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Tagged with bessel-functions differential-equations
6 questions
0
votes
2
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1k
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Orthogonality of Bessel function $\int_0^bxJ_a(\ell x)J_a(\ell' x)=0$ for $\ell\neq\ell'$
How do I show the above relation with Sturm-Liouville theory (assume the usual boundary conditions for the identity)? Here is what I have tried: if we start with
$$
\big(xJ_a'(\ell x)
\big)'+\left(\...
- 1,755
0
votes
1
answer
300
views
Alternate forms of the Bessel equation
I have a question regarding an alternate form of the Bessel equation and how that alternate form translates to the modified Bessel equation and its solution. The modified form is from:
http://...
- 4,713
4
votes
0
answers
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On modified Bessel solutions to complex ODE's using Kummer's series
I am trying to reduce the following ODE to Bessel's ODE form and hence solve it:
$$x^{2}y''(x)+x(4x^{3}-3)y'(x)+(4x^{8}-5x^{2}+3)y(x)=0\tag{1} \, .$$
I tried to solve it via the standard method, i.e.,...
- 449
3
votes
0
answers
108
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Convergence of Bessel (Sturm-Liouville) Expansions at the End Points
I have asked this question before on MSE but received no answer at all. So I assume that it is proper to ask it here. I am not a mathematician so my language may not be too precise, please correct me ...
CommunityBot
- 1
3
votes
0
answers
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Green's Function for a Kernel with Symmetric Fourier Transform $\nabla^2-x^2$
I am trying to find the inverse of the following kernel in 3 dimensions
$$
\nabla^2-x^2,
$$
where,
$$
x^2=\vec{x}.\vec{x}
$$
It seems quit simple and one would think there should already be solutions ...
- 459
0
votes
0
answers
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Energy Oscillations in a One Dimensional Crystal
Good day!
Can anyone help me find articles on similar topics "Energy Oscillations in a One Dimensional Crystal" (I have links to one article on this subject)?
article, that I have
Especially ...
