MATH2065: INTRO TO PDEs
Summer School 2015
Tutorial Questions 7
1. Solve Laplaces equation
2 2
+
=0
x2 y 2
for (x, y) in the square 0 < x < L, 0 < y < L, subject to the boundary conditions
(x, 0)
(x, L)
(0, y)
(L, y)
=
=
=
=
0 for 0 x L,
0 for 0 x L,
0 f
MATH2065: INTRO TO PDEs
Summer School 2015
Tutorial Questions 3
Questions marked with the dagger symbol are intentionally more challenging.
1. Find the Laplace transforms of each of the following functions.
(a) t3 e2t
(b) t sin 4t
(c) H(t 3)
(d) e3t sin 4
MATH2065: INTRO TO PDEs
Summer School 2015
Tutorial Questions 5
1. To practice your partial dierentiation, verify that
u(x, t) = sin (2x) e4
2 kt
is a solution to the heat equation
u
2u
=k 2.
t
x
2. Verify that u(x, t) = sin (x ct) is a solution to the w
MATH2065: INTRO TO PDEs
Summer School 2015
Tutorial Questions 2
Questions marked with the dagger symbol are intentionally more challenging.
1. Use the method of undetermined coecients to nd a particular solution to each of the
following inhomogeneous dier
MATH2065: INTRO TO PDEs
Summer School 2015
Tutorial Questions 6
1. Determine the eigenvalues and corresponding eigenfunctions of the dierential equation
d2
+ = 0
dx2
d
with boundary conditions (0) = 0 and
(L) = 0. Analyze the three cases with real :
dx
MATH2065: INTRO TO PDEs
Summer School 2015
Tutorial Questions 9
Questions marked with the dagger symbol are intentionally more challenging.
1. For the following functions, sketch the Fourier series of f (x) on the interval L x L
and determine the Fourier
MATH2065: INTRO TO PDEs
Summer School 2015
Tutorial Questions 13
This Tutorial Set contains a selection of questions covering material over the semester.
1. Consider the wave equation
2u
2u
= c2 2
;
< x < ,
t2
x
u(x, 0) = f (x) ,
u
(x, 0) = 0 .
t
(a) L
MATH2065: INTRO TO PDEs
Summer School 2015
Tutorial 8
There is no new material in this tutorial; Instead, you are being given a chance to revise
previous work.
1. Consider the initial value problem
d
5 = sin t + (t + 1) e5t
dt
;
(0) = 0 ,
where the unkno
MATH2065: INTRO TO PDEs
Summer School 2015
Tutorial Questions 10
Questions marked with the dagger symbol are intentionally more challenging.
1. (a) Calculate the Fourier series for the 2-periodic extension of the function
f (x) = x2 ,
Answer:
2
3
+4
(1)n
MATH2065: INTRO TO PDEs
Summer School 2015
Tutorial Questions 1
1. Show that the ordinary dierential equation
dx
= x,
dt
= constant
has the general solution
x = A et ,
where A is a constant.
Hence solve
(a) dx/dt = 3x,
x = 2 when t = 0.
(b) dy/dz = 5y,
y