Differential equation in Annulus

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SUMMARY

The discussion focuses on solving the differential equation \(\nabla^{2}\psi = A\psi\) for an annulus defined by inner diameter \(a\) and outer diameter \(b\). The solution involves using the Laplacian in polar coordinates, leading to a Bessel function as the result. The boundary conditions specified are \(\psi = 0\) at \(r = a\) and \(r = b\). The participants confirm that the approach is correct without needing to transform coordinates to account for the annulus structure.

PREREQUISITES
  • Understanding of differential equations, specifically Laplace's equation
  • Familiarity with Bessel functions and their applications
  • Knowledge of polar coordinates and their use in solving partial differential equations
  • Basic grasp of boundary value problems in mathematical physics
NEXT STEPS
  • Study the derivation and properties of Bessel functions in detail
  • Learn about boundary value problems and their significance in physics
  • Explore coordinate transformations in solving differential equations
  • Investigate applications of Laplace's equation in annular geometries
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Mathematicians, physicists, and engineering students interested in solving differential equations, particularly in contexts involving annular geometries and boundary value problems.

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Hi,

I want to solve a differential equation which goes like this..

\nabla^{2}\psi = A\psi

For an annulus having inner diameter as a and outer diameter as b. (A is some constant)

I can write down the laplacian in polar co-ordinates and carry on and get a bessel function as the solution.. And then apply the boundary conditions that at r=a and r=b \psi = 0

Will this do? Or will i have to incorporate the structure of the annulus in the equation like making a co-ordinate transform from r to r-a etc..
 
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What you say you are doing is perfectly correct.
 

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