All posts by admin

Florida Tech REU-2014, Final Presentations

FIT REU-2014, Final Presentations

National Science Foundation funded Research Experience for Undergraduates Site on Partial Differential Equations and Dynamical Systems has been successfully established at Florida Institute of Technology. In the first summer session in 2014 10 undergraduate students representing University of Chicago, City University of New York, University of Michigan, Williams College, University of Illinois at Urbana-Champaign, University of Auburn, University of Pennsylvania, California State University at Long Beach, University of Redlands and Florida Institute of Technology pursued research under my supervision.

Students worked in there groups working on Nonlinear Partial Differential Equations, Optimal Control and Inverse Problems and Dynamical Systems and Chaos Theory. All 10 students are selected as presenters in YMC-2014 in Ohio State University, August 22-24, 2014. Here is the link to abstracts of all three presentations.

Analysis of Interfaces for the Nonlinear Diffusion Equation with Linear Convection by Taylor P Schluter & Lauren T Lanier & Kev Johnson

On the Structure of Minimal $4(2k+1)$-orbits of Continuous Endomorphisms and Universality in Chaos by
Rashad U Abdulla & Batul Kanawati & Anders Ruden

On Some Inverse Free Boundary Problems for Second Order Parabolic PDEs by
Nicholas Crispi & Daniel Kassler & Paige Williams & Bruno Poggi & Scott Pelton-Stroud

Attached youtube video features final presentations by students at the end of the summer session of 2014.

NPDE Lecture 4 Discussion

IV. Nonlinear Diffusion in Non-smooth Domains
PDF Version

In this lecture I am going to talk about nonlinear diffusion equation in general non-smooth domains.
In a series of papers I developed the theory of nonlinear diffusion type equations in general non-smooth domains.
In this talk I am going to explain the results of my papers [1,2].
A particular motivation for this works arises from the problem about the evolution of interfaces in problems for porous medium equation.
Special interest concerns the cases when support of the initial data contains a corner or cusp singularity at some points.
What about the movement of these kind of singularities along the interface? To solve this problem, it is important at the first stage to develop general theory of boundary-value problems in non-cylindrical domains with boundary surfaces which has the same kind of behaviour as the interface. In many cases this may be non-smooth and characteristic. It should be mentioned that in the one-dimensional case Dirichlet and Cauchy-Dirichlet problems for the reaction-diffusion equations in irregular domains were studied in [3,4]. Primarily applying this theory a complete description of the evolution of interfaces was presented in [5,6].

Continue reading NPDE Lecture 4 Discussion

Optimal Control and Inverse Problems for PDEs

Development and investigation of the nonlinear models of the mathematical physics is one of the most important problems of the modern science. This project will shed light into modeling and control of an important class of nonlinear processes with phase transition, so called free boundary problems, arising in thermophysics and mechanics of continuous media, bioengineering and materials science. Free boundary problems involve the solution of the partial differential equations in domains that are unknown apriori: not only the solutions of the equations but also the domain of definition of the equations must be determined.
The main goal of this project is to gain insight into inverse free boundary problems. These problems arise when not only the solution of the PDEs and the free boundaries are unknown, but also some other characteristics of the media are not available. For example, temperature or heat flux on the fixed boundaries, intensity of the source term or phase transition temperature may not be available. Another important motivation for the analysis of the inverse Stefan problem arises in optimal control of processes with phase transitions.

In summer REU session of 2014 students will pursue research on the optimal control of the free boundary problem for the heat equation motivated by the new variational formulation suggested in a recent paper:

NPDE Lecture 2 Discussion

II. Weak Solutions and Uniqueness
PDF Version

Weak Solutions

Let me remind you what I did in a previous lecture. I considered the problem
[latex]
\begin{align}
u_{t}-\text{div}\big( u^{\sigma} \nabla u\big) &=0,~x \in\R^{N},~t>0; \ \sigma > 0,\label{eq:inst-pme-1}\\
\int_{\R^{N}}u(x,t)\,dx &=1,~t > 0\label{eq:inst-pme-2}\\
u(x,0) = \delta(x)\label{eq:inst-pme-3}
\end{align}
[/latex]
and constructed instantaneous point-source solution, also called the Zeldovich-Kompaneets-Barenblatt (ZKB) solution.
\begin{equation} u_*(x,t)=t^{-\frac{N}{2+N\sigma}}\left[\frac{\sigma}{2(2+N\sigma)} \left( \eta_{0}^{2}-\frac{|x|^{2}}{t^{\frac{2}{2+N\sigma}}}\right)_{+} \right]^{\frac{1}{\sigma}}\end{equation}
Note that the nonlinear diffusion equation is invariant under translation of time and space coordinate, and the solution to the problem
\[ \begin{cases}
u_{t}=\text{div}\big(u^{\sigma }u\big),~&(x,t) \in \R^{N}\times \R_{+}\\
u(x,0)=u_*(x,T),~&x \in \R^{N}
\end{cases}\]
is exactly $u_*(x,t+T)$, which has compact support for all $t\geq 0$.

However, several important questions are left open. First of all, ZKB solution is not a classical solution: it is not even differentiable on the boundary of the support. First important question which we need to answer is the following: In what sense is $u_*$ actually a solution of the problem? Hence, we need to define the notion of the weak solution. Second important question we need to answer is the following: May be there is a smooth solution of the same problem with different properties, and ZKB solution is just physically irrelevant mathematical example. Main goal of this lecture is to answer these important questions.
Continue reading NPDE Lecture 2 Discussion

NPDE Localization Effect in Diffusion-Absorption Equation

Reaction-diffusion equations can exhibit a phenomenon known as space localization: not only do they exhibit finite speed of propagation, but heat can be completely contained in a compact region for all time under certain conditions. This is a physically pertinent phenomena as well, having been observed in plasmas, but not encountered in solutions to the linear heat equation.

Consider the reaction-diffusion equation
\begin{equation} u_{t}=\text{div} \big( u^{\sigma}\nabla u\big)-\gamma u,\quad u \in \R^{N},~t>0\label{eq:pme-loc-1}\end{equation}
where $\sigma >0$, $\gamma>0$. That means nonlinear diffusion is accompanied with linear absorption.
Continue reading NPDE Localization Effect in Diffusion-Absorption Equation