Help with PDE in circular annulus(poisson eq)

[smoger]

what is the general solution of the poisson equation :

∂²A/∂r² + 1/r ∂A/∂r + 1/r² ∂²A/∂θ² = f(r,θ)

the function f(r,θ) is :
f(r,θ)=1/r (Ʃ X_ncos(nθ) + Y_nsin(nθ))

where the boundary is :

I(a<r<b, 0<θ<2pi)

the boundary condition is the netural boundary on (r=a) expressed as :

∂A/∂r=0 (r=a)


How can i find the A(r,θ)? i can not find any books related to this.
Most of them only consider laplace equation where f(r,θ)=0
someone help me.

 

[Thaakisfox]

Use the Green function for Neumann boundary conditions.

 

[AlephZero]

Using the idea of separating the variables, you should be able to see from the PDE that

A(r,θ) = Cn(r) cos(nθ) + Sn(r) sin(nθ)

is a solution for the right hand side terms (1/r)(Xn cos(nθ) + Yn sin(nθ))

That will give you ordinary differential equations to solve for Cn(r) and Sn(r).

 

[ivan_f_costa]

A(r,θ)=∫∫f(ρ,θ') g(r,θ,ρ,θ') dρ dθ' + cte' from eq. 5.0.19 Ref.1.
where
g(r,θ,ρ,θ') = -ln{[r^2 + ρ^2 - 2rρ cos(θ-θ')] [b^2 + (rρ/b)^2 - 2rρ cos(θ-θ')]}/4∏ + r^2/(4∏b^2)
from third line of page 68 of Ref.2.

Ref.1. https://rs5tl5.rapidshare.com/#!download|5tl6|1682244|Duffy_D._Green_s_functions_with_applications__CRC__2001__T__404s_.djvu|2827|R~0|0|0

Ref.2. http://www1.maths.leeds.ac.uk/~kersale/Teach/M3414/Notes/chap4.pdf

 

[blue_raver22]

The Poisson equation is a common differential equation that appears in many areas of science and engineering, including fluid dynamics, heat transfer, and electromagnetism. In your case, you are dealing with a Poisson equation in a circular annulus, which is a region between two concentric circles. The equation you provided is known as the polar form of the Poisson equation, and it describes the variation of a function A(r,θ) in terms of its derivatives with respect to the radial coordinate r and the angular coordinate θ.

To find the general solution to this equation, you can use the method of separation of variables. This involves assuming that the solution can be written as a product of two functions, one depending only on r and the other depending only on θ. Substituting this into the equation and rearranging terms, you will end up with two separate ordinary differential equations, one for each of the two functions. These can then be solved using standard techniques, such as the method of undetermined coefficients or the method of variation of parameters.

In your case, the function f(r,θ) is given in a Fourier series form, which means it can be expressed as a sum of sine and cosine terms with different coefficients. This is a common form for functions that have periodic properties, such as the boundary conditions in your problem. To find the specific solution for A(r,θ), you will need to determine the coefficients Xn and Yn in the Fourier series. This can be done by applying the boundary conditions at r=a and r=b, as well as the condition that A(r,θ) must be single-valued and continuous at all points.

It is true that most books on the Poisson equation only consider the case where f(r,θ)=0, which is known as the Laplace equation. However, the method of separation of variables can also be applied to the Poisson equation with a non-zero forcing term, as in your case. You may need to consult a textbook on partial differential equations or a book specifically on the Poisson equation to find more detailed explanations and examples of solving this type of problem.

In summary, to find the solution to the Poisson equation in a circular annulus with the given boundary conditions and forcing term, you will need to apply the method of separation of variables and solve two ordinary differential equations. You can then use the boundary conditions to determine the coefficients in the solution. I hope this helps guide you in finding a