DSpace at Texas State University
https://digital.library.txstate.edu:443
The Digital Collections digital repository system captures, stores, indexes, preserves, and distributes digital research material.2018-11-12T01:01:41ZPartial Regularity for Flows of H-Surfaces
https://digital.library.txstate.edu/handle/10877/7775
Partial Regularity for Flows of H-Surfaces
Wang, Changyou
This article studies regularity of weak solutions to the heat equation for H-surfaces. Under the assumption that the function H is Lipschitz and depends only on the first two components, the solution has regularity on its domain, except for a set of measure zero. Moreover, if the solution satisfies certain energy inequality, this set is finite.
1997-11-20T00:00:00ZSemilinear Hyperbolic Systems in one Space Dimension with Strongly Singular Initial Data
https://digital.library.txstate.edu/handle/10877/7774
Semilinear Hyperbolic Systems in one Space Dimension with Strongly Singular Initial Data
Travers, Kirsten E.
In this article interactions of singularities in semilinear hyperbolic partial differential equations in R^2 are studied. Consider a simple non-linear system of three equations with derivatives of Dirac delta functions as initial data. As the micro-local linear theory prescribes, the initial singularities propagate along forward bicharacteristics. But there are also anomalous singularities created when these characteristics intersect. Their regularity satisfies the following ``sum law'': the ``strength'' of the anomalous singularity equals the sum of the ``strengths'' of the incoming singularities. Hence the solution to the system becomes more singular as time progresses.
1997-08-28T00:00:00ZNumerical Solution of a Parabolic Equation with a Weakly Singular Positive-type Memory Term
https://digital.library.txstate.edu/handle/10877/7773
Numerical Solution of a Parabolic Equation with a Weakly Singular Positive-type Memory Term
Slodicka, Marian
We find a numerical solution of an initial and boundary value problem. This problem is a parabolic integro-differential equation whose integral is the convolution product of a positive-definite weakly singular kernel with the time derivative of the solution. The equation is discretized in space by linear finite elements, and in time by the backward-Euler method. We prove existence and uniqueness of the solution to the continuous problem, and demonstrate that some regularity is present. In addition, convergence of the discrete sequence of iterations is shown.
1997-06-04T00:00:00ZStable Evaluation of Differential Operators and Linear and Nonlinear Multi-scale Filtering
https://digital.library.txstate.edu/handle/10877/7772
Stable Evaluation of Differential Operators and Linear and Nonlinear Multi-scale Filtering
Scherzer, Otmar
Diffusion processes create multi--scale analyses, which enable the generation of simplified pictures, where for increasing scale the image gets sketchier. In many practical applications the ``scaled image'' can be characterized via a variational formulation as the solution of a minimization problem involving unbounded operators. These unbounded operators can be evaluated by regularization techniques. We show that the theory of stable evaluation of unbounded operators can be applied to efficiently solve these minimization problems.
1997-09-10T00:00:00Z