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.

In this thesis, we consider the Laplace quation on sector with various constant Dirichlet or Numann boundary conditions. Most of such problems have singularity in the solution. We first analyze the type of singularity on the corner and then survey some known methods to solve these problems. The boundary approximation method is used to compute some of their solutions with two singularities. Besides, a Laplace equation on a triangle with multiple solutions is solved by the method of separation of variables.

This thesis is on molecular dynamics studies of drying suspensions of bidisperse nanoparticle mixtures. I first use an explicit solvent model to investigate how the structure of the dry film depends on the evaporation rate of the solvent and the initial volume fractions of the nanoparticles. My simulation results show that the particle mixtures stratify according to their sizes when the suspensions are quickly dried, consistent with the prediction of recent theories. I further show that stratification can be controlled using thermophoresis induced by a thermal gradient imposed on the drying suspension. To model larger systems on longer time scales, I explore implicit solvent models of drying particle suspensions in which the solvent is treated as a uniform viscous background and the liquid-vapor interface is replaced by a potential barrier that confines all the solutes in the solution. Drying is then modeled as a process in which the location of the confining potential is moved. In order to clarify the physical foundation of this moving interface method, I analyze the meniscus on the outside of a circular cylinder and apply the results to understand the capillary force experienced by a spherical particle at a liquid-vapor interface. My analyses show that the capillary force is approximately linear with the displacement of the particle from its equilibrium location at the interface. An analytical expression is derived for the corresponding spring constant that depends on the surface tension and lateral span of the interface and the particle radius. I further show that with a careful mapping, both explicit and implicit solvent models yield similar stratification behavior for drying suspensions of bidisperse particles. Finally, I apply the moving interface method based on an implicit solvent to study the drying of various soft matter solutions, including a solution film of a mixture of polymers and nanoparticles, a suspension droplet of bidisperse nanoparticles, a solution droplet of a polymer blend, and a solution droplet of diblock copolymers. / PHD / Drying is a ubiquitous phenomenon. In this thesis, I use molecular dynamics methods to simulate the drying of a suspension of a bidisperse mixture of nanoparticles that have two different radii. First, I use a model in which the solvent is included explicitly as point particles and the nanoparticles are modeled as spheres with finite radii. Their trajectories are generated by numerically solving the Newtonian equations of motion for all the particles in the system. My simulations show that the bidisperse nanoparticle mixtures stratify according to their sizes after drying. For example, a “small-on-top” stratified film can be produced in which the smaller nanoparticles are distributed on top of the larger particles in the drying film. I further use a similar model to demonstrate that stratification can be controlled by imposing a thermal gradient on the drying suspension. I then map an explicit solvent system to an implicit one in which the solvent is treated as a uniform viscous background and only the nanoparticles are kept. The physical foundation of this mapping is clarified. I compare simulations using the explicit and implicit solvent models and show that similar stratification behavior emerge in both models. Therefore, the implicit solvent model can be applied to study much larger systems on longer time scales. Finally, I apply the implicit solvent model to study the drying of various soft matter solutions, including a solution film of a mixture of polymers and nanoparticles, a droplet of a bidisperse nanoparticle suspension, a solution droplet of a polymer blend, and a droplet of a diblock copolymer solution.

We consider the problem of analytic continuation of the solution of the multidimensional Lame system in infinite domains through known values of the solution and the corresponding strain tensor on a part of the boundary, i.e,the Cauchy problem.

In this paper we consider the regularization of the Cauchy problem for a system of second order differential equations with constant coefficients.

In the first two Chapters I outline the theory and background of separation of variables as an ansatz for solving fundamental partial differential equations (pdes) in Mathematical Physics. Two fundamental approaches will be highlighted, and more modern approaches discussed. In Chapter 3 I calculate the general trace-free conformal Killing tensor defined in Euclidean space - from the sum of symmetric tensor products of conformal Killing vectors. In Chapter 4 I determine the subcases with rotational symmetry and recover known examples pertaining to classical rotational coordinates. In Chapter 5 I obtain the induced action of the conformal group on the space of trace-free conformal Killing tensors. In Chapter 6 I use the invariants of trace-free conformal Killing tensors under the action of the conformal group to characterize, up to equivalence, the symmetric R-separable webs in E^3 that permit conformal separation of variables of the fundamental pdes in Mathematical Physics. In Chapter 7 the asymmetric R-separable metrics are obtained via a study of the separability conditions for the conformally invariant Laplace equation.

