Differential equation in Annulus

In summary, the speaker is discussing solving a differential equation in polar coordinates for an annulus with inner diameter a and outer diameter b. They mention using a Bessel function as the solution and applying boundary conditions at r=a and r=b. They also question whether they need to incorporate the structure of the annulus in the equation. The other person confirms that their approach is correct.
  • #1
FedEx
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Hi,

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

[itex]\nabla^{2}\psi = A\psi[/itex]

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 [itex]\psi = 0[/itex]

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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  • #2
What you say you are doing is perfectly correct.
 

Related to Differential equation in Annulus

What is a differential equation in annulus?

A differential equation in annulus is a mathematical equation that involves derivatives and is defined on a circular region with an inner and outer boundary. It is used to model physical phenomena such as heat transfer, fluid flow, and electrical fields in annular-shaped objects.

How is a differential equation in annulus different from a regular differential equation?

A differential equation in annulus differs from a regular differential equation in that it is defined on a circular region rather than a straight line. This means that the solutions to the equation will vary with respect to both radius and angle.

What are the applications of differential equations in annulus?

Differential equations in annulus have various applications in physics and engineering, including modeling heat transfer in cylindrical objects, analyzing fluid flow in pipes and tubes, and understanding the behavior of electrical fields in annular-shaped systems.

What are the common methods for solving a differential equation in annulus?

Some common methods for solving a differential equation in annulus include separation of variables, Laplace transform, and numerical methods such as finite difference and finite element methods.

What are the challenges in solving a differential equation in annulus?

Solving a differential equation in annulus can be challenging due to the complex geometry of the region and the need to consider both radial and angular variations in the solutions. Additionally, the boundary conditions may be more complicated compared to regular differential equations.

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