Then is [tex] u(\sqrt{6}, \theta) = \sqrt{6} \sin(\theta) + 6\sin^2(\theta) [/tex] a boundary condition?harsh said:You are trying to solve the Laplace equation on a disk. Try separation of variables, then break it down to 2 ODE's. Here is a start for you..
You will probably need to solve the PDE in polar coordinates.
- harsh
Tony11235 said:Then is [tex] u(\sqrt{6}, \theta) = \sqrt{6} \sin(\theta) + 6\sin^2(\theta) [/tex] a boundary condition?
I know. In an earlier problem I had to compute the laplacian in polar. Oh and one more thing, is there anything else I need to know about [tex] \theta [/tex]? Other than [tex] 0 < \theta < 2\pi [/tex] ?harsh said:Looks right. Make sure you solve the correct PDE, the laplacian in r,theta is not as simple as U_rr and U_theta*theta
- harsh
A harmonic function is a mathematical function that satisfies the Laplace equation, which is a partial differential equation. In simpler terms, a harmonic function is a function whose value at any point is the average of its values on the surrounding points.
To find a harmonic function on a disk, we need to know the boundary conditions, which are the values of the function on the boundary of the disk. Then, we can use the Laplace equation and solve for the function's values within the disk.
Yes, a harmonic function can have multiple solutions on a disk. This is because the Laplace equation is a second-order differential equation, which means it has two independent solutions. Therefore, we can have multiple harmonic functions that satisfy the same boundary conditions on a disk.
The function U(x,y)=y+y^2 is a specific example of a harmonic function on a disk. It represents a parabolic function that increases in a particular direction on the disk. This function can be used to model various physical phenomena, such as heat flow or fluid flow.
Harmonic functions are closely related to complex analysis. In fact, harmonic functions can be represented as the real part of a complex analytic function. This allows us to use powerful techniques from complex analysis to solve problems involving harmonic functions.