Lecture 5: Weak Solutions. We have seen that a general solution to the equation
(5.1)
utt uxx = 0
is given by
(5.2)
u(t, x) = h(t x) + k (t + x)
If h, k C 2 then (5.2) is C 2 so the equation (5.1) makes sense point wise. This
is called a classical solutio
Lecture 3: The Fourier transform. The Fourier transform F : f f is
dened to be
( ) =
(3.1)
f
f (x) eix dx.
Rn
The Fourier transform is invertible, in fact we will prove Fouriers inversion formula:
1
(3.2)
f (x) =
f ( ) eix dx
n
(2 ) Rn
The Fourier transf
Lecture 2: Convergence of Fourier series. Suppose that f (x) is a period
function f (x + 2 ) = f (x). Suppose also that f (x) is either in C 1 or L2 . Then the
Fourier series converges uniformly respectively in L2 to f (x):
f (x) =
ck eikx ,
k=
where the
Math 631 Partial Differential Equations I: Linear Equations
Lecture 1: Introduction.
1.1 Denition. A Partial Dierential Equation (PDE) of order k for a function
u(x) of x Rn is an equation involving u and its derivatives up to order k
(1.1)
F (x, u(x), u(
Lecture 3: The evolution equations and Fourier series. Let us consider the
simplest case of solving the linear wave equations on a circle:
(2.1)
2
2
t u x u = 0,
u(0, x) = f (x),
ut (0, x) = g (x),
were data are assumed to be periodic f (x + 2 ) = f (x) a
Lecture 14: Analytic Solutions. For the rst part below we are following Section 4.6 in Evans but for the proof of convergence we are following Taylor.
The simplest pde
(1)
t u(t, x) = cx u(t, x),
u(0, x) = g (x)
can be solved in the class of real analytic
Lecture 6:Weak limits. Any weak limit of a distribution is a distribution. We
say that fn f weakly if
fn dx
f dx,
C0 .
Moreover, it follows directly from the denitions that fn f if fn f .
Any distribution f is the weak limit of a sequence of fn C0 .
In
Lecture 7: Fundamental solutions using distribution theory. The funda
mental solution E of a partial dierential operator P (D) = a is dened by
P (D )E =
Using the fundamental solution one can solve the equation
P (D)u = F,
In fact u = E F satises
P (D)(E
Lecture 10: Heat Equation. Sections 2.2.5a and
2.3.1 Fundamental solution for the Heat eq., 2.3.4a Energy methods,
Problems 2.5:12a, 13, 14, page 87.
1
Lecture 12: Wave Equations. we did section 2.4.2 wave eq. in two dim and
2.4.2 wave eg. with in homogeneous term and related it to the previous fundamental
solutions we obtained with distribution theory.
1
Lecture 11: Wave Equations. We did section 2.3.4 Heat eq. with energy methods and section 3.4.1 Wave eq. rst radial solutions and then in general with
spherical means in 3 dimensions. Section 2.4.1abc.
Problems 2.5: 18,19, 24 page 88-90.
1
Lecture 8: Harmonic Functions.
We did sections 2.2 Introduction, 2.2.2 and 2.2.3ab(def) in Evans book, giving the
classical properties of harmonic functions. We already done 2.2.1 Fundamental
solution.
Physical interpretation of Laplace equation.
Mean val