Analytical solution of Laplace's equation with horrendous boundary conditions

[Nardis]

Hi,

I'm trying to find an analytical solution of Laplace's equation:

$$\phi_{xx}$$ + $$\phi_{tt}$$ = 0

with the tricky boundary conditions:

1. $$\phi(x=0,|t|>\tau)= 0$$
2. $$\phi(x\neq0, |t|>>\tau)=0$$
3. $$\phi_{x}(x=0, |t|<\tau)=-1$$
4. $$\phi_{t}(x, |t|>>\tau)=0$$

I have the following ansatz(I think that's the correct term):

$$\phi(x,t)=\int^{\infty}_{0}A(k)e^{-kx}cos(kt)dk$$

i.e. a Fourier integral. It has the form that it has since I don't want the solution blowing up at infinity (I should also add that I'm only interested in $$x\geq0$$) and that the solution has to be even in time (this is required by the physics of the problem). My attempts to extract $$A(k)$$ using standard Fourier methods have failed, due to the difficulty of the b.cs.

Can anyone help me come up with an analytical solution to this problem?

PS, it IS possible to solve the problem using conformal mapping, but I'm trying to find another analytical way of solving it, mainly for the purposes of extension to another related problem.

 

[coomast]

Hi,

I'm trying to find an analytical solution of Laplace's equation:

$$\phi_{xx}$$ + $$\phi_{tt}$$ = 0

with the tricky boundary conditions:

1. $$\phi(x=0,|t|>\tau)= 0$$
2. $$\phi(x\neq0, |t|>>\tau)=0$$
3. $$\phi_{x}(x=0, |t|<\tau)=-1$$
4. $$\phi_{t}(x, |t|>>\tau)=0$$

I have the following ansatz(I think that's the correct term):

$$\phi(x,t)=\int^{\infty}_{0}A(k)e^{-kx}cos(kt)dk$$

i.e. a Fourier integral. It has the form that it has since I don't want the solution blowing up at infinity (I should also add that I'm only interested in $$x\geq0$$) and that the solution has to be even in time (this is required by the physics of the problem). My attempts to extract $$A(k)$$ using standard Fourier methods have failed, due to the difficulty of the b.cs.

Can anyone help me come up with an analytical solution to this problem?

PS, it IS possible to solve the problem using conformal mapping, but I'm trying to find another analytical way of solving it, mainly for the purposes of extension to another related problem.

Welcome to the forum Nardis.

I have a few questions regarding this problem.

*) Is the time t running from $$-\infty$$ to $$+\infty$$, or is the problem supposed to be for positive t only? If the latter is true, why the absolute value?
*) What is the condition $$\phi(x=0, |t|<\tau)=?$$

I assume that you need an analytic solution? I am looking for a series solution, assuming this is also valid...

 

[Nardis]

Hi coomast,

1. Yes, time is running from $$-\infty$$ to $$\infty$$. Sorry I didn't make that clearer.

2. Unfortunately $$\phi(x=0, |t|<\tau)$$ is unknown. This and the fact that $$\phi_x(x=0)$$ is unknown for all time is what makes the problem hard.

Yep, it'd be great if you could find a series solution.

 

[coomast]

Hi coomast,

1. Yes, time is running from $$-\infty$$ to $$\infty$$. Sorry I didn't make that clearer.

2. Unfortunately $$\phi(x=0, |t|<\tau)$$ is unknown. This and the fact that $$\phi_x(x=0)$$ is unknown for all time is what makes the problem hard.

Yep, it'd be great if you could find a series solution.

There is still something that I don't understand. How can the Laplace equation be with time?
You either have the Laplace equation in some region bounded by two (or more) space variables or the equation for describing waves (with time). This here seems to be something mixed?

In solving these equations, the boundary must be completely given in some way, p.e. a function, it's derivative or some mixed conditions, but at least at the whole boundary which is not the case here.

Can you tell me more on what it is for or where it is coming from?

 

[coomast]

Nardis?