這篇論文的目的是討論下述半線性橢圓方程的穩定解。 / -Δu = f (u) in Ω, / 這裡Ω是R{U+207F}中的光滑區域。對於非線性項f (u) = / [附圖]. / 我們得到了關於穩定解的Liouville-type結果。對於帶有更一般非線性項的半線性橢圓方程，在二到四維的情況下，我們得到一個關於穩定解的先驗估計。 / 最後，在維數很大的情形下，我們具體的給出一個關於以下雙調和方程的指數P的上界P，使得當P滿足 <1 P < Pc ‘下述雙調和方程不存在穩定解。 / Δ²u=u{U+1D56}, u>0 in R{U+207F} / The main aim of this thesis is to review recent results on stable solutions to the following semilinear elliptic equation / -Δu = f (u) in Ω, / where Ω ⊆ R{U+207F}.For the special case f (u) = / [With mathematic formula]. / For the general nonlinearity f in a bounded domain, we obtain a priori estimate for the stable solution when 2 ≤ n ≤ 4 / Finally, we give a explicit bound on the exponent for the nonexistence of stable solutions to the following biharmonic problem in large dimensions. / Δ²u=u{U+1D56}, u>0 in R{U+207F} / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Yang, Wen. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 66-69). / Abstracts also in Chinese. / Chapter 1 --- Introduction --- p.5 / Chapter 2 --- On the Lane-Emden equation --- p.12 / Chapter 2.1 --- On stable solutions to the Lane-Emden equation --- p.12 / Chapter 2.2 --- On nite Morse index solutions to the Lane-Emden equation --- p.19 / Chapter 3 --- On general semilinear elliptic equation --- p.28 / Chapter 4 --- On the biharmonic equation --- p.44 / Chapter 4.1 --- The rst part of the proof of Theorem 1.6 --- p.44 / Chapter 4.2 --- The second part of the proof of Theorem 1.6 --- p.54 / Bibliography --- p.66

Lee, Chak Shing. / "August 2011." / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 60-62). / Abstracts in English and Chinese. / Chapter 1 --- Model Problems --- p.1 / Chapter 1.1 --- Introduction --- p.1 / Chapter 1.2 --- The curl-curl operator --- p.2 / Chapter 1.3 --- The convection-diffusion equation --- p.6 / Chapter 2 --- Staggered DG method for the Curl-Curl operator --- p.8 / Chapter 2.1 --- Introduction --- p.8 / Chapter 2.2 --- Discontinuous Galerkin discretization --- p.8 / Chapter 2.3 --- Stability for aligned fields --- p.14 / Chapter 2.4 --- Error estimates --- p.17 / Chapter 2.5 --- Numerical experiments --- p.21 / Chapter 2.6 --- Concluding Remarks --- p.32 / Chapter 3 --- Staggered DG method for the convection-diffusion equation --- p.33 / Chapter 3.1 --- Introduction --- p.33 / Chapter 3.2 --- Method description --- p.33 / Chapter 3.3 --- Preservation of physical structures --- p.38 / Chapter 3.4 --- Stability and convergence --- p.42 / Chapter 3.4.1 --- Static problem --- p.42 / Chapter 3.4.2 --- Time-dependent problem --- p.46 / Chapter 3.5 --- Fully discrete scheme --- p.49 / Chapter 3.6 --- Numerical examples --- p.55 / Chapter 3.6.1 --- The static problem --- p.55 / Chapter 3.6.2 --- Time dependent problem --- p.56 / Chapter 3.7 --- Concluding Remark --- p.59 / Bibliography --- p.60

