Homework Answers
Solve part (d) 6. Consider the eigenvalue problem 2"xy3y Ay 0 y(1)0, y(2)= 0. + 1 < x< 2, (a) Write the proble...
Solve part (d)
6. Consider the eigenvalue problem 2"xy3y Ay 0 y(1)0, y(2)= 0. + 1 < x< 2, (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 ln x, 1 < 2, in terms of these (e) Find the orthogonal expansion of f(x) eigenfunctions _
6. Consider the eigenvalue problem 2"xy3y...
6. Consider the eigenvalue problem 1 < x < 2, y(1) = 0, y(2) = 0. (a) Write the problem in Sturm-Liouville form,...
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...
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 explai...
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...
II. 1. Find the eigenvalues and the eigenfunctions for the following Sturm-Liouville problem X"+AX=0, x(0)=0, X'(TT)...
II. 1. Find the eigenvalues and the eigenfunctions for the following Sturm-Liouville problem X"+AX=0, x(0)=0, X'(TT) = 0
Consider the following second order PDE Uit – 9Uxx = 0, 0<x< < t > 0,...
Consider the following second order PDE Uit – 9Uxx = 0, 0<x< < t > 0, (A) and the following boundary value/initial conditions: Ux(t,0) = uſt, 5) = 0, t>0, u(0, x) = 44(0, x) = 4 cos’ x, 0<x< (BC) (IC) for the function u= u(t, x). a. (5 points) Find ordinary differential equations for functions T = T(t) and X = X(x) such that the function u(t, x) = T(t)X(x) satisfies the PDE (A). b. (5 points) Find...
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.
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.
(a) Consider the following ODE ф" — 4ф + 4ф+ 0 (1) - with (0)(1)0 i....
(a) Consider the following ODE ф" — 4ф + 4ф+ 0 (1) - with (0)(1)0 i. Put (1 into standard Sturm-Liouville form ii. Find the corresponding eigenvalue relation and eigenfunctions. Note that you do not have to normalise the eigenfunctions. (b) Solve the heat equation (2) 0<х<1 t>0 Ut u(0, t) u(1,t) u(x, 0) sin(2тx) + 1
(a) Consider the following ODE ф" — 4ф + 4ф+ 0 (1) - with (0)(1)0 i. Put (1 into standard Sturm-Liouville form ii....
Solve the heat equation 4,0 < x < 3,1 > 0 kou det u(0, 1) =...
Solve the heat equation 4,0 < x < 3,1 > 0 kou det u(0, 1) = 0, u(3,t) = 0,1 > 0 S2, 0<x< } u(x,0) = { 10, { <x<3 are the eigenfunctions You will need to apply separation of variables to obtain a family of product solutions un(x, t) = x (x)Ty(t) where X of a Sturm-Liouville problem with eigenvalues an (as in Section 12.1). Using the explicit expressions for un(x, t) gives (8,0) = ŠA, n=0 Then...
1. Consider the non-Sturm-Liouville differential equation Multiply this equation by H(x). Determi...
#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...
2) For the Sturm-Liouville eigenvalue problem + λφ-0, dt2 do 0, dc (a) 0 verify the...
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.
Solve part (d)
6. Consider the eigenvalue problem 2"xy3y Ay 0 y(1)0, y(2)= 0. + 1 < x< 2, (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 ln x, 1 < 2, in terms of these (e) Find the orthogonal expansion of f(x) eigenfunctions _
6. Consider the eigenvalue problem 2"xy3y...
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...
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...
II. 1. Find the eigenvalues and the eigenfunctions for the following Sturm-Liouville problem X"+AX=0, x(0)=0, X'(TT) = 0
Consider the following second order PDE Uit – 9Uxx = 0, 0<x< < t > 0, (A) and the following boundary value/initial conditions: Ux(t,0) = uſt, 5) = 0, t>0, u(0, x) = 44(0, x) = 4 cos’ x, 0<x< (BC) (IC) for the function u= u(t, x). a. (5 points) Find ordinary differential equations for functions T = T(t) and X = X(x) such that the function u(t, x) = T(t)X(x) satisfies the PDE (A). b. (5 points) Find...
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.
(a) Consider the following ODE ф" — 4ф + 4ф+ 0 (1) - with (0)(1)0 i. Put (1 into standard Sturm-Liouville form ii. Find the corresponding eigenvalue relation and eigenfunctions. Note that you do not have to normalise the eigenfunctions. (b) Solve the heat equation (2) 0<х<1 t>0 Ut u(0, t) u(1,t) u(x, 0) sin(2тx) + 1
(a) Consider the following ODE ф" — 4ф + 4ф+ 0 (1) - with (0)(1)0 i. Put (1 into standard Sturm-Liouville form ii....
Solve the heat equation 4,0 < x < 3,1 > 0 kou det u(0, 1) = 0, u(3,t) = 0,1 > 0 S2, 0<x< } u(x,0) = { 10, { <x<3 are the eigenfunctions You will need to apply separation of variables to obtain a family of product solutions un(x, t) = x (x)Ty(t) where X of a Sturm-Liouville problem with eigenvalues an (as in Section 12.1). Using the explicit expressions for un(x, t) gives (8,0) = ŠA, n=0 Then...
#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...
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.
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