# Thread: Half Line Wave Equation

1. ## Half Line Wave Equation

Hi,

Solve $\displaystyle \ \ \ \frac{\partial{u}}{\partial{t}} = k(\frac{\partial^2{u}}{\partial{t}^2}) \ \ \ \ \ \ \ u(x,0) = e^{-x} \ \ \ \ \ \ \ \ u(0,t) = 0 \ \ \ \ \$ on the half line $\displaystyle 0<x<\infty$.

I know you are supposed to put this into the equation $\displaystyle \frac{1}{\sqrt{4\pi*kt}}\int_0^\infty[e^\frac{-(x-y)^2}{4kt} - e^\frac{-(x+y)^2}{4kt}]\Phi(y)dy$ but i'm not sure how to simplify it.

Thanks

2. Originally Posted by tbyou87
Hi,

Solve $\displaystyle \ \ \ \frac{\partial{u}}{\partial{t}} = k(\frac{\partial^2{u}}{\partial{t}^2}) \ \ \ \ \ \ \ u(x,0) = e^{-x} \ \ \ \ \ \ \ \ u(0,t) = 0 \ \ \ \ \$ on the half line $\displaystyle 0<x<\infty$.

I know you are supposed to put this into the equation $\displaystyle \frac{1}{\sqrt{4\pi*kt}}\int_0^\infty[e^\frac{-(x-y)^2}{4kt} - e^\frac{-(x+y)^2}{4kt}]\Phi(y)dy$ but i'm not sure how to simplify it.

Thanks
What is $\displaystyle \Phi (y)$?

3. Basically $\displaystyle \Phi(y) = e^{-y}$
So you "just" need to simplify the integrand. I know you need to use the gaussian at some point and maybe competing the square for the exponents.
It thus becomes:
$\displaystyle \frac{1}{\sqrt{4\pi*kt}}\int_0^\infty[e^\frac{-(x-y)^2}{4kt} - e^\frac{-(x+y)^2}{4kt}]e^{-y}dy$

4. Originally Posted by tbyou87
Basically $\displaystyle \Phi(y) = e^{-y}$
So you "just" need to simplify the integrand. I know you need to use the gaussian at some point and maybe competing the square for the exponents.
It thus becomes:
$\displaystyle \frac{1}{\sqrt{4\pi*kt}}\int_0^\infty[e^\frac{-(x-y)^2}{4kt} - e^\frac{-(x+y)^2}{4kt}]e^{-y}dy$
Open up the paranthesis for $\displaystyle (x-y)^2$ and $\displaystyle (x+y)^2$ then distribute over by $\displaystyle e^{-y}$. That should do it.