All Questions
Tagged with legendre-polynomials partial-differential-equations
13
questions
2
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0
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Fourier-Legendre series for $x^n$
I'm struggling to find the Legendre expansion for $x^n$ (exercise 15.1.17 from Mathematical Methods for Physicists).
I'm trying to evaluate the following integral:
$$a_m = \frac{2m+1}{2} \int_{-1}^{1} ...
- 29
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votes
1
answer
119
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Prove of a Legendre's polynomial problem
I have a question that I need to prove that for $n≥1$,
$$\frac{1}{2n}\int_{-1}^{1}x\frac{d}{dx}(P_n(x)^2)dx=\frac{2}{2n+1}$$
I have to evaluate the integral instead of using the orthogonality property ...
0
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1
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165
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What will be the explicit formula for Shifted Legendre's polynomial in interval $x\in[a,b]$
If I defined shifted Legendre polynomial $\tilde P_{n}(x)=P_{n}(\frac{2x-b-a}{b-a}) for\;all x\in[a,b]$ Then what will be the explicit formula $P_{n}(x)=\sum_{k=0}^{n} (-1)^{n+k} \frac{(n+k)!}{(n-k)! ...
1
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1
answer
179
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Eigenfunction expansion of a heat equation with Legendre polynomials
I am trying to solve the following PDE by performing an eigenfunction expansion:
$$
\frac{\partial p}{\partial t} = -\cos \varphi \frac{\partial p}{\partial x} + D\frac{\partial^2 p}{\partial \varphi^...
- 185
1
vote
2
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344
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Orthogonality of Legendre polynomials using specific properties
I'm having significant issues with a problem and would appreciate any help at all with it. It is regarding proving the orthogonality of Legendre polynomials using a specific recursion formula and ...
3
votes
0
answers
212
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Solving PDE with Legendre functions
I'm studying a paper which has a PDE of the form
$$\frac{\partial p}{\partial L}(L,n) = A\bigg((n^2-1)\frac{\partial p}{\partial n}(L,n)\bigg),\quad n>1,$$
with a Dirac delta initial condition $p(...
- 457
2
votes
1
answer
380
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Since the solution of the Legendre’s equations are the Legendre polynomials $P_n(\mu)$, write the general solution for $u(r, \theta)$.
I am working on the problem
Consider the steady-state of the heat equation in a ball of radius a centred at the origin. In spherical coordinates, the ball occupied the region $0 \le r \le a$, $0 \...
user575678
0
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1
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741
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Distributive properties of Laplacian operator to solve Poisson-like equation?
As I understand it, the Laplacian operator is linear, and thus $\nabla^2(f +g) =\nabla^2f + \nabla^2g.$ I was wondering if this might be exploitable to solve Poisson-like equations in spherical / ...
- 367
1
vote
2
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477
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Which approach is correct to find $a_n$'s for Legendre Equation solution with boundary conditions?
Imagine we have a solution to a differential equation of the form $$T(r,x)=\sum_{n=0}^\infty a_nr^nP_n(x)$$ where $P_n(x)$ is the legendre polynomial satisfying the legendre equation of index $2$.
We ...
- 5,247
2
votes
2
answers
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Proof orthogonality of Legendre polynomials using Sturm Liouiville theory only
I want to prove that Legendre Polynomials corresponding to different eigenvalues are orthogonal, using (if possible) only Sturm Liouville Theory. I had a look here but they seem to prove something ...
- 5,247
0
votes
1
answer
1k
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Laplace Equation on Prolate Spheroidal Coordinates
I'm currently trying to solve Laplace's equation outside of a prolate spheroid. The scalar field has to vanish far from the spheroid. Because of the geometry I thought it might be convenient to use ...
- 530
0
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0
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217
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Closure for Legendre Polynomials
Suppose I have a PDE which can be solved by an expansion on the Legendre Polynomial basis, for an axisymmetric problem in spherical coordinates.
For the derivation for such a problem, see these ...
- 441
1
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1
answer
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Writing the squared sine as a Legendre polynomial of cosine
I'm just getting learning about how Legendre polynomials come about when considering product solutions in spherical coordinates with azimuthal symmetry. I'm trying a problem on my own, and I'm a bit ...
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