Reference no: EM132374936
Consider the following linear operator: Lu = d2u/dx2 + 2du/dx + u.
1. What is the corresponding adjoint problem of the boundary value problem
Lu = 0, x ∈ (0, π), u(0) = 0, u'(π) + u(π) = 0.
2. Comment on the existence and uniqueness of solutions for the following boundary value problem
Lu = f (x), x ∈ (0, π), u(0) = 0, u'(π) + u(π) = 0. (Note that the boundary conditions are the same as in problem 1.)
3. Consider the eigenvalue problem
Lu + λu = 0, x ∈ (0, π), u(0) = 0, u(π) = 0.
(a) Determine the possible eigenvalues and eigenfunctions.
(b) Use an integrating factor to put the eigenvalue problem in Sturm-Liouville form.
(c) What is the appropriate inner product for this system?
4. Use the results from problem 3 to express f (x) = e-x as an eigenfunction expansion.