Reference no: EM13916150
1. Solve the initial value problem
{ ut + cos(kt)ux = ax2
u(x, 0) = 0
Plot the solution for k = Π/3 and a = 1, as well as the projected characteristic curves for this PDE, finding a way to plot them both on the same graph.
2. Show that, for functions obeying homogeneous Dirichlet conditions on the boundary of a domain D, the differential operator ζ
ζ = a(x)d2/dx2 + b(x) d/dx + c(x)
is self-adjoint if a' (x) = b(x).
3. Prove that if the weight function w(x) in a Sturm-Liouville problem is increased for all values of x in a domain D, then the smallest eigenvalue decreases.
4. Find all solutions to the Sturm-Liouville problem on the domain D = {lx| < d}:
[d2/dx2 - β2 ]u = λu on D
du/dx = 0 on ∂D
for β < Π/d. Plot the 5 solutions with the smallest values of λ. Prove that all solutions are orthogonal.
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: Plot the solution for k = Π/3 and a = 1, as well as the projected characteristic curves for this PDE, finding a way to plot them both on the same graph - Prove that if the weight function w(x) in a Sturm-Liouville problem is increased for all valu..
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