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Document Preview: Homework 1 (due September 7)
While you are free to discuss homework problems among yourselves, the
homework solutions submitted must be your own . If two solutions are
identical, then both solutions will receive partial credit.
(1) Find a formula that implicitly denes the solution u =u(t;x) of the
initial value problem for the reaction-advection equation
u
u + 2u =  ; x2R; t> 0
t x
u + 2
with initial condition
 x=2
u(0;x) =e for x2R:
Show from this implicit formula that you can always solve for u in
terms of x and t. Also show that u(t;x)> 0 for all x2R and t> 0
so that the implicit formula makes sense.
x
(2) Solve the equation yu +xu = 0 with u(x; 0) =e . In which region
x y
of the xy plane is the solution uniquely determined?
2 x
(3) Solve u   3u   4u = 0; u(x; 0) =x ; u (x; 0) =e .
xx xt tt t
Hint: Factor the operator as we did for the wave equation.
1

Robert · Accepted Answer

3) uxx - 3uxt – 4utt = 0
3) uxx - 3uxt – 4utt = 0
1 - 3µ - 4µ2 = 0
The solutions are µ1 = -1/4 , µ2 =1
dx/dt = 1/4 and dx/dt = -1
whose solutions are,
x – 1/4t = C1 and x + t = C2
u(x,t) = F(x-1/4t) + G(x+t)
the derivative ut(x,t) = -1/4F1(x-1/4t) + G1(x+t)
the initial conditions given are
F(x) + G(x) = x2……………..

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