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Another Existence Theorem for Weak Solutions
Barry Fadness
Abstract.
Last term we saw that given an elliptic operator L there exists a nonnegative constant c such that the problem Lu + cu = g with u = 0 on the boundary has a weak solution for any g and m >= c. I shall discuss a second existence result (from Evans) that is based on the Fredholm alternative. In addition, I want to briefly mention Fredholm operators, which have an interesting relationship to compact operators. I plan to define the index of a linear map, and list some properties.
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Complex Fluid Flow
Samuel Mohler
Abstract. The assumptions of 2-D Incompressible Inviscid fluid motion are useful for engineering applications. With these assumptions, the flow across cylinders or spheres can be modeled analytically, but for any complex geometry real valued functions fail to produce simple results. Luckily the assumptions allow a connection to using complex valued function and conformal transforms to solve more geometries. The theory of the origins of the complex velocity function and the basics of the conformal transform will be discussed.
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APPLICATION OF PDEs IN CONSUMER THEORY
Tuyen Tran
Abstract. In the microeconomic theory the consumer is an economic agent who wants to maximize
the utility of the set of bundles of goods subject to the budget constraint and thereafter
deduce the law of demand that states quantities demanded as functions of their market price
and income. This project will present the mathematical formulation of consumer’s problem
and introduce a different approach which apply the methods of the theory of PDEs to
recover Indirect Utility function from the Marshallian Demand function.
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An Introduction to Spectral Methods for the Numerical Solution of Partial Differential Equations
Ganesh Gunaji
Abstract.
The theory of Galerkin spectral methods for the numerical solution of partial differential equations is introduced along with a discussion of the advantages and disadvantages of these spectral methods over finite element and finite difference methods. Convergence of Galerkin spectral methods will be discussed along with an example demonstrating the implementation of a Fourier Galerkin spectral method.
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The Black-Scholes PDE: A Mathematical Excursion
Sebastian Baldivieso
Abstract.
Last term we derived the Fundamental Black-Scholes PDE by forming risk-free portfolios
to eliminate the random component presented in the market. Financial applications of
payments to dividends were presented as well as numerical solutions to the PDE. This term,
using the Hille-Yoshida Theorem, we wish to present a formal proof for the existance and
uniqueness of the solution to the Fundamental Black-Scholes PDE that pays no dividends.
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Existence and Uniqueness of Solution of Schrodinger Equation
Tathagata Goswami
Abstract.
In this presentation I will prove "If A and -A are maximal monotone
operators in a Hilbert space H, then they together generate a
group of isometries". Also as an application, I will show existence
of solution of 1-D Schrodinger equation using this property.
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Wave Scattering in Inhomogeneous Strings
Ahmed Elsakori
Abstract. A scattering, or diffraction, problem consists of an incoming wave, an interaction, and an outgoing wave. Because the interaction itself may have very complicated effects, we focus our attention on its consequence, the incoming? outgoing process, known as the scattering process. In this paper, I present the case where there is discontinuity in the density of the string where the form of the solution will be given in finite sums of reflected and transmitted waves over finite time.
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Simulating fluid for computer animation
Rogerio Mendes
Abstract.
In this talk I will present a fast algorithm to simulate fluids in real time (or almost) for
animation. Starting with the Momentum Equation we will derive the necessary features to
animate fluids efficiently, and show why some terms are more important than others when it
comes to visualization for entertainment only.