Laplace Equation Polar Form

Your Name]In summary, the problem provided is a BVP with a circular domain and periodic boundary conditions. The general solution can be written as u(r,ψ) = C + r^{n}Ʃ_{n}a_{n}cos nψ + b_{n}sin nψ, and the Fourier constants can be determined by plugging in the boundary values and solving for a and b. An alternative approach using separation of variables may also be used to simplify the solution.
  • #1
middleramen

Homework Statement



Solve the BVP:

r[itex]^{2}[/itex]u[itex]_{rr}[/itex] + ru[itex]_{r}[/itex] + u[itex]_{ψψ}[/itex] = 0

0 ≤ r ≤ 1, 0 < ψ < 2π

u(1,ψ) = 0.5(π - ψ)



Homework Equations





The Attempt at a Solution



I've derived the general solution of u(r,ψ) = C + r[itex]^{n}[/itex]Ʃ[itex]_{n}[/itex]a[itex]_{n}[/itex]cos nψ + b[itex]_{n}[/itex]sin nψ, where a,b, C are constants.

Attempts to determine Fourier constants don't give meaningful results.

Also, it was my understanding that the boundary condition must be periodic, i.e. u(1,0) = u(1,2π), which is untrue for this problem.
 
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  • #2






Thank you for posting your question. I am a scientist and I would be happy to help you solve the BVP provided.

Firstly, I would like to confirm that the boundary conditions for this problem are indeed periodic, as you suspected. This means that u(1,0) = u(1,2π) and u(1,ψ) = 0.5(π - ψ) for 0 < ψ < 2π. This is because the problem is defined on a circular domain, and the solution must be periodic in order to accurately represent the physical system.

Based on your attempt at a solution, it seems like you have correctly derived the general solution for this BVP. However, in order to determine the Fourier constants, you will need to use the boundary conditions. You can do this by plugging in the boundary values for r and ψ into your general solution and solving for the constants. This will result in a set of equations that can be solved simultaneously to find the values of a and b.

I would also like to mention that the general solution you derived may not be the most efficient way to solve this problem. An alternative approach would be to use separation of variables, where you assume that the solution can be written as a product of two functions, one that depends only on r and one that depends only on ψ. This will result in a simpler form of the solution, which can then be used to determine the Fourier constants.

I hope this helps you solve the BVP. If you have any further questions or need more clarification, please do not hesitate to ask. Good luck!




 

1. What is Laplace Equation in Polar Form?

The Laplace equation in polar form is a partial differential equation that describes the relationship between a function and its second derivatives. It is often used in the field of physics and engineering to model various phenomena such as heat diffusion, electrostatics, and fluid flow.

2. What is the difference between Laplace Equation in Cartesian Form and Polar Form?

The Laplace equation in Cartesian form is written in terms of the independent variables x, y, and z, while the polar form is written in terms of the independent variables r and θ. The polar form is often more convenient for problems involving circular or cylindrical symmetry, while the Cartesian form is more useful for problems with rectangular or Cartesian coordinates.

3. How is Laplace Equation in Polar Form derived?

The Laplace equation in polar form can be derived from the Cartesian form using the chain rule and the definition of polar coordinates. It is also possible to derive it directly from fundamental principles of physics, such as the conservation of energy or the principle of least action.

4. What are the boundary conditions for solving Laplace Equation in Polar Form?

The boundary conditions for solving Laplace equation in polar form depend on the specific problem being solved. In general, they specify the values of the function or its derivatives at the boundaries of the region in which the equation is being solved. These boundary conditions are necessary to ensure a unique solution to the equation.

5. What are some real-world applications of Laplace Equation in Polar Form?

Laplace equation in polar form has a wide range of applications in physics, engineering, and other fields. Some examples include modeling the flow of heat in a circular metal plate, analyzing the electric potential around a charged cylindrical conductor, and predicting the pressure distribution in a rotating fluid. It is also used in image processing, signal analysis, and other areas of mathematics and computer science.

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