Numerical PDE's II - Circular Domain

In summary, the conversation is about approximating the solution of a partial differential equation with given boundary conditions in a circular annulus. The homework equations and attempt at a solution are also mentioned. The question is whether this problem can be applied elsewhere and when it would be appropriate to model something with a circular annulus.
  • #1
Nusc
760
2

Homework Statement


Approximate the solution of:
[tex]

\frac{\partial^{2}u }{{\partial r^{2} }} + \frac{1}{r}\frac{\partial u}{\partial r} + \frac{1}{r^{2}}\frac{\partial^{2}u}{\partial \theta^{2}} = 0, 1<r<3, 0<\theta<\pi,

u(1,\theta) = 1-\cos(\theta),u(2,\theta)=u(r,0)=0 u(r,\pi)=4-2r

[/tex]

Homework Equations


[tex]

\delta r = \frac{r_{outer} - r_{inner}}{N}
\delta \theta = \frac{\theta_{end} - \theta_{start}}{M}

[/tex]
n,m are positve integers

[tex]
\delta r_i = r_{inner} + i\delta r
[/tex]
[tex]
\delta theta_j = \theta_{start} + j\delta \theta
[/tex]

The Attempt at a Solution


[tex]
\frac{u_{i+1,j} - 2U_{i,j} + U_{i-1,j}}{(\delta r)^2} + \frac{1}{r_i}\frac{u_{i+1,j}-u_{i-1,j}}{2\delta r} +\frac{u_{i,j+1} - 2U_{i,j} + U_{i,j-1}}{(\delta \theta)^2} = f_{i,j}
[/tex]

Multiply this equation by [tex] (\delta r)^2
[/tex]
[tex]
u_{i+1,j} - 2u_{i,j} + u_{i-1,j} + \frac{1}{r_i}\frac{u_{i+1,j}-u_{i-1,j}}{2}\delta r + \frac{1}{r_i}\frac{\delta r}{\delta \theta}(u_{i+1,j}-2u{i,j}+u{i,j-1} = f_{i,j}(\delta r)^2

[/tex]

What do I do next>
 
Last edited:
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  • #2
And the question is?
 
  • #3
Nevermind, I got it.

But does anyone know if such a problem with those given BC's can be applied elsewhere?
When would you model something with a circular annulus?
 

1. What is a numerical PDE?

A numerical PDE (partial differential equation) is a mathematical equation that describes the behavior and change of a physical system over time or space. It involves multiple variables and their derivatives, and can be solved using numerical methods on a computer.

2. What is a circular domain in the context of numerical PDE's?

A circular domain refers to a two-dimensional region or area that is bounded by a circle. In numerical PDE's, this circular domain is often used to represent physical systems with circular symmetry, such as a circular membrane or a circular pool of liquid.

3. What are some common numerical methods used to solve PDE's on a circular domain?

Some common numerical methods for solving PDE's on a circular domain include finite difference methods, finite element methods, and spectral methods. These methods involve discretizing the circular domain into a grid or mesh and using iterative calculations to approximate the solution.

4. How do boundary conditions affect the solution of a PDE on a circular domain?

Boundary conditions specify the behavior of the solution at the edges of the circular domain. They can significantly affect the solution of a PDE, as they introduce constraints and information about the physical system being modeled. Inaccurate or incorrect boundary conditions can lead to erroneous results.

5. What are some real-world applications of PDE's on circular domains?

PDE's on circular domains have a wide range of applications in various fields such as physics, engineering, and biology. Some examples include modeling heat transfer in a circular metal plate, simulating the flow of fluids in a circular pipe, and predicting the spread of diseases in a population with circular movement patterns.

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