Hey guys,
I am stuck on solving the following semi-infinite wave equations:
Ut = Uxx 0<x<infinity, 0<t
Ux (0,t) = 0
U(x,0) = f(x)
Ut(x,0) = 0
I have no idea how to get started. Can anybody get me started in the right direction, thanks.
Hey guys,
I am stuck on solving the following semi-infinite wave equations:
Ut = Uxx 0<x<infinity, 0<t
Ux (0,t) = 0
U(x,0) = f(x)
Ut(x,0) = 0
I have no idea how to get started. Can anybody get me started in the right direction, thanks.
What you give is a "diffusion equation", not a "wave equation". Wasn't the equation really $\displaystyle U_{tt}= U_{xx}$? How you would solve it depends on how much you already know.
The most basic method would be to "separate" the variables by looking for a solution of the form U(x,t)= X(x)T(t) and getting two ordinary differential equations for X and T separately. The general solution, for a finite interval, would be a sum of such things and, for an infinite interval, an integral.
Or you could "shortcut" that process by assuming the result- that the solution can be written as a "Fourier Transform" of the form $\displaystyle \frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty T(t)e^{-isx}ds$. Then $\displaystyle U_{tt}= \frac{1}{\sqrt{2\pi}\int_{-\infty}^\infty T' e^{-isx}ds$ and $\displaystyle U_{xx}= \frac{-ix}{\sqrt{2\pi}}\int_{-\infty}^\infty Te^{-isx}ds$ so that your equation is $\displaystyle \frac{1}{\sqrt{2\pi}\int_{-\infty}^\infty T' e^{-isx}ds= \frac{-is}{\sqrt{2\pi}}\int_{-\infty}^\infty Te^{-isx}ds$ or $\displaystyle \int_{-\infty}^\infty (T"- T)e^{-isx}ds= 0$. In order for that to be true you must have T"+ isT= 0 for all t. Solve that for T and then use the conditions to determine the constants involved.