I have been trying to understand the process of solving partial differential equations - specifically related to electrostatics, but I think the issue is mathematical.
I would like to solve a 2D Laplace equation where three sides of an area under consideration are bound by "walls" and one side by a half-circle. Imagine a cross section of a tennis ball sitting in a cup.
Along the boundary the equation takes the same value, call it $A_0$. How can I specify these boundary conditions for the 2D Laplace equation? The majority of examples I have come across use a "rectangle" area with boundary conditions such as:
- $u(x,0) = C_1$
- $u(x,H) = C_2$
- $u(0,y) = C_3$
- $u(W,y) = C_4$
For the equation
$ \nabla^2 u = 0 $
In cartesian coordinates. But a condition for a circle would be either parametric or conditional, wouldn't it? For example, a circle of radius R could be described by the condition
$ x^2 + y^2 = R^2 $
Which brings to my mind a condition such as
$ u(x,y)=A_0 , \forall x,y: x^2 + y^2 = R^2$
But this does not fit with any solution methods I have come across yet. Does this make sense, and are there resources for problems like this?
The majority of examples you have come across use a rectangle area because this allows separation of variables in cartesian coordinates.
A full circle also works when using polar coordinates, for example.
A general problem will not be separable under any coordinate system. In that case there no analytical method to help you. You must solve your equation numerically and this can be very hard.