MAT311H5F
Assignment 8
Problem 1
Prove Greens rst identity: For every pair of functions X1 (x), X2 (x) on
(a, b),
b
b
f (x) g (x) dx = f (x) g (x)
a
a
b
f (x) g (x) dx.
a
Problem 2
Use the previous problem to show that the following eigenvalue problem has
MAT311H5F
Assignment 7
Problem 1
Any function dened on a symmetric interval [l, l] can be (uniquely) written
as = e + o where e is an even function and o is an odd function. Find
e and o in terms of . (Hint: you should write the above equality at x and
at
MAT311H5F
Assignment 6
Problem 1
Show that the following identity holds for any two functions f , g in L2 [a, b]:
f +g
2
2
+ f g
2
2
= 2( f
2
2
+ g 2 ).
2
Problem 2
Find the Fourier Sine Series of x2 on [0, ].
Problem 3
Let en (x) = sin (n + 1/2)x. Check
MAT311H5F
Assignment 5
Problem 1
Solve the Heat equation ut = kuxx for 0 < x < , t > 0 with the initial
condition
u(x, 0) = 1 + 2 sin x
and the boundary conditions u(0, t) = u(, t) = 1.
Hint: Notice that the boundary condition is not homogeneous.
Problem
MAT311H5F
Assignment 4
Problem 1
Show that if u(x, t) and v (x, t) are solutions to the Dirichlet problems for
the Heat equation
ut (x, t) kuxx (x, t) = f (x, t),
u(x, 0) = 1 (x), u(0, t) = u(1, t) = g1 (t)
vt (x, t) kvxx (x, t) = f (x, t),
v (x, 0) = 2 (
MAT311H5F
Assignment 3
Problem 1
Consider the problem
u (x) + u(x) = f (x),
u(0) = u( ) = 0,
where f (x) is a given function.
Does this equation have a solution for every function f ? If the equation
has a solution for a given f , is it unique?
Problem 2
MAT311H5F
Assignment 2
Problem 1
Find the solution of the equation
2ux + 5uy = 0
which satises the condition u(x, 1) = 10x + 1.
Problem 2
Solve the equation
3ux + 2uy = 4y.
Hint: First you need to nd one solution.
Problem 3
Solve the equation
yux + 2xuy =
MAT311H5F
Assignment 1
Problem 1
Which of the following operators are linear?
1. L(u) = sin xu(x, y ) cos x.
2. L(u) = uxy ex u.
3. L(u) = u + ex (u )2 u.
Problem 2
Are the polynomials x, 1 + x, x2 2x linearly dependent or independent?
Do they span the sp