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%This function plots the solution to the linearly
dispersive wave equation. First, save the file in a
suitable folder and change the matlab environment to that
particular folder. One can call the function by giving a
suitable value for the time and positi
REU
Outline and Ideas
Ive included my ideas and possible approaches to take:
1. Establish the theoretical results of [1] with the Schrodinger Wave
Equation.
Approach: In a sequential manner, prove the theorems and corollaries
associated with the linearly
%This function plots the solution to the Schr\cfw_odinger
wave equation. First, save the file in a suitable folder
and change the matlab environment to that particular
folder. One can call the function by giving a suitable
value for the time and position,
% This BibTeX bibliography file was created using BibDesk.
% http:/bibdesk.sourceforge.net/
% Created for Sudarshan Balakrishnan at 2011-06-10 08:54:46 -0400
% Saved with string encoding Unicode (UTF-8)
@bookcfw_dft,
Author = cfw_J.O Smith III,
Date-Added
Schrdinger Equation
o
Sudarshan Balakrishnan
September 1, 2011
Analytic Solution
The time-dependent Schrdinger Equation in one dimension given by :
o
(x, t)
h2 2 ( (x, t)
+ U (x, t) (x, t) = ih
2m
x2
t
(1)
Here, we consider h, m both be equal to one and
Splitting Algorithm
Sudarshan Balakrishnan
August 23, 2011
Linear Ordinary Dierential Equations
In order to understand the splitting associated with the logistic equation, let us rst
consider the linear ODE given by:
du
du
du
= Au,
= Bu,
= Cu.
(1)
dt
dt
d
% This BibTeX bibliography file was created using BibDesk.
% http:/bibdesk.sourceforge.net/
% Created for Sudarshan Balakrishnan at 2011-08-19 07:10:47 -0400
% Saved with string encoding Unicode (UTF-8)
@articlecfw_tao,
Author = cfw_Terence Tao, Helge Hol
Chapter 16: Fourier series
Sudarshan Balakrishnan
July 26, 2011
Periodic Functions
Exercise 16.1.1
Show that every real number x can be written in exactly one way in the
form x = k + y , where k is an integer and y [0, 1).
Proof
In order to prove this sta
Dispersive Quantization of the linear Schrdinger equation
o
Sudarshan Balakrishnan
September 18, 2011
Abstract. The study of linear dispersive partial dierential equations with piecewise
constant periodic initial data leads to quantized structures at time
Quantization
Theorem 1
Fast Fourier Transform
Dispersive Quantization
Sudarshan Balakrishnan1
Gong Chen2
1 Department
of Mathematics
University of Michigan
2 Department
of Mathematics
University of Minnesota
Research Experience for Undergraduates, 2011
Su
Chapter 16: Fourier series
Sudarshan Balakrishnan
June 17, 2011
Periodic Functions
Exercise 16.1.1
Show that every real number x can be written in exactly one way in the
form x = k + y , where k is an integer and y [0, 1).
Proof
In order to prove this sta
Schrdinger Equation
o
Sudarshan Balakrishnan
June 14, 2011
Lemma 2.
The Fourier coecients of a function f (x) are q -periodic in their indices, so
ck+q = ck for all k , if and only if the series represents a linear combination
of q (periodically extended)
Research
Sudarshan Balakrishnan
June 28, 2011
Linearly dispersive wave equation
The purpose of this write-up is to examine the solution to the linearly dispersive wave equation at dierent times and observe any number-theoretic
results that might arise. Us
Dispersive Quantization
Sudarshan Balakrishnan, Gong Chen
August 19, 2011
The study of linear dispersive partial dierential equations with piecewise constant
periodic initial data leads to quantized structures at times which are rational multiples
of and
% This BibTeX bibliography file was created using BibDesk.
% http:/bibdesk.sourceforge.net/
% Created for Sudarshan Balakrishnan at 2011-04-28 08:02:24 -0400
% Saved with string encoding Unicode (UTF-8)
@articlecfw_Quantum,
Author = cfw_Kapitanski, L and
Comments
Numerical Methods in Matlab - Brandon
1. Explain/dene terms in the overview such as DFT, FFT and heat equations. It would
be good for the reader to review these terms if he/she isnt familiar with them.
2. Expand more on how the equations are sepa
Dispersive Quantization
Peter J. Olver
School of Mathematics
University of Minnesota
Minneapolis, MN 55455
olver@math.umn.edu
http:/www.math.umn.edu/olver
Abstract. The evolution, through linear dispersion, of piecewise constant periodic
initial data lead
Dispersive Quantization of the Linear Schrdinger Equation
o
Sudarshan Balakrishnan
September 20, 2011
Abstract. The study of linear dispersive partial dierential equations with piecewise
constant periodic initial data leads to quantized structures at time
Fast Fourier Transform
Sudarshan Balakrishnan
August 23, 2011
The purpose of this write up is to notice the similarity between the exact solution of
the linearly dispersive wave equation and the solution computed using the fast fourier
transform. The comm
% This BibTeX bibliography file was created using BibDesk.
% http:/bibdesk.sourceforge.net/
% Created for Sudarshan Balakrishnan at 2011-10-02 21:17:12 -0400
% Saved with string encoding Unicode (UTF-8)
@articlecfw_oskolov,
Author = cfw_Oskolov K.I,
Date-
Dispersive Quantization of the Linear Schrdinger Equation
o
Sudarshan Balakrishnan
October 3, 2011
Abstract. The study of linear dispersive partial dierential equations with piecewise
constant periodic initial data leads to quantized structures at times w
Introduction to Non-linear dispersive wave
equations: Linares and Ponce
Sudarshan Balakrishnan
August 8, 2011
Chapter 1
Exercise 1.1
Let n 1 and f (x) = e2|x| . Show that:
f ( ) =
[(n + 1)/2]
+ | |2 )(n+1)/2
(n+1)/2 (1
Proof
In order to prove this statem
Numerical Solution
Sudarshan Balakrishnan
August 24, 2011
Logistic Equation
Given the logistic equation u = u(u 1) we notice that the associated operators were
A(u) = u and B (u) = u2 [1].The exact solutions associated with A, B and C = A + B
are given by
Dispersive Quantization of the linear Schrdinger equation
o
Sudarshan Balakrishnan
September 20, 2011
Abstract. The study of linear dispersive partial dierential equations with piecewise
constant periodic initial data leads to quantized structures at time
The Ohio State University, August 19-21
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