Analyses And Applications For Three Types Of Models In Mathematics And Physics

This paper is composed of three parts. In chapter 1, we focus on the generalization of a class of third order nonlinear differential equations. It arises naturally in astronomic and hydromechanical models. The main concern in chapter 2 is one class of dual phase lagging heat conduction equation using alternating-direction finite element method with moving grid. Chapter 3 centers on a TDOA passive location algorithm based on constrained total least squares.Generalized from third order equations, a class of higher order boundary value equations is concerned by lower and upper solution method. A new modified problem is established by introducing an auxiliary function. By using Nagumo condition, Leray-Schauder degree theory and a priori estimate technique, the existence of solution to the modified problem is discerned. Then through the bridge of the property of auxiliary function, the modified problem is converted to the higher order equations and the main theorem is achieved.In chapter 2, the finite element is put forward to the class of 3-dimensional dual phase lagging heat conduction equations. We adopt different finite element grids for different domain area. By using L~2 projection, the finite element method with moving grid is given. For saving the computation, the alternating direction method is availed and the multidimensional difference systems are decomposed to a set of independent one-dimensional problems. Finally using the theory and technique of partial differential, we obtain the optimal L~2 error estimates.In chapter 3, a TDOA passive location algorithm based on constrained total least squares is proposed, which can get the closed form solution. The algorithm analyzes the model of TDOA passive location and transfers them to the constrained total least squares problem. The polynomial function of the Lagrange multiplier is obtained from the technique of multiplier method, and the closed form estimator of source location is obtained through the root of the Lagrange multiplier. Simulation results demonstrate that the accuracy of estimator has been improved compared with LS method and is close to the Cramer-Rao lower bound at sufficiently high signal-to-noise ratio conditions.