This thesis investigates the computation of piecewise polynomial solutions to ill- conditioned linear systems of equations when noise on the linear measurements is observed. Specifically, we enhance our understanding of and provide qualifications on when such ill-conditioned systems of equations can be solved to a satisfactory accuracy. We show that the conventional condition number of the coefficient matrix is not sufficiently informative in this regard and propose a more relevant conditioning measure that takes into account the decay rate of singular values. We also discuss interactions of several factors affecting the solvability of such systems, including the number of discontinuities in solutions, as well as the distribution of nonzero entries in sparse matrices. In addition, we construct and test an approach for computing piecewise polynomial solutions of highly ill-conditioned linear systems using a randomized, SVD-based truncation, and L1-norm regularization. The randomized truncation is a stabilization technique that helps reduce the cost of the traditional SVD truncation for large and severely ill-conditioned matrices. For L1-minimization, we apply a solver based on the Alternating Direction Method. Numerical results are presented to compare our approach that is faster and can solve larger problems, called RTL1 (randomized truncation L1-minimization), with a well-known solver PP-TSVD.

In this thesis, development process of de-icing and anti-icing solutions and their flow instability characteristics are presented. In the beginning, the chemical additives in the solutions and their effects on the most critical physical properties of the solutions were investigated. Firstly, chemical additives were added to glycol and water mixtures at different weight ratios one by one in order to see their individual effects. Then, the changes in physical properties were observed when the chemicals were added to water-glycol mixture together. After that, study was focused on effect of polymer which makes the solution pseudoplastic. Further investigations on viscosity behavior of the solution at different pH values, glycol and water mixtures and surfactant weight ratios, which is used for surface tension reduction, were performed. For the investigation of flow instability characteristics of the solutions&rsquo / flows, linear stability analysis for two-layer flows is a basic tool. Firstly, the effects of main parameters on the stability of two-layer flows were observed with a parametric study. Then, the commercially available and developed de-icing and anti-icing solutions were compared according to the characteristics of unstable waves. According to the results, unstable waves on developed de-icing fluids are observed at a lower critical wind speed compared to the commercially available de-icing solution. Moreover, it flows off the wing faster due to a higher value of critical wave speed. Developed anti-icing solution has similar wave characteristics compared to commercially available anti-icing solution, except critical wave speed, which is significantly lower. However, this problem can be overcome by decreasing the viscosity of developed anti-icing solution at very low shear rates.

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.

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 consider soltions to the two-dimensional Euler equations with uniformly continuous initial vorticity in a critical or subcritical Besov space. We use paradifferential calculus to show that the solution will lose an arbitrarily small amount of smoothness over any fixed finite time interval. This result is motivated by a theorem of Bahouri and Chemin which states that the Sobolev exponent of a solution to the two-dimensional Euler equations in a critical or subcritical Sobolev space may decay exponentially with time. To prove our result, one can use methods similar to those used by Bahouri and Chemin for initial vorticity in a Besov space with Besov exponent between 0 and 1; however, we use different methods to prove a result which applies for any Sobolev exponent between 0 and 2. The remainder of this thesis is based on joint work with J. Kelliher. We study the vanishing viscosity limit of solutions of the Navier-Stokes equations to solutions of the Euler equations in the plane assuming initial vorticity is in a variant Besov space introduced by Vishik. Our methods allow us to extend a global in time uniqueness result established by Vishik for the two-dimensional Euler equations in this space. / text

We have directed much attention towards developing quantum trajectory methods which can accurately predict the transmission probabilities for a variety of quantum mechanical barrier scattering processes. One promising method involves solving the complex quantum Hamilton-Jacobi equation with the Derivative Propagation Method (DPM). We present this method, termed complex valued DPM (CVDPM(n)). CVDPM(n) has been successfully employed in the Lagrangian frame to accurately compute transmission probabilities on 'thick' one dimensional Eckart and Gaussian potential surfaces. CVDPM(n) is able to reproduce accurate results with a much lower order of approximation than is required by real valued quantum trajectory methods, from initial wave packet energies ranging from the tunneling case (E[subscript o]=0) to high energy cases (twice the barrier height). We successfully extended CVDPM(n) to two-dimensional problems (one translational degree of freedom representing an Eckart or Gaussian barrier coupled to a vibrational degree of freedom) in the Lagrangian framework with great success. CVDPM helps to explain why barrier scattering from "thick" barriers is a much more well posed problem than barrier scattering from "thin" barriers. Though results in these two cases are in very good agreement with grid methods, the search for an appropriate set of initial conditions (termed an 'isochrone) from which to launch the trajectories leads to a time-consuming search problem that is reminiscent of the rootsearching problem from semi-classical dynamics. In order to circumvent the isochrone problem, we present CVDPM(n) equations of motion which are derived and implemented in the arbitrary Lagrangian-Eulerian frame for a metastable potential as well as the Eckart and Gaussian surfaces. In this way, the isochrone problem can be circumvented but at the cost of introducing other computational difficulties. In order to understand why CVDPM may give better transmission probabilities than real valued counterparts, much attention we have been studying and applying numerical analytic continuation techniques to visualize complex-extended wave packets as well as the complex-extended quantum potential. Numerical analytic continuation techniques have also been used to analytically continue a discrete real-valued potential into the complex plane for CVDPM with very promising results.

