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Maths 361: Partial Differential Equations

Assignment 2, due Friday, 12th May, 11:59pm

Full marks:  50 marks.

2.  (15 marks) Consider the following PDE:

ut  = u北北 + γu,    0 < x < 1,    t > 0

u(0, t) = 0,    u(1, t) = 0,    t > 0

u(x,0) = 1,    0 < x < 1

with the parameter γ ∈ R.

(a)  Solve using separation of variables.   You may leave the eigenfunction expansion

coefficients in inner product form.

(b) Explain how γ affects the behaviour of the solution.

3.  (10 marks) Using separation of variables, solve the PDE

u北北 + uuu  = 0

on x ∈ [0, 2], y ∈ [0, 3] with the boundary conditions

u(x, 0) = 0,                                         u(x,3) = 0,

u北 (0, y) = sin(πy),                               u(2, y) = 0.

Plot the solution that you obtain.

4.  (10 marks) Consider the equation

^xu北 + uu  = 0, 0 < x < o, 0 < y < o

u(0, y) = eu , u(x, 0) = 1

(a) If you wanted to use Laplace transforms to solve the above PDE, which coordinate

should I transform in? Why?

(b) Laplace transform the PDE above, and solve the ODE in the new coordinate system

to find U .  (You will need to rearrange and use integrating factors.  Your final solution will include two terms, one purely dependent on s, and the other with a  “constant” of integration C(s)).

(c) Use the initial value of your PDE to determine the remaining “constant” C(s).

(d) Finally, use your table of Laplace transforms to find u(x, y).