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This study focuses ontransient heat flow behavior in which centralizing energy on themetal makes metal surface come to aheat molten state with centralized heat source . This flow field is two-dimensional transient model, using Phase-field method and Two-phase flow to simulatemetal surface. In this study is under considerations of the mass conservation equation, momentum equation, energy equation and the level-set equation, regardless of the impact due to the concentration diffusion. At last it will show the flow of the molten zone caused by temperature, and the flows in molten zone forced by thermocapillary which is caused byvariation of temperature.

This dissertation presents the Generalized Finite Element Method (GFEM) for the scalar Helmholtz equation, which describes the time harmonic acoustic wave propagation problem. We introduce several handbook functions for the Helmholtz equation, namely the planewave, wave-band, and Vekua functions, and we use these handbook functions to enrich the Finite Element space via the Partition of Unity Method to create the GFEM space. The enrichment of the approximation space by these handbook functions reduces the pollution effect due to wave number and we are able to obtain a highly accurate solution with a much smaller number of degrees-of-freedom compared with the classical Finite Element Method. The q-convergence of the handbook functions is investigated, where q is the order of the handbook function, and it is shown that asymptotically the handbook functions exhibit the same rate of exponential convergence. Hence we can conclude that the selection of the handbook functions from an admissible set should be dictated only by the ease of implementation and computational costs. Another issue addressed in this dissertation is the error coming from the artificial truncation boundary condition, which is necessary to model the Helmholtz problem set in the unbounded domain. We observe that for high q, the most significant component of the error is the one due to the artificial truncation boundary condition. Here we propose a method to assess this error by performing an additional computation on the extended domain using GFEM with high q.

The particular solutions of the Laplace equations and their singularities are fundamental to numerical partial di erential equations in both algorithms and error analysis. We first review the explicit solutions of Laplace¡¦s equations on sectors with the Dirichlet and the Neumann boundary conditions. These harmonic functions clearly expose the solution¡¦s regularity/singularity at the vertex. So we can analyze the singularity of the Laplace¡¦s solutions on polygons at di erent domain corners and for various boundary conditions. By using this knowledge we can designed many new testing models with di erent kind of singularities, like discontinuous and mild singularities, beside the popular singularity models, Motz¡¦s and the cracked beam problems, We use the boundary approximation method, i.e. the collocation Tre tz method in engineering literatures, to solve the above testing models of Laplace boundary value problems on polygons. Suppose the uniform particular solutions are chosen in the entire domain. When there is no singularity on all corners, this method has the exponential convergence. However, its rate of convergence will deteriorate to polynomial if there exist some corner singularities. From experimental data, we even have three type of convergence, i.e. exponential, polynomial or their mixed types. We will study these convergent behaviors and their causes. Finally, we will uncover the relation between the order of convergence and the intensity of corner singularities.

The elliptic equation $Delta u+K(|x|)|u|^{p-1}u=0,xin mathbf{R}^{n}$ is studied, where $p>1$, $n>2$, $K(r)$ is smooth and positive on $(0,infty)$, and $rK(r)in L^{1}(0,1)$. It is known that the radial solution either oscillates infinitely, or $lim_{r ightarrow infty}r^{n-2}u(r;al) in Rsetminus {0}$ (rapidly decaying), or $lim_{r ightarrow infty}r^{n-2}u(r;al) = infty (or -infty)$ (slowly decaying). Let $u=u(r;al)$ is a solution satisfying $u(0)=al$. In this thesis, we classify all the radial solutions into three types: Type R($i$): $u$ has exactly $i$ zeros on $(0,infty)$, and is rapidly decaying at $r=infty$. Type S($i$): $u$ has exactly $i$ zeros on $(0,infty)$, and is slowly decaying at $r=infty$. Type O: $u$ has infinitely many zeros on $(0,infty)$. If $rK_{r}(r)/K(r)$ satisfies some conditions, then the structure of radial solutions is determined completely. In particular, there exists $0<al_{0}<al_{1}<al_{2}<cdots<infty$ such that $u(r;al_{i})$ is of Type R($i$), and $u(r;al)$ is of Type S($i$) for all $al in (al_{i-1},al_{i})$, where $al_{-1}:=0$. These works are due to Yanagida and Yotsutani. Their main tools are Kelvin transformation, Pr"{u}fer transformation, and a Pohozaev identity. Here we give a concise account. Also, I impose a concept so called $r-mu graph$, and give two proofs of the Pohozaev identity.

