Essential Dierential Equations
Solutions 6
1. The generic solution of the ODE system du = Au is given by u(t) =
dt
eAt u(0). In this example we have the symmetric matrix
1 2
2 1
A=
.
The simplest way to solve the ODE system is to diagonalise A by computin

Essential Dierential Equations
Time 15.0515.55
Closed book test: 05112013
Attempt all six questions
No Calculators
(1) Consider a vector space X. Give the denition of a norm
If X = R2 , which of the following are norms?
x
1
=
1
(|x1 | + |x2 |),
2
x
2
= mi

Examples 6
Essential Dierential Equations
1. Write down the generic solution of the following ODE system: nd u(t)
R2 such that
du
=
dt
1 2
2 1
u;
t > 0;
with
u(0) =
1
2
.
Can you determine an explicit solution? (Hint: to do this you need to
calculate the

Solutions 5
Essential Dierential Equations
1. Note that u(j) (x) = sin(jx) satises the boundary condition u(j) (1) = 0
if and only if j is an integer. Moreover, if j = 0 then the solution is
identically zero and if j is positive then eigenfunctions u(j) c

Examples 5
Essential Dierential Equations
Consider the following eigenvalue problem: nd the pair cfw_, u, where R
and u : (0, 1) R, such that
u (x) = u(x)
x (0, 1),
(E)
together with the boundary conditions u(0) = 0 = u(1).
1. Show that there are innitely

Solutions 1
Essential Partial Dierential Equations
1. The six functions f1 , f2 , . . . , f6 are plotted below. The functions with
positive exponents are continuous over [0, 1]. The functions with negative
exponent are unbounded in the limit x 0 so are no

Examples 4
Essential Dierential Equations
1. Let V := cfw_v|v H 1 (0, 1); v(0) = 0, v(1) = 0 and consider the following
variational problem: given
f (x) =
1, 0 < x 1/2,
0, 1/2 < x < 1,
nd u V such that
a(u, ) = (),
where
V,
()
1
1
u
a(u, ) =
and
f.
()

Solutions 2
Essential Partial Dierential Equations
Ax
Ax
1. Let x = 0 be the function that satises x = supx=0 x . Note that
Ax / x A for any specic function x X. To show that A is a
valid norm we simply check the axioms:
positivity
A =
Ax
( 0)
=
0
x
(> 0

Solutions 4
Essential Dierential Equations
1
1. (a) We have u = 1 for x 1/2 and u(0) = 0. Hence, u(x) = 2 x2 +bx
= 0 for x > 1/2 and u(1) = 0 so that
for some b. Similarly, u
u(x) = a(x 1) for some constant a.
If we enforce the conditions that u and u ar

Solutions 3
Essential Dierential Equations
1. Given that Xh X we know that vh X so that
a(u, vh ) = (vh ) = a(uh , vh ) a(u, vh ) a(uh , vh ) = 0.
Since a(, ) is bilinear the result follows.
2. First, since a is coercive on X, we know that there exists >

Examples 1
Essential Partial Dierential Equations
1. Using MATLAB or otherwise, plot the following functions f : (0, 1) R
and list the ones that are continuous on [0, 1].
(i) f1 (x) = x2
(ii) f2 (x) = x3/2
(iii) f3 (x) = x1/2
(iv) f4 (x) = x1/4
(v) f5 (x)