MODULE 8: THE FOURIER TRANSFORM METHDOS FOR PDES
Lecture 4
15
Vibration of an Innite String
In this lecture we shall learn how Fourier Transforms can be used to solve one dimensional
wave equations in an innite (or semi-innite) interval. More precisely, w

Module 8: The Fourier Transform Methdos for PDEs
In the previous modules (Modules 5-7), the method of separation of variables was used
to obtain solutions of initial and boundary value problems for partial dierential equations given over bounded spatial r

Module 7: The Laplace Equation
In this module, we shall study one of the most important partial dierential equations in
physics known as the Laplace equation
2 u = 0 in Rn ,
where 2 u :=
n
2u
i=1 x2
i
(1)
is the Laplacian of the function u. The theory of

MODULE 7: THE LAPLACE EQUATION
17
Lecture 4
The Mixed BVP for a Rectangle
In this lecture we shall consider solving the mixed BVP for the Laplace equation. To
begin with, let us consider the following the Neumann problem for a rectangle:
PDE:
BC:
uxx + uy

13
MODULE 6: THE WAVE EQUATION
Lecture 4
The Finite Vibrating String Problem
In this lecture, we shall study the transverse vibrations of a nite string. If u(x, t) represents the displacement (deection) of the string and the ends of the string are held xe

10
MODULE 6: THE WAVE EQUATION
Lecture 3
The Semi-Innite String Problem
Before we introduce the semi-innite string problem, let us look at some special cases of
DAlemberts formula derived in the previous lecture.
EXAMPLE 1. Consider the problem for the se

Module 3: Second-Order Partial Dierential
Equations
In Module 3, we shall discuss some general concepts associated with second-order linear
PDEs. These types of PDEs arise in connection with various physical problems such as
the motion of a vibration stri

16
MODULE 5: HEAT EQUATION
Lecture 4
Time-Independent Homogeneous BC
The boundary conditions in previous lecture are assumed to be homogeneous, where we
are able to use the superposition principle in forming general solutions of the PDE. We
now turn to th

MODULE 2: FIRST-ORDER PARTIAL DIFFERENTIAL EQUATIONS
Lecture 3
16
Quasilinear First-Order PDEs
A rst order quasilinear PDE is of the form
a(x, y, z )
z
z
+ b(x, y, z )
= c(x, y, z ).
x
y
(1)
Such equations occur in a variety of nonlinear wave propagation

Module 6: The Wave Equation
In this module we shall study the one-dimensional wave equation which describes transverse vibrations of an elastic string. This module is organized as follows. In the rst
lecture, we shall discuss the mathematical formulation

MODULE 5: HEAT EQUATION
5
Lecture 2
The Maximum and Minimum Principle
In this lecture, we shall prove the maximum and minimum properties of the heat equation.
These properties can be used to prove uniqueness and continuous dependence on data of
the soluti

Module 2: First-Order Partial Dierential Equations
The mathematical formulations of many problems in science and engineering reduce to
study of rst-order PDEs. For instance, the study of rst-order PDEs arise in gas ow
problems, trac ow problems, phenomeno

Steven R. Dunbar
Department of Mathematics
203 Avery Hall
University of Nebraska-Lincoln
Lincoln, NE 68588-0130
http:/www.math.unl.edu
Voice: 402-472-3731
Fax: 402-472-8466
Stochastic Processes and
Advanced Mathematical Finance
Transformations of the Wien

An Introduction to Hill Ciphers
Using Linear Algebra
Brian Worthington
University of North Texas
MATH 2700.002
5/10/2010
Hill Ciphers
Created
by Lester S. Hill in 1929
Polygraphic
Uses
Substitution Cipher
Linear Algebra to Encrypt and
Decrypt
Simple Subst

Mathematical Database
PIGEONHOLE PRINCIPLE
Pigeonhole principle is a fundamental but powerful tool in combinatorics. Unlike many other
strong theorems, the principle itself is exceptionally simple. Unless you have looked into it
thoroughly, it is hard to

Homework 1
Math 375
Spring 2007
Discrete Structure
Remark 1: This work covers the following topics.
1) Addition Principle.
2) Multiplication Principle.
3) Inclusion-Exclusion Principle.
Remark 2: The graded part is due Friday, January 19, 2007.
Examples1

Free to photocopy and distribute
August 19, 2007 Version
David A. SANTOS
[email protected]
Contents
Preface
ii
Legal Notice
iii
To the Student
iv
1 Preliminaries
1.1 Sets . . . . . . . . . . . .
1.2 Sample Spaces and Events
1.3 Combining Events . . . .

DEPARTMENT OF MATHEMATICS
Indian Institute of Technology Guwahati
MA501: Discrete Mathematics
Mid Semester Exam
September 23, 2012
Instructor: Bikash Bhattacharjya
Time: 2 hours
Maximum Marks: 30
Model Solutions
1. Let A = cfw_n : n N and i =
1. Find a bi