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Maxwell’s equations are a set of partial differential equations that describe the classic electromagnetic problems, electrodynamics etc. Effective numerical methods are derived to solve the equations in the past decades, and continued to be of great interest to be developed to its completion. In this thesis, we introduce and propose numerical methods using staggered meshes that deal with both two dimensional and three dimensional space problem in polygonal and general curved domains. / Finite difference method, finite volume method, spectral method and staggered discontinuous Galerkin method are discussed in the thesis. A forth order finite difference method using Taylor expansion technic is proposed. The integral form of the original Maxwell’s equations give rise to methods based on more general domain. For the finite volume method, covolume methods both on the cyclic polygon elements and noncyclic polygon elements are derived. To derive a higher order accurate method, staggered discontinuous Galerkin method based on the same domain decomposition present in the finite volume method use Nedelec elements is derived in two dimensional space, and spectral method using nodal high-order method operate on a general domain in 3D with flexible domain geometry is introduced. Numerical results are shown to show the performance oft he above mentioned approximation methods in 2D case. / 麥克斯韋方程組是一組描述經典電磁問題，電磁力學的偏微分方程。在過去數十年，行之有效的偏微分方程數值解已經被推導出並用於求解該方程，該問題現在仍然吸引著學者極大的興趣，並日臻完善。在這篇論文中，我們介紹並提出一些運用曲域交錯網格數值方法在二維和三維的多面體和更一般幾何體處理麥克斯韋方程組問題。 / 本論文對有限差分法，有限體積法，光譜法和交錯間斷有限元方法進行了討論。利用泰勒展開式這一方法推導出一個二維的四階有限差分方法。基於原來的麥克斯韋方程組的積分形式所得到的數值方法更適用於更普遍的域。對於有限體積法，對循環多邊形元素和非環狀多邊形元素的有限體積方法都將被導出。為了得到一個更高階準確的方法，基於有限體積法中使用的域分解方法，使用Nedelec元素，推導了二維空間的高階有限元方法。基於頂點高階數值方法的光譜法對於三維一般定義域的幾何形態更為靈活適用。在二維的定義域中，數值模擬結果驗證上述數值方法的精確性。 / Jian, Fangqiong. / Thesis M.Phil. Chinese University of Hong Kong 2014. / Includes bibliographical references (leaves 62-65). / Abstracts also in Chinese. / Title from PDF title page (viewed on 07, October, 2016). / Detailed summary in vernacular field only. / Detailed summary in vernacular field only.

Cheung, Man Wah. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2007. / Includes bibliographical references (leaves 89-91). / Abstracts in English and Chinese. / Chapter 1 --- Introduction to Inverse Problems --- p.1 / Chapter 1.1 --- Major properties --- p.1 / Chapter 1.2 --- Typical examples --- p.3 / Chapter 1.3 --- Thesis outline --- p.5 / Chapter 2 --- Some Operator Theory --- p.6 / Chapter 2.1 --- Fredholm integral equation of the first kind --- p.6 / Chapter 2.2 --- Compact operator theory --- p.8 / Chapter 2.3 --- Singular system --- p.12 / Chapter 2.4 --- Moore-Penrose generalized inverse --- p.14 / Chapter 3 --- Regularization Theory for First Kind Equations --- p.19 / Chapter 3.1 --- General regularization theory --- p.19 / Chapter 3.2 --- Tikhonov regularization --- p.24 / Chapter 3.3 --- Landweber iteration --- p.26 / Chapter 3.4 --- TSVD --- p.28 / Chapter 4 --- Multilevel Algorithms for Ill-posed Problems --- p.30 / Chapter 4.1 --- Basic assumptions and definitions --- p.31 / Chapter 4.2 --- Multilevel analysis --- p.33 / Chapter 4.3 --- Applications --- p.37 / Chapter 4.3.1 --- Preconditioned iterative methods with nonzero regularization parameter --- p.38 / Chapter 4.3.2 --- Preconditioned iterative methods with zero regularization parameter --- p.38 / Chapter 4.3.3 --- Full multilevel algorithm --- p.40 / Chapter 5 --- Numerical Experiments --- p.41 / Chapter 5.1 --- Integral equations --- p.41 / Chapter 5.1.1 --- Discretization --- p.42 / Chapter 5.1.2 --- Test problems --- p.43 / Chapter 5.1.3 --- "Singular values, singular vectors and condition numbers" --- p.45 / Chapter 5.1.4 --- Effect of condition numbers on numerical accuracies --- p.49 / Chapter 5.2 --- Differential equations --- p.50 / Chapter 5.2.1 --- Discretization --- p.51 / Chapter 5.2.2 --- "Singular values, singular vectors and condition numbers" --- p.53 / Chapter 5.3 --- Numerical experiments by classical methods --- p.55 / Chapter 5.3.1 --- Tikhonov regularization --- p.55 / Chapter 5.3.2 --- TSVD --- p.56 / Chapter 5.3.3 --- Landweber iteration --- p.63 / Chapter 5.4 --- Numerical experiments by multilevel methods --- p.63 / Chapter 5.4.1 --- General convergence --- p.63 / Chapter 5.4.2 --- Numerical results --- p.65 / Chapter 5.4.3 --- Effect of multilevel parameters on convergence --- p.76 / Bibliography --- p.89

