The Wavelet Method For Differential Equation

From the 1930's, very many mathematical models from physics, engineering, chemistry, biology, economics. ect., were displayed as plane autonomous systems with limit cycles. The problem of limit cycles has become more and more important and has attracted the attention of many pure and applied mathematicians. A lot of problems in science can be eventually presented in the form of ordinary differential equation (ODE). It is possible for wavelet methods to solve plenty of questions, which are described by ODE, in autonomous control theory, Architectural engineers begin to pay close attention to wavelet analysis and its application. While the methods now we using to analysis and compute ODE have their own advantages, they also have some shortcomings. In this dissertation, based on the research of wavelet transformation and theory, the following problems are discussed: 1. The theory frame of wavelet approximation for the solution of ODE is put forward. The regularity of a class of compactly supported orthogonal wavelet is analyzed. A new method for the estimation of exponent regularity of the kind wavelet and related scale function is proposed. Based on the optimal approximation of spline interval wavelet, the interpolation error estimation is given. An approach to the study of multiresolution analysis and wavelet on finite interval with free knots is presented. A sort of algorithm of fast wavelet transformation is obtained to find Out wavelet approximation function. 2. The wavelet function is very suitable for solving the boundary value problem of ODE, not only for it characteristic of finite basis function, but has advantage of frequency method. We research the singularity of linear boundary value problem by making use of localizable distinguishing feature of wavelet. The improved wavelet-collocation method is given. 3. Combining with Lyapunov stability theory and taking advantage of wavelet approximation method, we propose a new algorithm of continuous adaptive control without system recognition on lines. To a kind of unknown nonlinear system, our algorithm can insurance the stability of closed loop. The trace error can be convergence into a neighborhood of zero. 4. How to manage the output signal analysis of architectural systems under the earthquake with wavelet analysis is a new subject. The systems of simple elastic particle (water tower, house, ect.) and multiple elastic particles(factory building, chimney. ect.) are analyzed with wavelet transformations, the relationship between the wavelet transformation of output signal with the pulse respond of architectural system and input signal of that system is put forward. The ideal theory basis is offered to analysis output signal of architectural under the pulse respond. 5. It is of significance to find out the limit cycles of plane autonomous systems both in theory and in application. First, we sum up three methods to look for the equation of limit cycle, two of them are our earliest results of researching limit cycles. Then, we study the place of limit cycle by means of harmonic wavelet . A sort of algorithm to compute the equation of limit cycle is designed. A new path leading to the limit cycle theory is hewed out.