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PDEs Problem: STURM-LIVOUVILLE PROBLEMS 1. Determine the normalized eigenfunctions of the given problem 2 Ans:()-sin
2. Determine the normalized eigenfunctions of the given problem y" + ly = 0, y(0) = 0, y'(1)=0)
Use the solutions obtained in Problem 3 to find the eigenvalues and normalized eigenfunctions of the Sturm-Liouville problem
What are the eigenvalues and eigenfunctions of the Sturm-Louiville problem ODE: X" + X = 0, 0 < x < 1 BCs: ⇢ X(0) = 0 X(0) = 0 What are the functions p(x), q(x), and r(x) in the general Sturm-Louiville problem? We were unable to transcribe this imageWe were unable to transcribe this image
6. Determine the sequence of eigenvalues and corresponding eigenfunctions for each of the following Sturm-Liouville systems: (a) + Ap = 0, 9 (0) - 0) , 3º (2) + 5 °' (2) - 0) (b) 1 (0) + R -0, R'(1) - 0, R' (eº) - 0
II. 1. Find the eigenvalues and the eigenfunctions for the following Sturm-Liouville problem X"+AX=0, x(0)=0, X'(TT) = 0
#2 ONLY PLEASE 1. Consider the non-Sturm-Liouville differential equation Multiply this equation by H(x). Determine H(x) such that the equation may be reduced to the standard Sturm-Liouville form: do Given a(z), 3(2), and 7(2), what are p(x), σ(x), and q(x) 2. Consider the eigenvalue problem (a) Use the result from the previous problem to put this in Sturm-Liouville form (b) Using the Rayleigh quotient, show that λ > 0. (c) Solve this equation subject to the boundary conditions and determine...
Determine the eigenvalues and eigenfunctions for the eigenvalue problem Hint: this is not a Sturm Liouville problem since the equation is not self-adjoint. Suggest a transformation of the dependent variable to reduce the problem to a self-adjoint one. We were unable to transcribe this image0 < x < π, y'(0) 1/ ( π) = 0 0
(20 points) For the following problem use separation of variables to identify the Sturm-Liouville Problem and its eigenvalues and eigenfunctions. DO NOT solve it. This is a steady state temper- ature problem in a cylinder where the temperature depends on ρ and z only. 4. Up(1,2) = 0, 0<z<1 a(p, 0) 0, u(p, 1) 5, 0 < ρ < 1 (20 points) For the following problem use separation of variables to identify the Sturm-Liouville Problem and its eigenvalues and eigenfunctions....
and determine in which regions the given PDE’s are elliptic/ hyperbolic/ parabolic Problem 3 Find all solution of the following PDEs 1. 2. --= y, u=u(z, y). Problem 3 Find all solution of the following PDEs 1. 2. --= y, u=u(z, y).
6. Consider the eigenvalue problem 1 < x < 2, y(1) = 0, y(2) = 0. (a) Write the problem in Sturm-Liouville form, identifying p, q, and w. (b) Is the problem regular? Explain |(c) Is the operator S symmetric? Explain. (d) Find all eigenvalues and eigenfunctions. Discuss in light of Theorem 4.3 (e) Find the orthogonal expansion of f(x) = ln x, 1 < x < 2, in terms of these eigenfunctions. (f) Find the smallest N such that...