Consider the Sturm-Liouville problem
X′′(x) + λX(x) = 0 subject toX′(0) = 0, X(l) = 0.
Are the boundary conditions symmetric?
Do these boundary conditions yield negative eigenvalues?
Determine the eigenvalues and eigenfunctions, Xn(x). (It is
enough in some cases to provide the equation that determines the
eigenvalues rather than an explicit formula.)
Are the eigenfunctions orthogonal?
Determine the eigenvalues and the corresponding normalized
eigenfunctions of the following Sturm–Liouville problem: y''(x) +
λy(x) = 0, x ∈ [0;L], y(0) = 0, y(L) = 0,
Find the eigenvalues and eigenfunctions of the Sturm-Liouville
system
y"+ lamda y = 0 o<x<1
y(0) = 0
y'(1) = 0
(b) Show that the eigenfunctions Yn and Ym you obtained from the
above
are orthogonal if n not= m.
Consider a semi-infinite square well: U(x)=0 for 0 ≤ x ≤ L,
U(x)=U0 for x > L, and U(x) is infinity otherwise.
Determine the wavefunction for E < Uo , as far as possible,
and
obtain the transcendental equation for the allowable energies E.
Find the necessary condition(s) on E for the solution to exist.
Consider the equation uux + uy = 0 with the initial
condition
u(x, 0) = h(x) = ⇢ 0 for x > 0
uo for x < 0, with
uo< 0.
Show that there is a second weak solution with a shock along the
line x = uo y / 2
The solution in both mathematical and graphical presentation
before and after the shock.