Determine the eigenvalues and eigenfunctions of the following operator (assume σ(x) ≡ 1): L(u) = u''−2u...

Determine the eigenvalues and eigenfunctions of the following operator (assume σ(x) ≡ 1): L(u) = u''−2u x ∈ (−1,1) with periodic boundary conditions u(−1) = u(1), u'(−1) = u'(1). Box your ﬁnal answer

Expert Solution

Colby Messinger answered 1 year ago

Show that the following are eigenfunctions of the Laplacian operator and determine the eigenvalues: a. (r^-1)sinkr...

Show that the following are eigenfunctions of the Laplacian operator and determine the eigenvalues: a. (r^-1)sinkr b.( r^-3)[(sin^2)θ]sin2φ

Consider the following linear operator: L u = d^2u/dx^2+ 2 du/dx + u. Consider the eigenvalue...

Consider the following linear operator: L u = d^2u/dx^2+ 2 du/dx + u. Consider the eigenvalue problem L u + 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?

Determine the eigenvalues and the corresponding normalized eigenfunctions of the following Sturm–Liouville problem: y''(x) + λy(x)...

Determine the eigenvalues and the corresponding normalized eigenfunctions of the following Sturm–Liouville problem: y''(x) + λy(x) = 0, x ∈ [0;L], y(0) = 0, y(L) = 0,

Find the eigenvalues and eigenfunctions of the given boundary value problem. Assume that all eigenvalues are...

Find the eigenvalues and eigenfunctions of the given boundary value problem. Assume that all eigenvalues are real. (Let n represent an arbitrary positive number.) y''+λy= 0， y（0）= 0， y'（π）= 0

Determine the eigenvalues and eigenfunctions of the following regular Sturm–Liouville systems: (a) y′′ + λy =...

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Solve the wave equation, a2 ∂2u ∂x2 = ∂2u ∂t2 0 < x < L, t...

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Find the eigenvalues and eigenfunctions for the following boundary value problem. y"+6y'-(L-8)=0, y(0)=0, y(2)=0 L ==...

Find the eigenvalues and eigenfunctions for the following boundary value problem. y"+6y'-(L-8)=0, y(0)=0, y(2)=0 L == Lambda

Find the eigenvalues and eigenfunctions of the Sturm-Liouville system y"+ lamda y = 0 o<x<1 y(0)...

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Si U=(x2+y2+z2)-1/2 , demuestre que ∂2U/∂x2+∂2U/∂y2+∂2U/∂z2=0

Find the eigenvalues λn and eigenfunctions yn(x) for the given boundary-value problem. (Give your answers in...

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