MATH-UA 263 PARTIAL DIFFERENTIAL EQUATIONS
HOMEWORK 1 (DUE 2/10)
(1) Classify the following differential equations in term of whether they are ordinary or
partial, linear or nonlinear and what their o
Vishnu Bachani
Apr 6 Partial Differential Equations
Distributions
How do we make the concept of a delta function rigorous?
Test Functions
A test function is C function (all derivates of any order are
Vishnu Bachani
Review Session Mar 7 Partial Differential Equations
(
PDE
domain
boundary conditions
I Uniqueness
1) Energy method
- heat equation, wave equations, transport
- bounded or unbounded doma
Vishnu Bachani
Mar 23 Partial Differential Equations
The Laplace equation is invariant under rigid transforms: translations and rotation. It is trivial to see if
xx u + yy u = 0
and
x0 = x + a,
y0 = y
Vishnu Bachani
PDE Lecture 6 - 23 Feb 2017
The Robin Condition
Consider the eigenvalue problem
X 00 = X
With boundary conditions
X 0 a0 X = 0
at x = 0
0
at x = `
= 2 > 0,
>0
X a` X = 0
Consider the c
MATH-UA 263 PARTIAL DIFFERENTIAL EQUATIONS
HOMEWORK 3 (DUE 2/24)
(1) Consider the solution 1 x2 2kt to the heat equation ut = kuxx . Find the locations
of its maximum and its minimum in the closed rec
Vishnu Bachani
PDE Lecture 8 - 2 Mar 2017
A Fourier sine series is the 2`-periodic extension of the odd extension of functions defined on (0.`]. The Fourier cosine
series is the 2`-periodic extension
Vishnu Bachani
PDE Lecture 6 - 16 Feb 2017
4.1 - Separation of Variables
Let us consider the one-dimensional wave equation with Dirichlet boundary conditions, i.e. the string does not move at
each end
Vishnu Bachani
PDE Lecture 4 - 7 Feb 2017
The minimum principle
Given a solution to the heat equation ut = kuxx , k > 0, the minimum is assumed either at t = 0 or x = 0 or x = `.
Proof
u satisfies the