,and theoretical methods consist of quantitative analysis method and qualitative analysis method.For the nonlinear system,if the exact analytical solution of the differential equation can not be obtained,we often need to seek the numerical solutions.For ordinary differential systems,commonly used numerical method is Euler method and Runge-Kutta method,etc.For partial differential systems,commonly used numerical methods are finite difference method,finite element method,etc.Though the method of numerical solution for the ordinary differential system is well developed so far,the result is often discrete,which is not convenient for the subsequent analysis.Seeking the approximate exact solution of the nonlinear differential system has always been a difficult problem in nonlinear science and its application.However,the nonlinear systems which can seek approximate analytical solutions are limited to deterministic systems.In science and engineering,uncertain nonlinear systems are often encountered,such as stochastic systems,fuzzy systems and fuzzy stochastic systems.For this kind of nonlinear systems,it is impossible to seek an approximate exact solutiondirectly.In this thesis,several classes of strongly nonlinear systems are studied.The approximate solutions of several deterministic strongly nonlinear ordinary/partial differential equations are obtained by using the artificial neural network method;the reliability problem of a class of three-degrees-of-freedom strongly nonlinear systems under broadband random excitation is studied by using the stochastic averaging method of generalized harmonic function.We obtain the conditional reliability function and mean first-exit time,and the theoretical results are verified by numerical simulation.