In the thesis, the error and stability analysis is made for the 2D Helmholtz equation by the method of fundamental solutions (MFS) using both Bessel and Neumann functions. The bounds of errors in bounded simply-connected domains are derived, while the bounds of condition number are derived only for disk domains. The MFS using Bessel functions is more efficient than the MFS using Neumann functions. Interestingly, for the MFS using Bessel functions, the radius R of the source points is not necessarily larger than the maximal radius r_max of the solution domain. This is against the traditional condition: r_max < R for MFS. Numerical experiments are carried out to support the analysis and conclusions made.

Most of reports deal with bounded simply-connected domains; only a few involve in exterior and annular problems (Chen et al. [3], Katsuroda[10] and Ushijima and Chibu [30]). For exterior problems of Laplace's equations, there exist two kinds of infinity conditions, (1) |u|≤C and (2) u=O( ln r), which must be complied with by the fundamental solutions chosen. For u=O(ln r), the traditional fundamental solutions can be used. However, for |u|≤C, new fundamental solutions are explored, with a brief error analysis. Numerical experiments are carried out to verify the theoretical analysis made. Numerical experiments are also provided for annular domains, to show that the method of fundamental solutions (MFS) is inferior to the method of particular solutions (MPS), in both accuracy and stability. MFS and MPS are classified into the Trefftz method (TM) using fundamental solutions (FS) and particular solutions (PS), respectively. The remarkable advantage of MFS over MPS is the uniform $ln|overline{PQ_i}|$, to lead to simple algorithms and programming, thus to save a great deal of human power. Hence, we may reach the engineering requirements by much less efforts and a little payment. Besides, the crack singularity in unbounded domain is also studied. A combination of both PS and FS is also employed, called combination of MFS. The numerical results of MPS and combination of MFS are coincident with each other. The study in this thesis may greatly extend the application of MFS from bounded simply-connected domains to other more complicated domains.

In this thesis, the analysis of the method of fundamental solution(MFS) is expanded for biharmonic equations. The bounds of errors are derived for the traditional and the Almansi's approaches in bounded simply-connected domains. The exponential and the polynomial convergence rates are obtained from highly and finite smooth solutions, respectively. Also the bounds of condition number are derived for the disk domains, to show the exponential growth rates. The analysis in this thesis is the first time to provide the rigor analysis of the CTM for biharmonic equations, and the intrinsic nature of accuracy and stability is similar to that of Laplace's equation. Numerical experiment are carried out for both smooth and singularity problems. The numerical results coincide with the theoretical analysis made. When the particular solutions satisfying the biharmonic equation can be found, the method of particular solutions(MPS) is always superior to MFS, supported by numerical examples. However, if such singular particular solutions near the singular points can not be found, the local refinement of collocation nodes and the greedy adaptive techniques can be used. It seems that the greedy adaptive techniques may provide a better solution for singularity problems. Beside, the numerical solutions by Almansi's approaches are slightly better in accuracy and stability than those by the traditional FS. Hence, the MFS with Almansi's approaches is recommended, due to the simple analysis, which can be obtained directly from the analysis of MFS for Laplace's equation.

In this thesis, we extend the spectral collocation methods(SCM) (i.e., pseudo-spectral method) in Quarteroni and Valli [27] for the semilinear, parameter-dependentproblems(PDP) in the square with the Dirichlet boundary condition. The optimal error bounds are derived in this thesis for both H1 and L2 norms. For the solutions sufficiently smooth, the very high convergence rates can be obtained. The algorithms of the SCM are simple and easy to carry out. Only a few of basis functions are needed so that not only can the high accuracy of the PDP solutions be achieved, but also a great deal of CPU time may be saved. Moreover, for PDP the stability analysis of SCM is also made, to have the same growth rates of condition number as those for Poisson¡¦s equation. Numerical experiments are carried out to verify the theoretical analysis made.

The Lagrange multiplier used for the Dirichlet condition is well known in mathematics community, and the Lagrange multiplier used for the Neumann condition is popular for the Trefftz method in engineering community, in particular for elasticity problems. The latter is called the Hybrid Trefftz method (HTM). However, it seems to export no analysis for HTM. This paper is devoted to error analysis of the HTM for −£Gu + cu = 0 with c = 1 or c = 0. Error bounds are derived to provide the optimal convergence rates. Numerical experiments and comparisons between two kinds of Lagrange multipliers are also reported. The analysis in this paper can also be extended to the HTM for elasticity problems.

The Lagrange multiplier used for the displacement (i.e., Dirichlet) condition is well known in mathematics community (see [1, 2, 10, 18]), and the Lagrange multiplier used for the traction (i.e., Neumann)condition is popular for the Trefftz method for elasticity problems in engineering community, which is called the Hybrid Trefftz method (HTM). However, it seems to export no analysis for HTM. This paper is devoted to error analysis of the HTM for elasticity problems. Numerical experiments are reported to support the analysis made.

<p>When speaking a language, whether it is our first or second language, grammatical mistakes will be made. The aim of this essay is to look into what kinds of mistakes some Swedish learners of English make when speaking English and to analyze why these mistakes are made. The essay also aims at looking into what grammar teaching can look like in Sweden and how some teachers look upon their students’ oral proficiency.</p><p>The method used for this study was a qualitative one, namely interviews. Twelve students, eight in grade seven and four in grade nine, and two teachers were interviewed. During the interviews with the students a dictaphone was used. When interviewing the teachers notes were taken, and these have been the foundation of the analysis.</p><p>The results showed that many of the mistakes made by the students seemed to originate in transfer from their first language. Preposition mistakes, for instance, were made in 20% of the cases and they mainly originated in interference with their first language.</p><p>Verbs turned out to be the area where most mistakes were made, followed by prepositions and pronouns. 50% of the mistakes made by students in grade nine were verb mistakes, whereas the students in grade seven made verb mistakes in 33% of the cases.</p><p>This study further shows that the teachers had a good grasp of what their students know, and do not know, but there were some mistakes the learners made which the teachers did not mention. Finally, the study showed that spoken language is in focus within the classroom. Students are allowed to make mistakes, even though the interviewed teachers find grammar important.</p>

Thesis (Ph. D.)--University of Washington, 2003. / Vita. Includes bibliographical references (leaves 88-93).

Thesis (M.S.)--Mississippi State University. Department of Computational Engineering. / Title from title screen. Includes bibliographical references.

In the design of digital signal processing systems, where single-precision results are required, the power dissipation and area of parallel multipliers can be significantly reduced by truncating the less significant columns and compensating to produce an approximate rounded product. This dissertation presents the design of truncated multiplications of signed inputs utilizing a new number system, the negative fractional two's complement number system which solves an inherent problem of the conventional two's complement number system. This research also presents a new truncated multiplication method to reduce the errors with only slightly more hardware. Error, area, delay and dynamic power estimates are performed at the structural HDL level. The new method is also applied to various conventional number systems. For division, which is the slowest and most complex of the arithmetic operations, a new truncated division method is described that yields the same errors as those of true rounding without additional execution time that is normally required for true rounding. The new method is also applied to various conventional number systems.