II. 1. Find the eigenvalues and the eigenfunctions for the following Sturm-Liouville problem X"+AX=0, x(0)=0, X'(TT)...
(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....
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
#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...
5. Consider the problem a2y"y _2.J 0 x1 = 0, y(0) 0, y(1= 0. (a) Put the problem in Sturm-Liouville form and explain the nature of any singular points. (b) State the appropriate modified boundary conditions (c) Find all eigenvalues and eigenfunctions for the modified problem 5. Consider the problem a2y"y _2.J 0 x1 = 0, y(0) 0, y(1= 0. (a) Put the problem in Sturm-Liouville form and explain the nature of any singular points. (b) State the appropriate modified...
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
Use the solutions obtained in Problem 3 to find the eigenvalues and normalized eigenfunctions of the Sturm-Liouville problem
4. Consider the following Sturm-Liouville problem with u(0)u'(0)-0 and u(1) u 0 ISTANBUL TECHNICAL UNIVERSITY, UZB218E, Return date: Before 16 (a) Find the eigenvalues. (b) Solve the problem. 4. Consider the following Sturm-Liouville problem with u(0)u'(0)-0 and u(1) u 0 ISTANBUL TECHNICAL UNIVERSITY, UZB218E, Return date: Before 16 (a) Find the eigenvalues. (b) Solve the problem.
Please show steps. Use an energy argument to show that the eigenvalues of the Sturm-Liouville Problem on (0, L) given by X"(x)--AX(x), X,(0) = 0, X'(L) = 0. 0 < x < L, are non-negative, i.e., course Moodle page. > 0. Note that an example of the energy method is available on the 1
2) For the Sturm-Liouville eigenvalue problem + λφ-0, dt2 do 0, dc (a) 0 verify the following properties: a) The nth eigenfunction has (n-1) zeros on the open interval 0<x<a b) There are an infinite number of eigenvalues with a smallest, but no largest. c) What does the Rayleigh Quotient say about negative and zero eigenfunctions.
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...