Introduction - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 11

1.11 Solve the heat equation for a rod with the temperature fixed only at the left
end. Suppose that at the right end, the rod is isolated, so that the heat flow, and
thus ∂T /∂x , vanishes there.
Hint: For k odd, the functions φ k (x) = sin kx satisfy the boundary conditions

0 ,ϕ ( 2 ) =

φ k ( 0 ) = 0 .

1.12 Suppose u is a solution of the wave equation, and that at time t = 0, u is

zero outside of a bounded set. Show that the energy

[ u t + c 2 ( grad u) 2 ] dx

( 1 . 19 )

d

R

is constant.
Hint: Write the wave equation in the symmetric form

u t = c div v,
v t = c grad u,

and represent the time derivative of the integrand in (1.19) as the divergence of an
appropriate expression.