there's something about these PDE:s that I dont understand, cant find out how it really works. Here comes a problem that we can discuss.(adsbygoogle = window.adsbygoogle || []).push({});

2 equal 0.2m think iron plates got the temperatures 100 and 0 degree C from the beginning. At the time t = 0 are these 2 plates laid next to eachother. Calculate the temperature where the plates touch after 10min (a_iron = 1.5*10^-5 m^2/s)

the hints in the book says:

du/dt - a d^2u/dx^2 = 0

u(0,t) = u(0.4,t) = 0

u(x,0) = 100 0<x<0.2

u(x,0) = 0 0.2<x<0.4

that sounds fair to me, but then they continue with, " Hopfully we reckognice the room-operator and know which eigenfunctions we have. If so we can directly try with a sinus serie and get the solution

u(x,t) = 200/pi sum( (1-cos(k*pi/2))/k * exp( -ak^2pi^2*t/0.16) * sin(k*pi*x/0.4)"

how do we get this?

I mean, with the Stum Liouville operator we write it as A = -d^2/dt^2 so we get

du/dt + aAu = 0

and A*fi_k = lamda_k * fi_k

where we assume that the solution looks like u(x,t)= sum(u_k(t)*fi_k(x))

and that u_k(t) = (fi_k|s)/(fi_k|fi_k) * exp(-a*lamda_k*t)

(|) is the scalar product and s is u(x,0).

when I calculate fi_k I get fi_k = sin(alfa_k*x) where alfa_k = k*pi/0.4

and lamda_k = alfa_k^2

but how do I get u_k(t). with the methos I know it depends on u(x,0). And here I got 2 values for that. I probably could write u(x,0) with heaviside, but I have no ide on how to take the scalar product then, so I choose not to :)

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: PDE problems

**Physics Forums | Science Articles, Homework Help, Discussion**