In this thesis, we study the optimal estimate of eigenvalue ratios £f_n/£f_m of the Sturm-Liouville equation with Dirichlet boundary conditions on (0, £k). In 2005, Horvath and Kiss [10] showed that £f_n/£f_m≤(n/m)^2 when the potential function q ≥ 0 and is a single-well function. Also this is an optimal upper estimate, for equality holds if and only if q = 0. Their result gives a positive answer to a problem posed by Ashbaugh and Benguria [2], who earlier showed that £f_n/£f_1≤n^2 when q ≥ 0. Here we first simplify the proof of Horvath and Kiss [10]. We use a modified Prufer substitutiony(x)=r(x)sin(£s£c(x)), y'(x)=r(x)£scos(£s£c(x)), where £s = ¡Ô£f. This modified phase seems to be more effective than the phases £p and £r that Horvath and Kiss [10] used. Furthermore our approach can be generalized to study the one-dimensional p-Laplacian eigenvalue problem. We show that for the Dirichlet problem of the equation -[(y')^(p-1)]'=(p-1)(£f-q)y^(p-1), where p > 1 and f^(p-1)=|f|^(p-1)sgn f =|f|^(p-2)f. The eigenvalue ratios satisfies £f_n/£f_m≤(n/m)^p, assuming that q(x) ≥ 0 and q is a single-well function on the domain (0, £k_p). Again this is an optimal upper estimate.

The Tschirnhaus Transform is a method to solve quartic equations. If the quartic equation has four distinct real roots, a low computation complexity algorithm is proposed and applied to the the 10-order Line Spectrum Pairs(LSP).However, during the procedures of calculation, the existence of the inverse trigonometric functions increase the computation load of the hardware implementation. We also propose some methods to solve this problem and increase the speed of the calculation. A table of comparison result with the previous proposed Complex-Free Farrari formula is also included.

Thesis (M.S.)--Worcester Polytechnic Institute. / Keywords: heterostructure; semiconductor; quantum engineering; self consistency. Includes bibliographical references (p. 30-33).

Through the review of creativity research I have found that studies lack certain crucial parts: a) a theoretical framework for the study of creativity in science, b) studies considering the unique components related to scientific creativity, and c) studies of the interactions among key components through simultaneous analyses. The primary purpose of this study is to explore the dynamic interactions among four components (scientific proficiency, intrinsic motivation, creative competence, context supporting creativity) related to scientific creativity under the framework of scientific creativity. A total of 295 Korean middle school students participated. Well-known and commonly used measurements were selected and developed. Two scientific achievement scores and one score measured by performance-based assessment were used to measure student scientific knowledge/inquiry skills. Six items selected from the study of Lederman, Abd-El-Khalick, Bell, and Schwartz (2002) were used to assess how well students understand the nature of science. Five items were selected from the subscale of the scientific attitude inventory version II (Moore & Foy, 1997) to assess student attitude toward science. The Test of Creative Thinking-Drawing Production (Urban & Jellen, 1996) was used to measure creative competence. Eight items chosen from the 15 items of the Work Preference Inventory (1994) were applied to measure students' intrinsic motivation. To assess the level of context supporting creativity, eight items were adapted from measurement of the work environment (Amabile, Conti, Coon, Lazenby, and Herron, 1996). To assess scientific creativity, one open-ended science problem was used and three raters rated the level of scientific creativity through the Consensual Assessment Technique (Amabile, 1996). The results show that scientific proficiency and creative competence correlates with scientific creativity. Intrinsic motivation and context components do not predict scientific creativity. The strength of relationships between scientific proficiency and scientific creativity (estimate parameter=0.43) and creative competence and scientific creativity (estimate parameter=0.17) are similar [Δx²(.05)(1)=0.670, P > .05]. In specific analysis of structural model, I found that creative competence and scientific proficiency play a role of partial mediators among three components (general creativity, scientific proficiency, and scientific creativity). The moderate effects of intrinsic motivation and context component were investigated, but the moderation effects were not found.

We study blowup of solutions of one-dimensional nonlinear heat equations (NLH). We consider two cases: a power nonlinearity and initial conditions having two equal absolute maxima and a polynomial nonlinearity and initial conditions having a single global maximum. We show in both cases that for a certain open set of initial conditions solutions of the NLH blow up in finite time and we find asymptotical behavior of blowup frofiles. In the first case the blowup occurs at two points while in the second case, at one point.