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  1. D

    I Help getting started with this differential equation

    I need to solve ∂2Φ/∂s2 + (1/s)*∂Φ/ds - C = 0 Where s is a radial coordinate and C is a constant. I know this is fairly simple but I haven't had to solve a problem like this in a long time. Can someone advise me on how to begin working towards a general solution? Is the method of...
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    I Finish and plot this separation of variables problem

    As I understand it the density is going to be calculated based on the enthalpy once psi is known for each iteration of a larger algorithm. So setting m = 0 and B = 0, and choosing ψ (s1) = ψouter = 0 as a boundary condition then I can state the R(r) solution as a sum, with λ0,n defined in terms...
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    I Finish and plot this separation of variables problem

    Ok after studying that link I might be a little closer to making sense of this. My domain does not include the origin. Should I discard the ##Y_k## anyway because the function needs to remain physical anyway? In any case I should treat the bessel functions the same way I would treat sine or...
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    I Finish and plot this separation of variables problem

    As I said I am not practiced with using bessel functions in any capacity (besides a few homework problems over a year ago) but I am getting the impression that they are difficult to work with unless you can state a boundary R(a) = 0. This had something to do with the fact that the bessel...
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    I Finish and plot this separation of variables problem

    There is a lot going in to this derivation but it is intended to give the stream function and therefore velocity at equilibrium of a rotating system. Ω is simply frame rotation (taken to be known for the purposes of this question). Indeed the more compact form of this PDE is: $$ξ_z = ρf(ψ) -...
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    I Finish and plot this separation of variables problem

    I will also elaborate on my recent attempts to solve this: If I want the solution to be axisymmetric (which physically, I think I do), then it seems I need to require k = 0, then I can solve for A and B using inner and outer boundary conditions ##(ψ(s_0) = ψ_{inner} , ψ(s_1) = ψ_{outer})## with...
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    I Finish and plot this separation of variables problem

    Fair enough!: The PDE: $$ \frac{1}{s}⋅\frac{∂}{∂s}(\frac{s}{ρ}\frac{∂ψ}{∂s}) + \frac{1}{s^2}⋅\frac{∂}{∂Φ}(\frac{1}{ρ}\frac{∂ψ}{∂Φ}) - 2Ω + ρc_0 + ρc_1ψ = 0$$ What I think the general solution is: $$ ψ = \frac{1}{ρc_1}[AJ_k(ρs\sqrt{c_1}) + BY_k(ρs\sqrt{c_1})]⋅[Ccos(kΦ) + Dsin(kΦ)] +...
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    I Finish and plot this separation of variables problem

    I have a PDE which I have solved numerically using a guass-seidel method, but I want to compare it to the analytical solution. I have used separation of variables to get the general solution, but I need help applying it. The PDE is (1/s)⋅(∂/∂s)[(s/ρ)(∂ψ/∂s)] + (1/s2)⋅(∂/∂Φ)[(1/ρ)(∂ψ/∂Φ)] - 2Ω...
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    I Best method to solve this discretized PDE

    A couple of things I'd like to clarify as I'm having trouble with this. Shouldn't b = ρ2c1 per the first substitution? For the second set of substitutions, if we replace Ψ in the eq by Ψ' I can't get a0 to vanish algebraically as you have. Sorry if this is obvious, I've only worked with...
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    I Best method to solve this discretized PDE

    As an update if anyone is still interested, I constructed a guass-seidel algorithm which converges for most of the examples I've tried. One of my next steps will be to test the solution for a constant density using the analytical solution, so thanks for helping with that!
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    I Best method to solve this discretized PDE

    This is a numerical problem solving exercise. You're suggesting solve the two second order ODEs for R(r) and Θ(θ) by discretizing them and generating a tridiagonal matrix? I'm not sure I can justify defining ρ as constant in the first step though. We are assuming equilibrium (dψ/dt) = 0 but the...
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    I Best method to solve this discretized PDE

    Sorry I did neglect to identify my variables in that long post. ρ is mass density, assumed to be a known function of s and Φ. The work I've done so far trying to solve this numerically has been in python.
  13. D

    I Best method to solve this discretized PDE

    I am attempting to solve the following PDE for Ψ representing a stream function on a 2D annulus grid: (1/s)⋅(∂/∂s)[(s/ρ)(∂ψ/∂s)] + (1/s2)⋅(∂/∂Φ)[(1/ρ)(∂ψ/∂Φ)] - 2Ω + ρ(c0 + c1ψ) = 0 I have made a vertex centered discretization: (1/sj)⋅(1/Δs2)⋅[(sj+1/2/ρj+1/2,l){ψj+1,l - ψj,l} -...
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    I Help discretizing this PDE (stream function)

