KUMUNU 2018 PDE, Dynamical Systems and Applications
Md Al Masum Bhuiyan, University of Texas, El Paso
A Dynamical Analysis of Earthquake Waves by Using Ornstein Uhlenbeck Type Model
This work is devoted to modeling of earthquake wave time series. We propose a stochastic differential equation arising on the superposition of independent Ornstein- Uhlenbeck processes driven by a Γ(m,n) process. Superposition of independent Γ(m, n) Ornstein-Uhlenbeck processes offers analytic flexibility and provides a dynamic class of continuous time processes capable of exhibiting long memory behavior. The stochastic differential equation is applied to the study of earthquake waves by fitting the superposed Γ(m, n) Ornstein-Uhlenbeck model to earthquakes sequences in Arizona, USA. We obtained very good fitting of the observed velocity proportional of earthquakes with the stochastic differential equations, which supports the use of this methodology for the study of non-linear sequences.
Jason Bramburger, Brown University
Snaking in the Swift-Hohenberg Equation in Dimension 1+Epsilon
The Swift-Hohenberg equation is a widely studied partial differential equation which is known to support a variety of spatially localized structures. The one-dimensional equation exhibits spatially localized steady-state solutions which give way to a bifurcation structure known as snaking. That is, these solutions bounce between two different values of the bifurcation parameter while ascending in norm. The mechanism that drives snaking in one spatial dimension is now well-understood, but recent numerical investigations indicate that upon moving to two spatial dimensions, the related radially-symmetric spatially-localized solutions take on a significantly different snaking structure which consists of three major components. To understand this transition we apply a dimensional perturbation in an effort to use well-developed methods of perturbation theory and dynamical systems. In particular, we are able to identify key characteristics that lead to the segmentation of the snaking branch and therefore provide insight into how the bifurcation structure changes with the spatial dimension.
Hong Cai, Brown University
We think about wave fronts in Rosenzweig-MacArthur system in two cases. One is prey diffuses at the rate much smaller than that of the predator, the other is both of them diffuse slowly.
Paula Egging, University of Nebraska
Uniform Decay of a Structural Acoustic Dynamics
This poster presents recently derived results of uniform rational decay for strong solutions of a canonical structural acoustic PDE which has previously appeared in the literature. Our stability proof depends upon an appropriate invocation of a now well- known resolvent criterion of A. Borichev and Y. Tomilov.
Fazel Hadadifard, University of Kansas
Optimal Time Decay Rates for the Generalized Surface Quasi-Geostrophic Equation
We are interested in finding the optimal decay rate for a generalized fractional surface quasi-geostrophic equation. This type of equation arises frequently in fluid dynamics. To achieve this goal we use a method called the “Scaling Variables”.
Laszlo Kindrat, University of New Hampshire
Numerical Spectral Analysis of the Euler-Bernoulli Beam Model with Non-conservative Boundary Control
Numerical spectral analysis to investigate the vibrational frequencies of a flexible beam model equipped with fully non-conservative linear feedback boundary conditions is presented. The Chebyshev collocation method is used and a novel technique is developed to handle the dynamic boundary conditions. The accuracy of the numerical scheme is demonstrated by showing the rate of convergence of the scheme and by comparing numerical and analytic results. The very good agreement between the asymptotic and numerical approximations of the vibrational spectrum is discussed.
Ang Li, Brown University
Linear Stability of Planar Spiral Waves
Planar spiral waves have been observed in many natural systems and also as solutions of reaction-diffusion equation. Our goal is to investigate the linear stability of spectrally stable spiral waves by establishing pointwise estimate of the associated Green's function. The essential spectrum of the linearization about a spiral wave has countably many branches that touch the imaginary axis and is therefore not sectorial: we plan to use the Gearhart-Prüss Theorem to prove the spectral mapping theorem for the linearization about a spiral wave. Afterwards, Laplace-transform techniques will be used to derive the pointwise estimates of the Green's function.