Description
1. Consider the wave equation:
,
which is a special case of the general quasi-linear equation:
, with .
Find the slope of each of the two characteristics:
Find the expression in terms of and for and , so that the wave equation simplifies
to
2. Use the Fourier transform method to solve the 2-D Laplace equation in the upper plane
for the bounded solution:
Assume is of compact support;
3. Solve the following problem in two ways:
(a) by the method of similarity transformation. Look for the value of such that the
PDE reduces to an ode in
(b) by an integral transform in t, in this case a Laplace transform (You can use a table of
Laplace transform to do the inverse transform).
2 0 in 0, .
( ,0) ( ), .
uy x
ux f x x
Ñ = > – ¥ < < ¥
= – ¥ < < ¥
f(x) u(x, y)→0 as x → ±∞.
∂
∂t
u = ∂2
∂x
2 u , 0 < t;0 < x < ∞
u(x,0) = 0, u(x,t) bounded as x → ∞.
u(0,t) = T0 , a constant, t > 0.
a
, / ; x ta h h =