Magistro darbe pateikti paprastojo raudonėlio (Origanum vulgare L.) rausvažiedės ir baltažiedės formų, augančių VDU Kauno botanikos sodo vaistinių ir prieskoninių augalų kolekcijoje, vegetatyvinių organų lapo ir stiebo anatominės ir morfologinės sandaros ypatumai, bei įvertinti generatyvinių ir vegetatyvinių organų morfologinių požymių parametrų skirtumai. Nustatyti koreliacijos koeficientai tarp stiebo aukščio ir žiedyno ilgio, priklausomybė mentūrių skaičiaus žiedyne nuo žiedyno ilgio ir lapalakščio plošio ir ilgio tarpusavio priklausomubė. Visur pasireiškė teigiami koreliacijos ryšiai, kai vienam rodikliui didėjant, didėja ir kitas rodiklis. Abiejų tirtų paprastojo raudonėlio (Origanum vulgare L.) formų stiebas daugiametis, medėjantis, ne kulelinio tipo. Apytakos audiniai išsidėsto sifonostelėje. Vandens indai pavieniai. Būdingi spiraliniai ir tinkliniai vandens indai. Spiraliniai vandens indai su snapeliu. Žiotelinis aparatas anizocitinio ir diacitinio tipo, išsidėsto adoksalinėje ir aboksalinėje lapo pusėse. Origanum vulgare L. raudonžiedžio viršutinėje lapo pusėje 1mm² vidutiniškai 40, o apatinėje 125 žiotelių. Origanum vulgare L. baltažiedžio lapo viršutinėje pusėje vidutiniškai 1mm² 35, o apatinėje 97 žiotelių. Raudonėlio (Origanum L.) genties augaluose yra šizogeninės kilmės liaukos. Origanum vulgare L. raudonžiedžio sėklų daigumas buvo tiriamas mirkant ir daiginant jas katolitiniame ir anolitiniame vandenyje. Nustatyta, kad katolitinis vanduo skatina paprastojo... [toliau žr. visą tekstą] / This MA thesis paper presents an overview of the specific characteristics of the anatomic and morphologic formation of the vegetative organs – the leaf and the stem – of the red and white blossom varieties of simple oregano (Origanum vulgare L.) that are cultivated as part of the medicinal plants and potherbs collection at the Kaunas botanical garden of the Vytautas Magnus University. The work also presents an evaluation of differences between the parameters of morphologic characteristics of generative and vegetative organs. Determined are the correlation ratios for the height of the stem and the length of the inflorescence, the relationship between the number of whorls in the inflorescence and the length of the latter, as well as the interrelation of the lenth and the width of the lamina. Positive correlation (when an increase in one variable resulted in the increase of another variable) was determined in all cases. The stem of both studied forms of simple oregano (Origanum vulgare L.) is perennial, lignificating, of a non-bundle type. Circulation tissues are arranged in a siphonostele. Water vessels are single. Spiral and cellular water vessels are common. The spiral ones feature a nozzle. The leaves of the plants belonging to the oregano genus (Origanum L.) are equifacial. The length of the static tissue’s cells varies from 83 µm to 87 µm, while the width varies from 33 µm to 44 µm. The stomatic apparatus is of the anisocitic and diacitic types, located on the adoxal... [to full text]

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The collocation method for solving ordinary differential equations is examined. A detailed comparison with other weighted residual methods is made. The orthogonal collocation method is compared to the collocation method and the advantage of the former is illustrated. The sensitivity of the orthogonal collocation method to different parameters is studied. Orthogonal collocation on finite elements is used to solve an ordinary differential equation and its superiority over the orthogonal collocation method is shown. The orthogonal collocation on finite elements is also used to solve a partial differential equation from chemical kinetics. The results agree remarkably with those from the literature. / Thesis (M.Sc.)-University of KwaZulu-Natal, 2008.