Math wiz! Partial DE help

[alphatarget1]

so I'm given a heat equation:

ut=uxx

with the BC
u(0,t)=0
u(1,t)=cos(t)
and IC
u(x,0)=x

I can't do the substitution because the 2nd boundary condition is a function of t. I just need some pointers to get started 🙂

 

[Unitary]

It may be a typo, but ut=uxx is the heat equation. Did you mean utt=uxx?

 

[alphatarget1]

Originally posted by: Unitary
It may be a typo, but ut=uxx is the heat equation. Did you mean utt=uxx?

Yeah sorry, he actually just gave us a PDE, I think you're right.

 

[Unitary]

I'm certainly no expert at solving problems like this, but here are my first thoughts:

I would try to use separation of variables, i.e. assume u(x,t)=u(x)v(t). Plugging this into the heat equation, you will get two ODE's, one first order and one second order. I would try to solve these and, hopefully, get a family of solutions. I would then (again hopefully) use these solutions to piece together, by some infinite series, a solution satisfying my initial data u(x,0)=x.

Good luck!