Student: | Charles Siegel |
School: | Rutgers University |
E-mail: | cmsiegel [@] eden.rutgers.edu |
Research Area: | Partial Differential Equations |
Project Name: | Error Analysis in PDEs |
Mentor: | Avraham Soffer, Rutgers University |
We begin with R^{3}, and then we delete a subset of R^{3} around the origin, representing an object, say, an airplane. To determine what this object is in the real world, we can do something like bounce radar or sonar off of it. Then, we analyze the signal we receive. That is, we project a wave at the object and study the behavior of what is reflected. To simulate this situation mathematically, we need to solve the wave equation, that is, u_{tt}=Δu, with a boundary condition on the surface of the object we're studying such that the wave will be reflected. However, the standard method of solving the wave equation in space uses Green's Functions, which isn't feasible here, as we'd need to know the Green's Function for our object, and finding Green's Functions is rather difficult. Thus, numerical solutions have been searched for. Now, we really don't care what happens as the reflected waves go to infinity, so we'll take a box around the origin for our computation. However, we need to find out what boundary condition we should impose on that box. So first, we will Fourier Transform our equation in y and z, and then we will Laplace Transform in t. Both of these transformations essentially turn a derivative into a multiplication. The new equation comes out to be s^{2} u=-|k|^{2}u+u_{xx}, and has solution u(x)=A(s,k)exp(√s^{2}+|k|^{2}x)+B(s,k)exp(-√s^{2}+|k|^{2}x)$ Looking at the two solutions, A(s, k)exp(√s^{2}+|k|^{2}x) is a nonphysical solution, as it roughly corresponds to a wave coming in from infinity and converging on our object, so we want a boundary condition on the box such that A(s,k)≡0. Such a boundary condition can be seen to be ∂x v+√s^{2}+|k|^{2}v=0 Unfortunately, the operator √s^{2}+|k|^{2} is hard to work with, as it corresponds roughly to √∂_{t}^{2}+Δ Thus, we have to approximate it to find a solution to the wave equation. noticing that √s^{2}+|k|^{2}≈s as s gets large, we do so by taking s+√s^{2}+|k|^{2}-s. Also, √s^{2}+|k|^{2}-s can be expanded as a continued fraction. So, if we take a finite continued fraction representation of √s^{2}+|k|^{2}-s, we get an approximate solution, however it will be simplifiable to an operator that is a sum of products of space and time derivatives, and that is something that we can deal with. This method was devised by Mijda and Enquist in 1977, but they didn't prove anything about the error bounds for this procedure. Hagstrom, however, did some error analysis and proved that the error in this solution was less than εe^{ct} where c is small and positive. The goal of my project is to prove that the error is, in fact, less than cε/t^{n} and to find c and n. |