1.   (a)    The velocity u(x, t) of a fluid on the interval 0 < x < L obeys the viscous Burger’s equation

∂u       ∂u        ∂2u

∂t       ∂x       ∂x2

where u(0, t) = 0, u(L, t) = 0 and u(x, 0) = u0 x(L < x)/L2 . You can assume that u0  and ν are positive real constants.

(i)    What are the dimensions of u0  and ν?

(ii)    Assuming a weak nonlinearity, nondimensionalize this problem. (b)    We now consider the inviscid Burger’s equation

∂u       ∂u

∂t       ∂x

subject to initial conditions

(2 marks) (4 marks)

,．2,       u(x, 0) =．．1 < x,

． ．

．．1,

for x - <1,

for  < 1 < x - 0,

for x > 0.

(i)    Sketch the initial conditions. Construct the characteristics diagram in the (x, t)-plane by sketching the characteristics emanating from each of the intervals x - <1, <1 < x - 0 and x > 0. Show that a shock forms at t = 1.                                                (5 marks)

(ii)    Find an explicit solution valid for 0 < t - 1 and sketch it for t = 1/2 and t = 1.

(4 marks)

(iii)    Find an explicit solution after the shock has formed, sketch an amended diagram of characteristics showing the shock path and sketch the solution at t = 2.         (5 marks)

(Total: 20 marks)

2.   (a)    Consider the following PDE

y2   + x2   = 2xy2

(i)    Find the general solution to this PDE.                                                            (4 marks)

(ii)    Show that the particular solution to this PDE if you are given that u(0, y) = exp(y3 ) and u(x, 0) = x2 + exp(<x3 ) is

u(x, y) = x2 + exp(y3 < x3 )

(3 marks)

(iii)    What is the particular solution to this PDE if now you are given that u(0, y) = <y6  and u(x, 0) = x2 < x6 ?                                                                                          (3 marks)

(b)    A flat hammer hits an infinite horizontal string. The vertical displacement of the string is the solution to the following initial value problem

∂2u      ∂2u

 (x, 0) = φ(x)

with

φ(x) =．, 1,   for |x| - 1

(i)    Give explicit solutions for t = 0, t = 1/2, t = 1 and t > 1.                              (7 marks) (ii)    Sketch on the same graph the vertical displacement of the string for t = 0, t = 1/2, t = 1, t = 3/2, t = 2 and t = 5.                                                                     (3 marks)

(Total: 20 marks)

3.   (a)    Find the regions of the (x, y)-plane where the equation

(1 + x) + 2xy + 2y2   + x2   < xy2   + (1 < y)u = 0

is parabolic, hyperbolic or elliptic. Sketch these regions.                                        (5 marks)

(b)    Consider a perfectly elastic and flexible string of length L with linear density ρ under uniform tension τ . The string is stretched horizontally and fixed at both ends. Initially, it is vertically displaced at its mid-point by an amount A. At t = 0, the string is released.

(i)    Show that the vertical displacement in the string is given by

u(x, t) = ī  sin ┌ ┐ cos ┌ ┐

where c = |τ /ρ is the wavespeed.                                                                  (7 marks)

(ii)    Show that the kinetic energy of the string when it passes through its rest position is given by

2A2 ρc2

K =

L

You may use the following result     1/(2n + 1)2  = π 2 /8. What is the work done in first displacing the string (no formal proof required)?                                      (8 marks)

(Total: 20 marks)

4.   (a)    In y3 , the Klein-Gordon equation governs the quantum mechanical wavefunction ψ(≠ ) of a relativistic spinless particle with mass m. It reads

V2 ψ < m2 ψ = 0

Show that the solution for the scalar field ψ(≠ ) in any volume V bounded by a surface S is unique if either Dirichlet or Neumann conditions are specified on the surface S.    (5 marks)

(b)    Now we wish to use Green’s functions to find solutions to the non-homogeneous Klein-Gordon

equation with a density of charge ρ(≠ ), ≠ e y3 .

(i)    By applying the divergence theorem to the volume integral

卜  ←u(V2 < m2 )v < v(V2 < m2 )u」dV = 0

(where u and v are twice differentiable scalar fields) obtain a Green’s function expression for the solution ψ to

V2 ψ < m2 ψ = ρ(≠ )

in a bounded volume V and which takes the value ψ(≠ ) = f(≠ ) on S , the boundary of

V . The Green’s function, G(≠ , ≠0 ) to be used satisfies the following equation V2 G < m2 G = δ(≠ < ≠0 )

and vanishes when ≠ e S.                                                                               (6 marks) (ii)    When V = y3 , the Green’s function G(≠ , ≠0 ) can be written as G(η) = g(η)/η, where η =  |≠ < ≠0 | and g(η) is bounded as η o o.  Show that G(η) is a solution to the

following ODE

ηG// + 2G/ < m2 ηG = 0

Thus, conclude that

G(η) = <   

(6 marks)

(iii)    Finally, find ψ(≠ ) in the half-space x > 0 in the case of a unique positive charge located at ≠ 1  such that ψ = 0 both on the plane x = 0 and as r o o.                      (3 marks)

(Total: 20 marks)