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The New Jersey Institute of Technology's
Electronic Theses & Dissertations Project

Title: Numerical detection of complex singularities in two and three dimensions
Author: Malakuti, Kamyar
View Online: njit-etd2009-060
(x, 74 pages ~ 4.6 MB pdf)
Department: Department of Mathematical Sciences
Degree: Doctor of Philosophy
Program: Mathematical Sciences
Document Type: Dissertation
Advisory Committee: Siegel, Michael (Committee chair)
Caflisch, Russel E. (Committee member)
Kondic, Lou (Committee member)
Luke, Jonathan H.C. (Committee member)
Papageorgiou, Demetrius T. (Committee member)
Date: 2009-05
Keywords: Partial differential equation
Complex singularity
Burger equation
Availability: Unrestricted

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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