MATHS 361 Partial Differential Equations Mid-semester Test
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Department of Mathematics
MATHS 361
Partial Differential Equations
Mid-semester Test
April 10, 2019
1. (15 marks) Consider the following Sturm-Liouville problem:
y\\ (x) + λy(x) = 0 for 0 < x < 3, with y(0) = 0, y\ (3) = 0.
(a) Find all the eigenvalues and eigenfunctions.
(b) Compute the eigenfunction expansion of f (x) = x2 . You may leave the inner products in integral form.
Blank page for your working.
2. (10 marks) Consider the partial differential equation
ut = u北北 (1)
for x e [0, 1] and t > 0, with boundary conditions
u北 (0, t) = 0
u(1, t) = 0.
Which of the following are solutions to the PDE and boundary conditions? In each case explain your answer. Note that initial conditions are not given. [Hint: it is not necessarily easiest to solve the problem above.]
(a)
o
u(x, t) = cn cos (nπx) e-n2 π 2 t
n=1
(b)
u(x, t) = n cn sin ┌╱n - 、πx┐ e- (n- )2 π 2 t
(c)
u(x, t) = o cn cos ┌╱n - 、πx┐ e- (n- )2 π 2 t
(d)
u(x, t) = n00 cn cos ┌╱n - 、πx┐ e- (n- )2 π 2 t
(e)
u(x, t) = n cn sin ┌╱n - 、πx┐ e+ (n- )2 π 2 t
3. (15 marks) Consider the following PDE with boundary and initial conditions:
?2u 1 ?u 1 ?2u
?T2 T ?T T2 ?θ 2
with
u(T, 0) = 0, u(T, π/2) = 0, u(1, θ) = 3 sin(3θ).
(a) Use separation of variables to derive the two ordinary differential equa- tions problems that you would need to solve, along with their bound- ary/initial conditions. You do not need to solve these problems.
(b) Describe the problems that you have found, and explain how you would use their solutions to construct the solution to the original problem.
4. (15 marks) Consider the following PDE with boundary and initial conditions: ut = u北北 - u北 , for 0 < x < 1 and t > 0
with u(0, t) = 1, u(1, t) = 0, u(x,0) = x
(a) Find the steady state solution, us (x), for the PDE.
(b) Let U (x, t) = u(x, t) - us (x). Derive a PDE plus boundary and initial conditions for U (x, t). Show your working.
(c) Assuming there is a solution of the form U (x, t) = X(x)T (t), derive a boundary value problem that is satisfied by X(x). You do not need to solve the boundary value problem.
2023-03-30