# Finite string Partial differential equation prob

• Feb 23rd 2009, 01:49 PM
r2dee6
Finite string Partial differential equation prob
Consider the following finite string problem

$\displaystyle \mu_{tt} = \mu_{xx}$ where $\displaystyle 0\le x\le 6$

$\displaystyle \mu(x,0) = f(x)$ and $\displaystyle \mu_t(6,t) = 0$ for $\displaystyle t\ge 6$ with $\displaystyle \mu(x,0) = f(x)$, and $\displaystyle \mu_t(x,0) = 0$

where
$\displaystyle f(x) = 1 - |x - 3|$, if $\displaystyle |x - 3|\le 1$ and is 0 otherwise.(sorry i dont know how to enter this as a array)

give the equation of the solution for t = 2
• Feb 23rd 2009, 03:30 PM
HallsofIvy
Quote:

Originally Posted by r2dee6
Consider the following finite string problem

$\displaystyle \mu_{tt} = \mu_{xx}$ where $\displaystyle 0\le x\le 6$

$\displaystyle \mu(x,0) = f(x)$ and $\displaystyle \mu_t(6,t) = 0$ for $\displaystyle t\ge 6$ with $\displaystyle \mu(x,0) = f(x)$, and $\displaystyle \mu_t(x,0) = 0$

where
$\displaystyle f(x) = 1 - |x - 3|$, if $\displaystyle |x - 3|\le 1$ and is 0 otherwise.(sorry i dont know how to enter this as a array)

give the equation of the solution for t = 2

What course is this for? It looks like a straightforward separble equation. If you let $\displaystyle \mu(x,t)= X(x)T(t)$ then the equation becomes XT"= X"T or T"/T= X"/X. Since the left hand side is a function of t only and the right hand side is a function of x only, each side must be a constant. Calling that constant $\displaystyle \lambda$, we have $\displaystyle X"/X= \lamba$ or $\displaystyle X"= \lambda X$ and $\displaystyle T"/T= \lamba$ or $\displaystyle T"= \lambda T$.

However, there is a problem with the additional conditionsl. Since this equation is second order in both x and t, you need two boundary and two initial conditions. You have only one of each.
• Feb 23rd 2009, 04:38 PM
r2dee6
Its for the partial differential equations course. Im not sure how to go about doing it as I am self teaching myself the course