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Traffic Flow
Barry Fadness
Abstract.
A simple model assumes that the velocity of the cars only depends on the traffic density. For the resulting partial differential equation there are situations in which shocks develop or are propagated. If characteristic curves spread apart, then we construct a weak solution. An example is when a long red light turns green. Jump and entropy conditions will be mentioned.
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Optimal Fish Harvesting
Samuel Mohler
Abstract. Fish are the only food source that is still hunted commercially. Using partial differential equations we can model how populations of fish react to commercial fishing in the space and time variable. With this state equation, we define an optimization problem that theoretically yields the largest harvest while maintaining equilibrium with fish populations. Physical constraints will be discussed for such a problem.
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USING CONFORMAL MAPPINGS TO SOLVE LAPLACE'S EQUATION
Tuyen Tran
Abstract. The theory of conformal maps can be applied to some physical problems. In doing so we
will solve the Dirichlet problem and related problems for several types of two-dimensional domains.
The method for solving the Dirichlet and Neumann problems in a given domain D is to transform
it by a conformal map to a simpler domain D' on which the problem can be solved. When we have
solved the problem on D', we can transform the answer back to D.
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An Introduction to the Finite Element Method
Ganesh Gunaji
Abstract. The variational formulation of the finite element method is presented utilizing the tools of modern functional analysis. The standard continuous Galerkin finite element method is introduced, along with a brief discussion on existence and uniqueness of the solution for a model problem. The key steps behind the implementation of the standard finite element method for the solution of partial differential equations are described, including triangulation of the domain, numerical integration, assembly of the stiffness matrix, and inversion of the linear system. The advantages and disadvantages of the finite element method when compared to finite difference methods are discussed. Finally, a presentation and explanation of a computer implementation of a simple two dimensional finite element method on MATLAB is shown for the solution of Poisson's Equation on a unit square domain.
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The Black-Scholes PDE
Sebastian Baldivieso
Abstract.
We will present a derivation of the Black-Scholes PDE and practical interpretations of
this PDE. We will finish with some numerical examples as well as practical implementations
of the numerical solutions that can be useful to everyday market practioners.
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Extension Operator in Sobolev Spaces
Tathagata Goswami
Abstract. It is often easy to prove properties of the Sobolev spaces starting with the case \Omega = Rn. We will first prove that any function u in W1,p(Rn+) can be extended
to W1,p(Rn). Now using this, combined with the notion of partition of unity we will
prove the existance of an extension operator for a "smooth" domain whose boundary
is bounded.
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Minimal Surface Equation
Ahmed Elsakori
Abstract. The minimal surface equation is an elliptic equation but it is nonlinear and is not uniformly elliptic; indeed I will shortly present a derivation of the minimal surface equation as the Euler-Lagrange equations of the area functional on graphs. In this project I will be concerned with establishing existence of solutions to the minimal surface equation.
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Adapting Navier-Stokes for computer animation
Rogerio Mendes
Abstract.
Water, fire and smoke are vital parts of life and so they have to be illustrated correctly when one is trying to tell a story. In this presentation we will show the challenges of adapting the Navier-Stokes equation for creation of animated movies. Starting with the Momentum Equation we will derive the necessary aspects for being able to animate fluids efficiently, and show why some terms are more important then others when it comes to visualization.