摘要
有限区域初边值问题的小波方法中，解的小波形式在边界上的构造以及对问题边界条件的处理一直是这一方法中所关注的关键问题，目前对此尚未建立统一的途径。为此本文开展了以下工作：
1)．首次提出了一种改进的、对于有限区域初边值问题均适用的小波方法。该方法对边值问题和初值问题的解给出了统一的小波形式，即，通过边界处和区域内小波系数构造Lagrange插值多项式对区域外的小波系数进行外插，从而实现了问题的解向区域外的连续延拓并且完全由区域内的小波系数表征。在此基础之上，对问题的解给出了显式含有所有边界条件(即函数值及其导函数值)的小波形式，以此实现了边界条件的准确处理以及力学问题的求解。
2)．对于梁板结构静／动力学问题，基于上述小波形式以及梁板结构的变分原理，建立了能够统一处理梁、板静／动力学问题的各类齐次、非齐次边界条件和屈曲问题的各类边界支撑条件的小波—变分法，并得到了统一形式的离散静／动力学方程和屈曲问题的特征值方程。由于本文提出的改进小波形式及其小波基函对任意形式的边界条件保持不变，对于给定的梁板结构的横截面和材料性质，离散静／动力学方程、特征值方程无论形式上，还是方程的系数矩阵的值对于各类齐次、非齐次边界条件均保持不变性；同时，本文提出的小波形式中，小波系数是独立的，从而由变分法所建立的离散静／动力学方程、特征值方程自动封闭而且是适定的，对于任意给定的齐次或者非齐次边界条件，离散静／动力学方程存在唯一解。这一方法解决了现有梁板结构静／动力学的小波—Galerkin方法和小波—有限元方法处理非齐次边界条件时存在的离散静／动力学方程不适定的问题，以及对于不同形式的边界条件，离散方程的形式和系数矩阵的值不统一的问题。
3)．基于本文提出的改进小波形式，通过配点方法，将有限时域[0，T]上的多自由度系统非线性振动的初值问题转化求解一组非线性代数方程组，从而建立了非线性振动初值问题的多分辨率、自适应小波配点方法，并针对非线性性给出了基于同伦算法的小波自适应算法和程序。基于本文的小波形式，多自由度系统的非线性振动方程(通常为二阶时间常微分方程组)的初值问题可被直接转化为一组适定的非线性代数方程，而无需事先将其转换为状态方程。因此，与现有的基于状态方程的小波配点方法相比，本文方法所得的非线性代数方程组的未知数个数仅为前者的1／2；从而大大降低了计算所需的存储空间，至多为现有小波配点方法所需存储空间的1／2；同时大大提高了计算效率，至少是现有小波配点方法的2倍。
4)．很多振动控制系统可以表示为具有时滞的多自由度动力系统，本文尝试了将小波自适应方法应用到时滞系统的求解和稳定性分析。首先基于现有的小波自适应分解和重构算法得到了Laplace逆变换的自适应小波数值计算公式，应用该公式将多自由度多时滞线性振动系统近似等价的转换成映射动力系统。由此不仅可求解原时滞系统的初值问题，同时可给出其最右特征根近似值，从而判定其稳定性，并且能够在选定的系统参数平面上给出近似的稳定区域及其边界。与已有的线性多步法相比，本文可避免将时滞振动方程转换成状态方程，从而所得的映射矩阵的阶数将小于前者。
For wavelet methods in solving initial-boundary-value problems on finite domains, the construction of wavelet formation of solutions and the treatment of boundary conditions of the problem are the key to solve problems. However, no general method has been found to handle the construction of wavelet formations and the treatment of boundary conditions of the problem. Therefore, the following works are carried out in this thesis:
1). A modified wavelet method is proposed for the first time, which is applicable for both initial- and boundary-value problems on finite domains. This method presents a general form of wavelet expansion for the solutions of initial- and boundary-value problems, by extrapolating external wavelet coefficients by boundary and inner ones, in which way the continuity of the solutions' wavelet expansions are preserved near the domain boundaries. Based on this, the values of the solution and its first-order derivatives at the domain boundaries are explicitly combined into the newly presented wavelet expansions of solutions, by which many mechanic problems can be solved with wavelet methods.
2). Based on the modified wavelet formation mentioned above, wavelet-variational methods are established for the static/dynamic problems of beams and plates, and discrete static/dynamic equations and characteristic equations are derived in a general form, respectively, in which all types of homogeneous and non-homogeneous boundary conditions and boundary support conditions are treated in a general way. Because the wavelet formation for the deflection of beams and plates is general for all boundary conditions, not only the form of discrete static/dynamic equations and the characteristic equations, but also the the coefficient matrices of these equations are invariant to boundary conditions; on the other hand, the wavelet coefficients of the proposed modified wavelet formation are independent from each other, so the derived discrete static/dynamic equations and characteristic equations are well-posed, respectively, and for any given boundary condition the discrete static/dynamic equation has unique solution. The proposed method overcomes the defficiencies of current wavelet-Galerkin methods and wavelet-FEMs for the static/dynamic problems of beams and plates: the non-uniformity of the discrete static/dynamic equations for different types of boundary conditions, and the ill-posedness of equations as non-homogeneous boundary conditions are considered.
3). On the base of the proposed modified wavelet formation in this thesis, an initial-value problem of the nonlinear vibration of a MDOF system on time domain [0, T] is equivalently reduced to a group of nonlinear algebraic equations by applying collocation scheme; thus, a modified multi-resolution and a modified adaptive wavelet collocation method are established, respectively, for the initial-value problems of the nonlinear vibration of MDOF systems; and an adaptive algorithm based on homotopic algorithm is specially designed to cope with the nonlinearity of the problem. Based on the proposed modified wavelet formation, the nonlinear vibration equation(generally second-order time-dependent ODEs) of a MDOF system are directly transformed to a group of well-posed nonlinear algebraic equations, without been tranformed to state equations in advance. Hence, the number of unkowns in the presented method in this thesis is only 1/2 of that in most current wavelet collocation methods that are based on state equations. Therefore, compared with current wavelet collocation methods, the amount of storage required by the proposed modified wavelet collocation methods is greatly reduced, about 1/2 of the former at most, while the computation efficiency is greatly impoved, at least twice faster that the former.
4). Ctihsiderihg that many control systems can be described by time-delayed MDOF systems, we have tried to applying wavelets in the stability analysis of time-delayed systems. An adaptive wavelet formula is obtained for numerical inverse (?)aplace transformation, based on a current adaptive decomposition and re-construction algorithm; then time-delayed linear MDOF systems are tranformed to discrete dynamic systems, by using numerical inverse (?)aplace transformation. Therefore, the solution of the linear time-delayed system can be solved, and its right-most eigen-value can be approximately calculated, from which its approximate stable regions can be recognized in a selected parameter plane.
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