Solving Boundary Problem for PDE: v(r,θ) in a Disk Centered at Origin

[sara_87]

Homework Statement

We are given the pde:
$$\partial^2{v}/\partial{r^2}$$+(1/r)$$\partial{v}/\partial{r}$$+(1/r^2)$$\partial^2{v}/\partial{\theta^2}$$=0
and we are given that a general solution is given by:
v(r,$$\theta$$)=A$$_{0}$$+B$$_{0}$$ln(r) + $$\sum{r^{n}(A_{n}cos(n\theta) + B_{n}sin(n\theta))}+\sum{(1/r^n)(C_{n}cos(n\theta)+D_{n}sin(n\theta))}$$

show that if $$\Omega$$ is a disk centered at the origin of radius r0, continuous solutions of the pde are of the type:
v(r,$$\theta$$)=A0+$$\sum{r^n(A_{n}cos(n\theta)+B_{n}sin(n\theta))}$$

Homework Equations

The Attempt at a Solution

i think we have to show that \sum{(1/r^n)(C_{n}cos(n\theta)+D_{n}sin(n\theta))}[/tex] and B0ln(r) are not continuous but i don't think that's right.
Any help would be very much appreciated.

 

[mighty2000]

Thank you for your post. I am a scientist and I would be happy to assist you with your question.

First of all, let me clarify that the general solution you have provided is a solution to the given PDE, but it is not the most general solution. The most general solution would also include terms involving ln(r) and ln(r)cos(n\theta) and ln(r)sin(n\theta). However, for the purpose of this problem, we can ignore these terms as they do not affect the continuity of the solution on the disk \Omega.

Now, to prove that the continuous solutions of the given PDE are of the type v(r,\theta)=A0+\sum{r^n(A_{n}cos(n\theta)+B_{n}sin(n\theta))}, we need to show that the terms B0ln(r) and \sum{(1/r^n)(C_{n}cos(n\theta)+D_{n}sin(n\theta))} are not continuous on the disk \Omega.

To do this, we can use the definition of continuity, which states that a function f(x) is continuous at a point x=a if the limit of f(x) as x approaches a is equal to f(a). In other words, the function must have the same value at x=a and at any point close to x=a.

Let's start with B0ln(r). We can see that as r approaches 0, ln(r) becomes infinitely large, and thus B0ln(r) also becomes infinitely large. This means that the limit of B0ln(r) as r approaches 0 does not exist, and therefore B0ln(r) is not continuous at r=0. Since the disk \Omega is centered at the origin, this term is not continuous on the disk \Omega.

Next, let's look at the term \sum{(1/r^n)(C_{n}cos(n\theta)+D_{n}sin(n\theta))}. We can see that as r approaches 0, the terms (1/r^n) become infinitely large, and thus the entire sum becomes infinitely large. This means that the limit of \sum{(1/r^n)(C_{n}cos(n\theta)+D_{n}sin(n\theta))} as r approaches 0 does not exist, and therefore this term is not continuous at r=0. Again, since the disk \