In the first two Chapters I outline the theory and background of separation of variables as an ansatz for solving fundamental partial differential equations (pdes) in Mathematical Physics. Two fundamental approaches will be highlighted, and more modern approaches discussed. In Chapter 3 I calculate the general trace-free conformal Killing tensor defined in Euclidean space - from the sum of symmetric tensor products of conformal Killing vectors. In Chapter 4 I determine the subcases with rotational symmetry and recover known examples pertaining to classical rotational coordinates. In Chapter 5 I obtain the induced action of the conformal group on the space of trace-free conformal Killing tensors. In Chapter 6 I use the invariants of trace-free conformal Killing tensors under the action of the conformal group to characterize, up to equivalence, the symmetric R-separable webs in E^3 that permit conformal separation of variables of the fundamental pdes in Mathematical Physics. In Chapter 7 the asymmetric R-separable metrics are obtained via a study of the separability conditions for the conformally invariant Laplace equation.

The contact angle defined by Young's equation depends on the ratio between solid and liquid surface energies. Young's contact angle is constant for a given system, and cannot explain the stability of fluid droplets in capillary tubes. Within this framework, large variations in contact angle and explained aassuming surface roughness, heterogeneity or contamination. This research explores the static and dynamic behavior of fluid droplets within capillary tubes and the variations in contact angle among interacting menisci. Various cases are considered including wetting and non-wetting gluids, droplets in inclined capillary tubes or subjected to a pressure difference, within one-dimensional and three-dimensional capillary systems, and under static or dynamic conditions (either harmonic fluid pressure or tube oscillation). The research approach is based on complementary analytical modeling (total energy formulation) and experimental techniques (microscopic observations). The evolution of meniscus curvatures and droplet displacements are studied in all cases. Analytical and experimental results show that droplets can be stable within capillary tubes even under the influence of an external force, the resulting contact angles are not constant, and bariations from Young's contact angle aare extensively justified as menisci interaction. Menisci introduce stiffness, therefore two immiscible Newtonian fluids behave as a Maxwellian fluid, and droplets can exhibit resonance or relaxation spectral features.

Artificial vision is the problem of creating systems capable of processing visual information. A fundamental sub-problem of artificial vision is image segmentation, the problem of detecting a structure from a digital image. Examples of segmentation problems include the detection of a road from an aerial photograph or the determination of the boundaries of the brain's ventricles from medical imagery. The extraction of structures allows for subsequent higher-level cognitive tasks. One of them is shape comparison. For example, if the brain ventricles of a patient are segmented, can their shapes be used for diagnosis? That is to say, do the shapes of the extracted ventricles resemble more those of healthy patients or those of patients suffering from schizophrenia? This thesis deals with the problem of image segmentation and shape comparison in the mathematical framework of partial differential equations. The contribution of this thesis is threefold: 1. A technique for the segmentation of regions is proposed. A cost functional is defined for regions based on a non-parametric functional of the distribution of image intensities inside the region. This cost is constructed to favor regions that are homogeneous. Regions that are optimal with respect to that cost can be determined with limited user interaction. 2. The use of direction information is introduced for the segmentation of open curves and closed surfaces. A cost functional is defined for structures (curves or surfaces) by integrating a local, direction-dependent pattern detector along the structure. Optimal structures, corresponding to the best match with the pattern detector, can be determined using efficient algorithms. 3. A technique for shape comparison based on the Laplace equation is proposed. Given two surfaces, one-to-one correspondences are determined that allow for the characterization of local and global similarity measures. The local differences among shapes (resulting for example from a segmentation step) can be visualized for qualitative evaluation by a human expert. It can also be used for classifying shapes into, for example, normal and pathological classes.

Boundary approximation method, also known as the collocation Trefftz method in engineering, is used to solve Laplace boundary value problem on rectanglular domain. Suppose the particular solutions are chosen for the whole domain. If there is no singularity on other vertices, it should have exponential convergence. Otherwise, it will degenerate to polynomial convergence. In the latter case, the order of convergence has some relation with the intensity of singularity. So, it is easy to design models with desired convergent orders. On a sectorial domain, when one side of the boundary conditions is a transcendental function, it needs to be approximated by power series. The truncation of this power series will generate an artificial singularity when solving Laplace equation on polygon. So it will greatly slow down the expected order of convergence. This thesis study how the truncation error affects the convergent speed. Moreover, we focus on the transition behavior of the convergence from one order to another. In the end, we also apply our results to boundary approximation method with enriched basis.