    No, the density depends on s and Φ, the entire equation is: (1/s)⋅(∂/∂s)[(s/ρ)(∂ψ/∂s)] + (1/s2)⋅(∂/∂Φ)[(1/ρ)(∂ψ/∂Φ)] - 2Ω + ρ(c0 + c1ψ) = 0 Where Ω is frame rotation rate and c0, c1 are arbitrary constants (from the first and second term of a taylor expansion).
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    I Help discretizing this PDE (stream function)

    Oh, I should probably mention that I am only giving the terms which I need help discretizing. The rest of the equation is straightforward to discretize and is mostly constant terms.
  16. D

    I Help discretizing this PDE (stream function)

    I have a PDE that I want to solve for a stream function ψ(j,l) by discretizing it on a 2D annulus grid in cylindrical coordinates, then solving with guas-seidel methods (or maybe a different method, not the point): (1/s)⋅(∂/∂s)[(s/ρ)(∂ψ/∂s)] + (1/s2)⋅(∂/∂Φ)[(1/ρ)(∂ψ/∂Φ)] Where s and Φ are...
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    Photovoltaics and time near a black hole (as a key story element)

    Thanks for the very thoughtful response! Indeed, I write stories for characters and adventures in interesting settings, but this story would be enhanced by at least some exploration of the physics I think. It was inspired by in actual discussion of GR with one of my professors after all. I...
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    Photovoltaics and time near a black hole (as a key story element)

    I just realized this thread belongs in writing and world building, not the general sci-fi section. Sorry! Can it be moved?
  19. D

    Photovoltaics and time near a black hole (as a key story element)

    I'm a physics student and science fiction writer, and I've never been to this section of PF before! But I have an idea I think is cool for a novel or short story and I'd like some input on the physics involved. As I'm not overly worried about this idea being stolen I'm going to provide a short...
  20. D

    Equation of motion for oscillations about a stable orbit

    Okay I got it. Start with radial equation and let r = r0 + α where α << r0. Then do a Taylor expansion about r0 and the first derivative of U is zero, the second is known. You end up with the equation of an oscillator and you can read off the frequency without even solving the differential...
  21. D

    Equation of motion for oscillations about a stable orbit

    Ok I did a little more work with 'the radial equation': μr⋅⋅ = d/dr(U) = γ/r2 - L2/μr3 L2 = μ2r4ω2 which implies r⋅⋅ = γ/r2 - ω2r Which looks a lot like a differential equation which gives simple harmonic motion except for the γ term. Now, when dU/dr = 0 which is the condition that I used...
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    Equation of motion for oscillations about a stable orbit

    Homework Statement A) By examining the effective potential energy find the radius at which a planet with angular momentum L can orbit the sun in a circular orbit with fixed r (I have done this already) B) Show that the orbit is stable in the sense that a small radial nudge will cause only...
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    Bound surface charge on a linear dielectric half-cylinder

    Terrific! Back to practicing electrostatics problems then. Thanks for your help!
  24. D

    Bound surface charge on a linear dielectric half-cylinder

    When considering the free charge only, we have cylindrical symmetry, but that symmetry does not apply to the entire problem. So for now I will abandon using Guass's law. As for χe, assuming that it is very small let's us say that ε ≈ ε0. This would suggest that the electric field in the...
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    Bound surface charge on a linear dielectric half-cylinder

    Homework Statement Problem statement in attached photo. This is an ungraded assigned problem which I am using to study for an exam, so I don't need the whole solution just help with a couple of points I am confused about. One: Part d) is really important to how I will answer part b). If we can...
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    How can I plot equipotentials from this series?...

    Thanks to both of you who replied. You are both correct. There is no 'a' in the denominator, my mistake. I plotted 1000 terms of the (correct) series in Mathematica in order to read off the equipotentials. The series does converge to a constant at y = a.
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    How can I plot equipotentials from this series?...

    Homework Statement The problem is given in the attached picture, but I already have a solution to part a) which I am confident in (I have checked it carefully, compared to other students and confirmed it with my graduate-TA). Part b) asks us to plot the equipotentials but I cannot figure out...
  28. D

    I Would going to the Moon delay going to Mars?

    The only advantage would be to test and refine a living structure on the lunar surface before taking one to Mars. Admittedly, we can test such a structure on Earth so I concede, there's little to know benefit as far as a Mars mission is concerned. The two aren't mutually exclusive though. Any...
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    I Would going to the Moon delay going to Mars?

    You have to consider a manned voyage to Mars to be a huge infrastructure task as well. So testing and developing part of that infrastructure on the moon is a valid idea I think. Stopping in Chicago to build a house might not help you get to LA faster, but you might be able to help the next guy...
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    Temperature as a function of time in black body radiation

    Wow, you are absolutely correct. That is an embarrassing rookie calculus mistake. There is no undefined expression, the sky is not falling. Thanks very much to you both.
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