Finite difference numerical methods are developed for the solution system in the biomedical sciences; namely, fox-rabies model. First-order methods and second-order method are developed to solve the fox-rabies equations. The fox-rabies model is extended to one-space dimension to incorporate diffusion. The reaction terms in these systems of partial differential equations contain non-linear expressions. It is seen that the numerical solutions are obtained by solving non-linear algebraic system at each time step, as opposed to solving anon-linear algebraic system which is often required when integrating non-linear partial differential equations. The numerical methods proposed for the solution of the initial-value problem for the fox-rabies model are characterized to be implicit. In each case, however, it seen that the numerical solutions are obtained explicitly. In a series of numerical experiments, in which the ordinary differential equations are solved first of all, it seen that the proposed methods have an identical stability properties to those of the well-known, first-order, Euler method. The proposed methods for the numerical solution of partial differential equations are seen to be economical and reliable. Error analysis for the methods, computer implementation and numerical results are discussed. The stability of the numerical method is analyzed using maximum principle analysis.

Graduate

No description available.

No description available.

The classical existence theorem for the steady Navier-Stokes equations, based on a bound for the solution's Dirichlet integral, provides little qualitative information about the solution. In particular, if a domain is unbounded, it is not evident that the solution will be unique even when the data are small. Inspired by the works of Odqvist for the interior problem and of Finn for the problem of flow past an obstacle, we give a potential theoretic construction of a solution of the steady Navier-Stokes equations in several domains with noncompact boundaries. We begin by studying a scalar quasilinear elliptic problem in a half space, which serves as a model problem for the development of some of the methods which are later applied to the Navier-Stokes equations. Then, we consider Navier-Stokes flow in a half space, modeling such phenomena as a jet emanating from a wall, with prescribed boundary values. The solution which is obtained decays like |x|⁻² at infinity and has a finite Dirichlet integral. Finally, we solve the problem of flow through an aperture in a wall between two half spaces, with a prescribed net flux through the aperture, or with a prescribed pressure drop between the two half spaces. A steady solution is constructed which decays like |x|⁻² at infinity. For small data, uniqueness is proven within the class of functions which decay like |x|⁻¹ at infinity and have finite Dirichlet integrals. / Science, Faculty of / Mathematics, Department of / Graduate

This thesis investigates the viability of two boundary element methods for solving steady state problems, the continuous least squares method and the Galerkin minimization technique. In conventional boundary element methods, the singularities of the fundamental solution involved are usually located at fixed points on the boundary of the problem's domain or on an auxiliary boundary. This leads to some difficulties: when the singularities are located on the problem domain's boundary, it is not easy to evaluate the solution for points on or near that boundary whereas if the singularities are placed on an auxiliary boundary, this auxiliary boundary would have to be carefully chosen. Hence the methods studied here allow the singularities, initially located at some auxiliary boundary, to move until the best positions are found. These positions are determined by attempting to minimize the error via the least squares or the Galerkin technique. This results in a highly accurate, adaptive, but nonlinear method. We study various methods for solving systems of nonlinear equations resulting from the Galerkin technique. A hybrid method has been implemented, which involves the objective function from the least squares method while the gradient is due to the Galerkin method. Numerical examples involving Laplace's equation in two dimensions are presented and results using the discrete least squares method, the continuous least squares method and the Galerkin method are compared and discussed. The continuous least squares method appears to give the best results for the sample problems tried. / Science, Faculty of / Computer Science, Department of / Graduate

In the present study, methodological approaches to assess the impact of climate change on the thermal and thermo-hydro-mechanical (THM) regimes of the ground in some selected Canadian no-permafrost areas (Ottawa, Sudbury, Toronto) is proposed. A modeling study to evaluate ground temperature variations due to global warming is conducted using TEMP/W software from Geoslope ltd. The effect of future climate change projections, up to 2100, on the ground freeze-thaw cycle frequencies, frost penetration depth, and frost duration is assessed in some selected sites located in the Canadian no-permafrost region. Moreover, three softwares (TEMP/W, SEEP/W and SIGMA/W from Geoslope Ltd) have been used to establish a numerical tool that enable to assess the effect of global warming on THM response of the grounds in the selected Canadian no-permafrost areas. TEMP/W and SEEP/W were coupled in a thermo-hydraulic analysis to assess the impact of global warming on the hydraulic regime of the ground. Afterwards, SEEP/W and SIGMA/W were coupled in a hydraulic-mechanical analysis to study the impact of climate change induced porewater pressures change on the mechanical regime of the ground in some no-permafrost regions. Simulation study to assess the effect ground temperature changes on key geotechnical properties of the soils in the selected sites is conducted by using the aforementioned numerical tool. The change of the porewater pressure changes and distributions in the soil induced by global warming is studied. The effect of climate change on the ground consolidation or settlement in the selected no-permafrost sites is also investigated. Finally, this study provides a simulation of a bridge pile foundation ground to detect the THM changes around the pile structure due to climate warming. The results indicate that climate change will affect the thermal regime of the ground in the selected Canadian no permafrost areas. Ground temperature in the studied no-permafrost regions will likely increase by 2 to 4 C by 2100 due to global warming. Furthermore, the frost penetration depth will be significantly reduced in all study areas. It is also found that the frost duration will experience a gradual reduction with time up to 2100. In addition, the simulation results showed minimal influence of global warming on the porewater pressure distribution and magnitude in the studied grounds. Aligned to this, climate change did not seem to have a significant effect on the consolidation behavior or settlement of the ground in the studied no-permafrost areas. The simulation of the foundation ground confirms the results mentioned above, as temperature changes around the pile structure falls within the same range found in the thermal analysis. Porewater pressure distributions and ground settlement are not significantly affected along the pile perimeter. Overall, the design of pile foundation in the Canadian no-permafrost region will not be significantly affected by climate change up 2100. The tools developed and results obtained will be useful for the geotechnical design of climate-adaptive civil engineering or transportation structures in Canadian no permafrost areas.

<p> Personalized PageRank with high teleportation probability enables exploring the environment of a seed. With this insight, one can use an orthogonal factorization of a set of personalized PageRank vectors, like SVD, to derive a 2-dimensional representation of the network. This can be done for the whole network or a smaller piece. The power of this method lies in the fact that only a few columns, compared to the size of the networks, can be used to generate a local representation of the part of the network we are interested in. This technique has the potential to be seamlessly used for higher order structures, such as hypergraphs which have found a great deal of use for real-world data. This work investigates the characteristics of personalized PageRank and how it compares to the transition probabilities on the graph in terms of their ability to develop low dimensional representations. A key focus of the thesis are the similarities between the embeddings generated due to PageRank and those generated by spectral methods.</p>

Thesis. 1976. M.S.--Massachusetts Institute of Technology. Dept. of Meteorology. / Microfiche copy available in Archives and Science. / Bibliography: leaf 38. / by Alan L. Adams. / M.S.

Thesis. 1978. M.S.--Massachusetts Institute of Technology. Dept. of Meteorology. / Microfiche copy available in Archives and Science. / Bibliography : leaves 64-65. / M.S.