Partial Differential Equations for Scientists and Engineers
MATH 462

Spring 2016
MATH462
1. Fourier transform
The Fourier Transform is one of the most useful mathematical tools
in science. It was originally employed by J. Fourier in order to solve
problems in Heat Conduction so it is appropriate to introduce it here
and use it to sol
Partial Differential Equations for Scientists and Engineers
MATH 462

Spring 2016
MATH462
1. The Wave Equation
Let us now consider the wave equation, which is written below
utt c2 uxx = 0 ;
u(0, x) = f (x) ;
t>0 ;
< x < +
ut (0, x) = g(x)
< x < + .
(1)
(2)
We have two variables (t, x) and we will think of t as the time variable
and
Partial Differential Equations for Scientists and Engineers
MATH 462

Spring 2016
MATH462 EXAM # 1, SPRING 2016
1 : (25 pts) Consider the following vectorfield :
F(x) :=
x
kxk3 + 1
x = (x, y, z)
and denote by R3 the 3 dimensional Euclidean space. Use Gausss theorem in order to compute
Z Z Z
(divF)dv .
R3
Hint : Integrate over a ball
Partial Differential Equations for Scientists and Engineers
MATH 462

Spring 2016
1. First order ODEs
The simplest type of ODE is the following,
x = c
c
is a constant
x(0) = x0 ,
which has the solution x(t) = x0 +ct i.e. the solution is a linear function
in t. The next ODE that appears in many applications is,
x + ax = 0
(1)
x(0) = x0
Partial Differential Equations for Scientists and Engineers
MATH 462

Spring 2016
MATH462
1. Transport equation
Suppose that we want to solve the equation below,
ut + cux = 0 ;
t>0 ;
u(0, x) = f (x) ;
< x < +
< x < + ,
(1)
(2)
which means we would like to find (compute) the unknown function
u(t, x), assuming that c is some given (kn
Partial Differential Equations for Scientists and Engineers
MATH 462

Summer 2017
We want to solve the system,
ut + cux = v
vt cvx = f
We know that the solutions of each separate are given by
Z
t
v(s, x c(t s)ds
u(t, x) =
Z
0
t
f (s, x + c(t s)ds
hence
Z s
v(s, x c(t s) =
f (p, x c(t s) + c(s p)dp
v(t, x) =
0
0
which we substitute int
Partial Differential Equations for Scientists and Engineers
MATH 462

Summer 2017
PROJECT, SUMMER 2017
Schrodingers equation describes Quantum particles in terms of a complex wavefunction which we will call (t, x). The variable t is time and x is space (one dimension for
simplicity). The wavefunction of a noninteracting particle satis
Partial Differential Equations for Scientists and Engineers
MATH 462

Summer 2017
THE FINITE ELEMENT METHOD
Abstract. This is a brief outline of some ideas from Calculus of
Variations and its connection to FEM (first version)
Suppose that we would like to solve the, admitedly simple, problem
below,
PROBLEM (S) : Find a function u(x) de
Partial Differential Equations for Scientists and Engineers
MATH 462

Fall 2012
Crfob)
Final Exam Math 462
hiay 13,2010
Problem 1. Solve the following inhomogeneous transport equation
at I 2% 2 U2,
1
u(0,:1;) = .
x
Determine the region of the 1590 plane where the solution is uniquely
determined.
Problem 2. Solve the problem
u
Partial Differential Equations for Scientists and Engineers
MATH 462

Summer 2017
MATH462 FOURIER SERIES
1. Fourier Series
Fourier Series follow a similar path to that of the Fourier Transform
(with some twists). Consider a function f (x) defined originaly on the
interval [L/2 , L/2] and susequently extended in a periodic manner
so th