The Sturm–Liouville equation is a particular second-order linear differential equation with boundary conditions that often occurs in the study of linear, separable partial differential equations.
Questions tagged [sturm-liouville]
481 questions
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Sturm-Lioville problem: Term by term differentiation of eigenfunction expansion
Say I want to solve an Sturm-Lioville problem with non-homogeneous term but homogeneous boundary conditions:
$y''+(1/x)y'+(1/x^2)y=x$
$y'(0)=0$
$y(1)=0$
One method of solution involves finding the eigenfunctions associated with the homogeneous…
user3199900
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3
votes
2 answers
Rayleigh quotient on circular region of radius 2
I' m struggling with the following problem.
We have the eigenvalue problem:
$$u'' + \lambda u = 0$$
with associated boundary condition:
$$u' + 3u = 0$$
Now by using the Rayleigh quotient for $0 \leq r \leq 2$and the trial function:
$$\psi_0(r)=…
Joe Goldiamond
- 1,094
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votes
1 answer
Solve S.L problem
$$ y'' + \lambda y = 0 , ~~0 < x<1$$
$$ y'(0) = 0, y'(1) - y(1) = 0.$$
I have shown that there are no eigenvalues corresponding to $\lambda=0$.
For $\lambda < 0$, I let $\lambda = - k^2, k^2>0.$
Then solving the ODE gives
$$ y =…
Btzzzz
- 1,113
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votes
1 answer
Question about $\lambda = 0$ case in Sturm-Liouville problem
Given
$u''(x) + \lambda u(x) = 0, u'(0) = u'(\frac{\pi}{2}) = 0$,
I wish to compute the eigenvalues and corresponding eigenfunctions.
If $\lambda = 0$,
$u''(x) = 0 \implies u'(x) = A \implies u(x) = Ax + B$.
If $\lambda < 0$,
using the…
table
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vote
2 answers
Solving a Sturm-Liouville problem
I want to solve the Sturm-Liouville problem:
$\mathcal{A}u=-u''$
with I.C:
\begin{cases}
u(0)=0\\
u'(L)=0
\end{cases}
where $\mathcal{A}$ is the Sturm-Liuville operator defined on $\mathscr{D}_A:\{u\in C^2([0,L])$.
By reading this post from SE , I…
Luthier415Hz
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Sturm-Liouville problem involving $dy/dx$
$$ \frac{d^2y}{dx^2} + 2\frac{dy}{dx} + \lambda y=0 \ , y(-1)=y(1) = 0 $$
I have 2 ideas
1- $ (\frac{dy}{dx} +2y )' + \lambda y = 0 \\ \text{which means weighted function is } w=1 $
2- $ \frac{dy}{dx} (p \frac{dy}{dx}) + \lambda w(x)y = 0 \\…
tt z
- 129
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System of Schrodinger equations
I know that Schrodinger equation for 1 dimension is given by:
$-u''+qu=\lambda u$ and if I want to work on a system for say 2 dimensions,
the system would be:
\begin{equation}
\left(\begin{array}{c} -u_1''\\ -u_2''…
S.N.A
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Sturm Liouville problem on entire line, substitution
Observe Sturm-Liouville problem on entire line
$$-(p(x)y'(x))' + l(x)y(x)= \lambda r(x)y(x), \hspace{3mm} -\infty
Nebojša Đurić
- 331
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vote
2 answers
Find all positive eigenvalues and eigenfunctions for regular Sturm-Liouville
Regular Sturm-Liouville boundary-value problem on the interval [-4,4] given by
$$y''+\lambda y=0$$
with $y(-4)=0$ and $y(4)=0$.
Given that $\lambda >0$ the general solution is
$$ a\cos\left(\sqrt{\lambda }x\right)+ b\sin\left(\sqrt{\lambda…
geeking4math
- 93
1
vote
1 answer
Sturm-Liouville and general solution
Help with any of this is much appreciated. I tried multiple ways to figure out the problems and not once was I getting anywhere.
i. To find the general solution, I tried to use the characteristic equation but I don't think that is the correct way.…
geeking4math
- 93
1
vote
1 answer
Reduce to Sturm-liouville $My=x^2y''+3xy+y$
I need to reduce to sturm-liouville using the 'integrating factor' method:
$$My=x^2y''+3xy+y$$ where y=y(x). I am looking at my notes and I know the Sturm-Liouville form is $$\left(P(x)y'\right)'+Q(x)y$$ However, I don't understand what to do. If…
geeking4math
- 93
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votes
1 answer
Converting differential equation into sturm -Liouville
Convert the following equation to the form of a Sturm-Louville equation.
$$3x^2y''(x)+4xy'+6y(x)+\lambda y(x)=0,x>0$$
I used this.
But the substitution was tedious and I couldn't find the sturm-Liouville form. Please help me out!
Unknown
- 3,073
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How do I know that the eigenvalues of a Sturm-Liouville problem are simple or multiple?
In a Sturm Liouville problem, how can we predict that it only has simple eigenvalues or it contains multiple eigenvalues? Do the boundary conditions such as 1) zero Dirichlet 2) periodic 3) semi-periodic matter? Do regularity/singularity…
0
votes
1 answer
Put each equation in self-adjoint form (Sturm-Liouvulle)
Put each equation in self -adjoint form
$x^{2}{y}''-x{y}'+\lambda y=0$
If I want to put it in the self-attached form of Sturm Liouvulle, I first do this:
$x{y}''-{y}'+\frac{\lambda }{x}y=0$
$e^{-\int…
Enrique-akatsuki
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show that $λ=0$ is an eigenvalue of the regular Sturm- Liouville system
show that $λ=0$ is an eigenvalue of the regular Sturm- Liouville system
$\frac{\mathrm{d} }{\mathrm{d} x}[p(x){y}']+\lambda r(x)y=0$
${y}'(0)=0$
${y}'(1)=0$
What I have done is this:
$\frac{\mathrm{d} }{\mathrm{d} x}[p(x){y}']+\lambda…
Enrique-akatsuki
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