Date of Award
Doctor of Philosophy in Mathematical Sciences - (Ph.D.)
Russel E. Caflisch
Jonathan H.C. Luke
Demetrius T. Papageorgiou
Singularities often occur in solutions to partial differential equations; important examples 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.
Malakuti, Kamyar, "Numerical detection of complex singularities in two and three dimensions" (2009). Dissertations. 912.