In this thesis we shall investigate the numerical solutions to several important practical static and dynamic optimization problems in engineering and physics. The thesis is organized as follows.In Chapter 1 a general literature review is presented, including motivation and development of the problems, and existing results. Furthermore, some existing computational methods for optimal control problems are also discussed.In Chapter 2 the design of a semiconductor device is posed as an optimization problem: given an ideal voltage-current (V - I) characteristic, find one or more physical and geometrical parameters so that the V-I characteristic of the device matches the ideal one optimally with respect to a prescribed performance criterion. The voltage-current characteristic of a semiconductor device is governed by a set of nonlinear partial differential equations (PDE), and thus a black-box approach is taken for the numerical solution to the PDEs. Various existing numerical methods are proposed for the solution of the nonlinear optimization problem. The Jacobian of the cost function is ill-conditioned and a scaling technique is thus proposed to stabilize the resulting linear system. Numerical experiments, performed to show the usefulness of this approach, demonstrate that the approach always gives optimal or near-optimal solutions to the test problems in both two and three dimensions.In Chapter 3 we propose an efficient approach to numerical integration in one and two dimensions, where a grid set with a fixed number of vertices is to be chosen so that the error between the numerical integral and the exact integral is minimized. For one dimensional problem two schemes are developed for sufficiently smooth functions based on the mid-point rectangular quadrature rule and the trapezoidal rule respectively, and another method is also developed for integrands which are not ++ / sufficiently smooth. For two dimensional problems two schemes are first developed for sufficiently smooth functions. One is based on the barycenter rule on a rectangular partition, while the other is on a triangular partition. A scheme for insufficiently smooth functions is also developed. For illustration, several examples are solved using the proposed schemes, and the numerical results show the effectiveness of the approach.Chapter 4 deals with optimal recharge and driving plans for a battery-powered electric vehicle. A major problem facing battery-powered electric vehicles is in their batteries: weight and charge capacity. Thus a battery-powered electric vehicle only has a short driving range. To travel for a longer distance, the batteries are required to be recharged frequently. In this chapter we construct a model for a battery-powered electric vehicle, in which driving strategy is to be obtained so that the total traveling time between two locations is minimized. The problem is formulated as an unconventional optimization problem. However, by using the control parameterization enhancing transformation (CPET) (see [100]) it is shown that this unconventional optimization is equivalent to a conventional optimal parameter selection problem. Numerical examples are solved using the proposed method.In Chapter 5 we consider the numerical solution to a class of optimal control problems involving variable time points in their cost functions. The CPET is first used to convert the optimal control problem with variable time points into an equivalent optimal control problem with fixed multiple characteristic times (MCT). Using the control parameterization technique, the time horizon is partitioned into several subintervals. Let the partition points also be taken as decision variables. The control functions are approximated by piecewise constant or piecewise linear functions ++ / in accordance with these variable partition points. We thus obtain a finite dimensional optimization problem. The CPET transform is again used to convert approximate optimal control problems with variable partition points into equivalent standard optimal control problems with MCT, where the control functions are piecewise constant or piecewise linear functions with pre-fixed partition points. The transformed problems are essentially optimal parameter selection problems with MCT. The gradient formulae are obtained for the objective function as well as the constraint functions with respect to relevant decision variables. Numerical examples are solved using the proposed method.A numerical approach is proposed in Chapter 6 for constructing an approximate optimal feedback control law of a class of nonlinear optimal control problems. In this approach, the state space is partitioned into subdivisions, and the controllers are approximated by a linear combination of the 3rd order B-spline basis functions. Furthermore, the partition points are also taken as decision variables in this formulation. To show the effectiveness of the proposed approach, a two dimensional and a three dimensional examples are solved by the approach. The numerical results demonstrate that the method is superior to the existing methods with fixed partition points.

Includes bibliographical references (leaves 158-163). Concerns the development of high-order finite-difference methods on a uniform rectangular grid for advection and diffuse problems with smooth variable coefficients. This technique has been successfully applied to variable-coefficient advection and diffusion problems. Demonstrates that the new schemes may readily be incorporated into multi-dimensional problems by using locally one-dimensional techniques, or that they may be used in process splitting algorithms to solve complicatef time-dependent partial differential equations.

Bibliography: leaves 264-267.

Includes bibliographical references (leaves 158-163). / vi, 166 leaves : ill. ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Concerns the development of high-order finite-difference methods on a uniform rectangular grid for advection and diffuse problems with smooth variable coefficients. This technique has been successfully applied to variable-coefficient advection and diffusion problems. Demonstrates that the new schemes may readily be incorporated into multi-dimensional problems by using locally one-dimensional techniques, or that they may be used in process splitting algorithms to solve complicatef time-dependent partial differential equations. / Thesis (Ph.D.)--University of Adelaide, Dept. of Applied Mathematics, 2001