Question

(Differential equations)

I need to solve a PDE (Heat equation) where the initial condition is a constant, T_i. Can I treat the initial condition as a function f(x) in order to solve via separation of variables; and then replace it with T_i after i'm done solving?

Answer 1

might need to remember the diffyQ for heat :/

Answer 2

Sure!

\[\frac{\partial u}{\partial t}=\alpha \frac{\partial^2 u}{\partial x^2}\]

Answer 3

soo, to neaten it up

a u'' - y' = 0

Answer 4

... typoed it lol

a u'' - u' = 0

Answer 5

to do separation, you might want to redefine

z = u'

z' = u''

a dz/du- z = 0

a dz/du = z

dz/z = du/a

ln(z) = u/a + C

z = Ce^(u/a) = u'

Answer 6

Don't forget that u is a function of two variables, not one

Answer 7

.... doh!! those are partials ;/

Answer 8

hehe

Answer 9

http://tutorial.math.lamar.edu/Classes/DE/TheHeatEquation.aspx

then thisll be more congent than i can be :)

Answer 10

http://tutorial.math.lamar.edu/Classes/DE/SeparationofVariables.aspx

and this is the "later one" for the separation of variables solution to it

Answer 11

Looks like yes, one of his examples did exactly what I thought; treating the constant initial condition as a function of x, allowing separation of variables.

thanks.

Answer 12

:) good luck