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

<p>Syftet med undersökningen är att ta reda på i vilken utsträckning lärare kräver av sina elever att de skriftligen redovisar lösningar på matematikuppgifter de arbetar med, samt i vilken utsträckning lärarna tror att eleverna går tillbaka och reflekterar över sina uppgifter där skriftliga lösningar saknas när de rättar själva. Undersökningen är utförd genom en enkät samt tillhörande möjligheter för respondenterna att lämna egna kommentarer till frågorna. Studien visar att lärarnas önskan är att eleverna på något vis ska redovisa sina lösningar skriftligen samt gå tillbaka och reflektera när de rättar, men de kräver det inte av eleverna i någon större utsträckning. Eleverna rättar oftast själva, men lärarna kontrollerar emellanåt.</p>

<p>This study aims to connect problem-based learning with the problematic aspects during English lessons in school. In this way, suggestions for solutions to these problems could be generated. The study took place at an upper secondary school located in the middle regions of Sweden. In this school six unstructured observations were done during the English lessons in order to locate the problematic aspects. The students who participated in this study were in the ages of sixteen to eighteen years old. The theoretical framework was based on literature on problem-based learning and learning in general. The essential features of problem-based learning have been summarized and connected with the problematic aspects and classroom activities from the observations. In this way, it has been possible to come up with suggestions for solutions to some problems, such as unwillingness to speak and lack of motivation among students. The conclusions in this essay are that the problematic aspects dealt with features, such as unwillingness to speak as well as that the students did not understand the learning materials and that the teacher was not supportive. The solutions for these problems dealt mostly with triggering motivation by presenting problem-solving tasks, working with interaction and metacognition and planning the tasks in accordance with the students’ zone of proximal development. Furthermore, the teacher should work as a guide in the classroom to help the students along the way. The students should also receive positive and constructive feedback from the teacher, which will improve the learning among the students.</p>

Graduation date: 1996

In this article we construct the fundamental solutions for the wave equation arising in the de Sitter model of the universe. We use the fundamental solutions to represent solutions of the Cauchy problem and to prove the Lp − Lq-decay estimates for the solutions of the equation with and without a source term.

This study aims to connect problem-based learning with the problematic aspects during English lessons in school. In this way, suggestions for solutions to these problems could be generated. The study took place at an upper secondary school located in the middle regions of Sweden. In this school six unstructured observations were done during the English lessons in order to locate the problematic aspects. The students who participated in this study were in the ages of sixteen to eighteen years old. The theoretical framework was based on literature on problem-based learning and learning in general. The essential features of problem-based learning have been summarized and connected with the problematic aspects and classroom activities from the observations. In this way, it has been possible to come up with suggestions for solutions to some problems, such as unwillingness to speak and lack of motivation among students. The conclusions in this essay are that the problematic aspects dealt with features, such as unwillingness to speak as well as that the students did not understand the learning materials and that the teacher was not supportive. The solutions for these problems dealt mostly with triggering motivation by presenting problem-solving tasks, working with interaction and metacognition and planning the tasks in accordance with the students’ zone of proximal development. Furthermore, the teacher should work as a guide in the classroom to help the students along the way. The students should also receive positive and constructive feedback from the teacher, which will improve the learning among the students.

Syftet med undersökningen är att ta reda på i vilken utsträckning lärare kräver av sina elever att de skriftligen redovisar lösningar på matematikuppgifter de arbetar med, samt i vilken utsträckning lärarna tror att eleverna går tillbaka och reflekterar över sina uppgifter där skriftliga lösningar saknas när de rättar själva. Undersökningen är utförd genom en enkät samt tillhörande möjligheter för respondenterna att lämna egna kommentarer till frågorna. Studien visar att lärarnas önskan är att eleverna på något vis ska redovisa sina lösningar skriftligen samt gå tillbaka och reflektera när de rättar, men de kräver det inte av eleverna i någon större utsträckning. Eleverna rättar oftast själva, men lärarna kontrollerar emellanåt.