Thesis (Ph. D.)--University of Wisconsin, 1966. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

The purpose of this paper is to study nth order integral equations. The integrals studied in this paper are of the Riemann type.

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<p>Boundary integral equation techniques are useful in thenumerical simulation of scattering problems for wave equations.Their advantage over methods based on partial di.erentialequations comes from the lack of phase errors in the wavepropagation and from the fact that only the boundary of thescattering object needs to be discretized. Boundary integraltechniques are often applied in frequency domain but recentlyseveral time domain integral equation methods are beingdeveloped.</p><p>We study time domain integral equation methods for thescalar wave equation with a Galerkin discretization of twodi.erent integral formulations for a Dirichlet scatterer. The.rst method uses the Kirchho. formula for the solution of thescalar wave equation. The method is prone to get unstable modesand the method is stabilized using an averaging .lter on thesolution. The second method uses the integral formulations forthe Helmholtz equation in frequency domain, and this method isstable. The Galerkin formulation for a Neumann scattererarising from Helmholtz equation is implemented, but isunstable.</p><p>In the discretizations, integrals are evaluated overtriangles, sectors, segments and circles. Integrals areevaluated analytically and in some cases numerically. Singularintegrands are made .nite, using the Du.y transform.</p><p>The Galerkin discretizations uses constant basis functionsin time and nodal linear elements in space. Numericalcomputations verify that the Dirichlet methods are stable, .rstorder accurate in time and second order accurate in space.Tests are performed with a point source illuminating a plateand a plane wave illuminating a sphere.</p><p>We investigate the On Surface Radiation Condition, which canbe used as a medium to high frequency approximation of theKirchho. formula, for both Dirichlet and Neumann scatterers.Numerical computations are done for a Dirichlet scatterer.</p>

Very accurate methods, based on boundary integral techniques, are developed for the study of multiple, interacting fluid interfaces in an Eulerian fluid. These methods are applied to the evolution of a thin, periodic layer of constant vorticity embedded in irrotational fluid. Numerical regularity experiments are conducted and suggest that the interfaces of the layer develop a curvature singularity in infinite time. This is to be contrasted with the more singular vorticity distribution of a vortex sheet developing such a singularity in a finite time.

The Ito-Stratonovich theory of stochastic integration and stochastic differential equations has several shortcomings, especially when it comes to existence and consistency with the theory of Lebesque-Stieltjes integration and ordinary differential equations. An attempt is made firstly, to isolate the path property, possessed by almost all Brownian paths, that makes the stochastic theory of integration work. Secondly, to construct a new concept of solutions for differential equations, which would have the required consistency and continuity properties, within a class of deterministic noise functions, large enough to include almost all Brownian paths. The algebraic structure of iterated path integrals for smooth paths leads to a formal definition of a solution for a differential equation in terms of generalized path integrals for more general noises. This suggests a way of constructing solutions to differential equations in a large class of paths as limits of operators. The concept of the driving noise is extended to include the generalized path integrals of the noise. Less stringent conditions on the Holder continuity of the path can be compensated by giving more of its iterated integrals. Sufficient conditions for the solution to exist are proved in some special cases, and it is proved that almost all paths of Brownian motion as well as some other stochastic processes can be included in the theory.

A cohesive zone model approach is used in order to study the behaviour of cracks in elasto-plastic materials. The cohesive zone model being studied is time-dependent, unlike standard cohesive zone models in elasto-plasticity. The stress distribution over the cohesive zone is related to the normalised equivalent stress functional, and is expressed in the form of an Abel-type integral equation. During the stationary crack stage as well as the propagating crack stage, the aim is to study the behaviour of the cohesive zone length with respect to time as well as the crack tip opening. To aid accomplishing this aim, the stress intensity factor was set to zero at the cohesive zone tip. As well as other material parameters, the external applied load participates in the model equations. We will consider two cases for the external load, namely the case when this load is constant in time, and the case when this load behaves linearly with time. We will implement numerical schemes to obtain the crack growth as well as the cohesive zone growth with respect to time for both the elastic case and the visco-elastic case while considering different sets of parameters. The numerical convergence rates are obtained for each of the problems solved. This justifies the suitability of the numerical schemes used.