Date of Award

Spring 2009

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematical Sciences - (Ph.D.)

Department

Mathematical Sciences

First Advisor

Michael Siegel

Second Advisor

Russel E. Caflisch

Third Advisor

Lou Kondic

Fourth Advisor

Jonathan H.C. Luke

Fifth Advisor

Demetrius T. Papageorgiou

Abstract

Singularities often occur in solutions to partial differential equations; important exam¬ples include the formation of shock fronts in hyperbolic equations and self-focusing type blow up in nonlinear parabolic equations. Information about formation and structure of singularities can have significant role in interfacial fluid dynamics such as Kelvin-Helmholtz instability, Rayleigh-Taylor instability, and Hele-Shaw flow. In this thesis, we present a new method for the numerical analysis of complex singularities in solutions to partial differential equations. In the method, we analyze the decay of Fourier coefficients using a numerical form fit to ascertain the nature of singularities in two and three-dimensional functions. Our results generalize a well known method for the analysis of singularities in one-dimensional functions to higher dimensions. As an example, we apply this method to analyze the complex singularities for the 2D inviscid Burger's equation.

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Mathematics Commons

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