On Solutions Of Several Classes Of Differential Dynamical Systems

In the research of science and engineering,nonlinear problems are ubiquitous,and often be represented by differential equations(including ordinary differential equation,partial differential equation,differentialalgebraic equation,delay differential equation,stochastic differential equation,etc.)To solve these nonlinear problems,the solution of the differential equations is needed.Quantitative and qualitative analysis are two kinds of theoretical methodologies to study the dynamical behavior of a differential system.A direct quantitative way is to solve the corresponding differential equation analytically or numerically.However,the exact analytical solutions of differential equations are hard to obtain.Particularly in the case of nonlinearity,only numerical solutions are obtainable.So far,various numerical methods have been well developed to solve the differential equations.For ordinary differential equations,for example,Euler method and Runge-Kutta method are commonly used.For partial differential equations,commonly used numerical methods are finite difference method,finite element method,etc.The numerical solutions are often discrete,which is not convenient for the subsequent analysis.So,in nonlinear science,it is always an important subject and research focus to find the approximate analytical solution of a differential system.In this thesis,the approximate analytical solutions of several classes of differential dynamical systems are studied by using intelligent method of soft computation.The approximate solutions of several special differential-algebraic equations are obtained by using the artificial neural network method.According to the structure of neural network model,the numerical method of approximate analytical solution is founded with support vector machine method to solve weight directly.According to the stochastic process theory and maximum likelihood estimation,the drift coefficient,diffusion coefficient and weak form of solution of It? stochastic differential equation are obtained by using the neural network method and genetic programming in the case of known stochastic process trajectory.The approximate analytical solutions of the above examples are obtained and compared with the exact solution or the result from the Runge-Kutta method.High accuracy of the proposed method is demonstrated.Also,the probability distribution and autocorrelation of the stochastic solution are verified.