Week 9 Lectures, Math 716, Tanveer 1
1.1
Greens function as a distribution
Laplace Operator
For the Poisson-Problem with homogeneous boundary condition: u = f for x , u = 0 on we know that u(x0 ) =
(1)
G(x, x0 )f (x)dx
(2)
On the otherhand, if u is a test
Week 7 Lectures, Math 716, Tanveer 1 Introduction
Recall we discussed completeness of Fourier Series. This is relevant for constant coecient partial dierential equations in simple rectangular geometries or sometimes in circular geometry as well. Separatio
Week 6 Lectures, Math 716, Tanveer 1 Fourier Series
n=1
In the context of separation of variable to nd solutions of PDEs, we encountered f (x) = or bn sin
nx for x (0, l) l
(1)
a0 nx f (x) = + an cos for x (0, l) 2 l n=1 and in other cases f (x) = nx nx
Week 5 Lectures, Math 716, Tanveer 1 Separation of variable method
The method of separation of variable is a suitable technique for determining solutions to linear PDEs, usually with constant coecients, when the domain is bounded in at least one of the in
Week 4 Lectures, Math 716, Tanveer 1 Diffusion in Rn
x2 1 exp - S(x, t) = 4t 4t is a special solution to 1-D heat equation with properties S(x, t)dx = 1 for
R
Recall that for scalar x, (1)
t > 0, and yet
t0+
lim S(x, t) = 0 for fixed x = 0
(2)
This was ca
Week 3 Lectures, Math 716, Tanveer 1 Wave Equation as 1st order homogeneous system of PDEs
utt - c2 uxx = 0 is to notice that for C2 functions, it is equivalent to a system of 1st order equation ut = cvx , vt = cux (2) (1)
Another approach to linear wave
Week 2, Math 716 1 Linear 1st order PDEs in two independent variables
a1 (x1 , x2 )ux1 + a2 (x1 , x2 )ux2 = c(x1 , x2 ) where a1 , a2 and c are continuous function in some domain R2 . Denote x = (x1 , x2 ). We will assume 1. On some dierentiable curve = c
Week 1 notes: Math 716 1 PDE: Order, Linear & Nonlinear
A differential equation that involves more than one independent variable is called a partial differential equation, abbreviated as PDE. The order of the highest derivative occurring in the PDE is def
Solution to homework Set 6: Math 716
1. Determine the Greens function G(x, x0 ) for Dirichlet condition for Laplaces equation in 3-D dimensions in the hemispherical domain: = cfw_x : |x| < 1, x3 > 0 Use this to determine an integral expression for u(r, ,
Solution to Set 5: Math 716
1. Show that the partial dierential operator A dened by: Au = (p u) + qu
in a bounded Rn for p(x) > 0 is symmetric, with respect to the usual L2 inner-product. What condition on q (x) makes A positive. Suppose we consider the e
Homework Set 3: Math 716, Due: Wednesday, February 7th
1. Use energy method to prove uniqueness of classical solution to the initial value problem for the damped wave equation ( > 0): utt +ut = u for x Rn , t > 0 , with u(x, 0) = (x), ut (x, 0) = (x) , u(
Homework Set 2: Math 716, Due: Friday, January 26th
1. Consider the wave equation utt = c2 uxx , u(x, 0) = (x), ut (x, 0) = (x) for x R with c = 0. Assume C2 and C1 have compact support (i.e. there is a number R > 0 such that (x) = (x) = 0 for |x| > R). D
Homework Set 6: Math 716, Due Friday, March 6th
1. Determine the Greens function G(x, x0 ) for Dirichlet condition for Laplaces equation in
3-D dimensions in the hemispherical domain:
= cfw_x : |x| < 1, x3 > 0
Use this to determine an integral expression
Homework Set 3: Math 716, Due: Wednesday, February 11th
1. Use energy method to prove uniqueness of classical solution to the initial value problem for the damped wave equation ( > 0): utt +ut = u for x Rn , t > 0 , with u(x, 0) = (x), ut (x, 0) = (x) , u
Math 716, JR 221, Wi, '09,: Introduction to PDE The emphasis in Math 716 is on solving linear partial differential equations (PDEs) using a variety of different methods. Topics that will be discussed are: (1.) Linear partial differential equations, and th