1. ## Heat equation??

Hello,
I don't know that here is right forum for my question.
I want to solve this Heat Equation (Solution by Fourier series)
du/dt=d(2)u/dx^2 + e^(-t) 0<x<pi
du(0,t)/dx=du(pi,t)/dx=0
u(x,0)=cos(x)

I want to solve this differential equation.
Just I know I should use
u(x,t)=v(x,t)+w(x,t)
But how to find w(x,t) ? and more????

2. Why do you want to do it that way? Since you said "Solution by Fourier series" why not write it out as a Fourier series?
u(x,t)= \sum_{n=0}^\infty A_n(t) cos(nx)
(The boundary conditions set the derivative to 0 at 0 and \pi and sines do not satisfy that so I know the Fourier series can be written in terms of cosine only. Essentially, this is the same as assuming u is symmetric about the origin.)

u_{t}= \sum_{n=0}^\infty A_n' cos(nx)
u_xx= \sum_{n=0}^\infty -n^2A_n cos(nx).

Expand e^{-t} (a constant with respect to x) in a cosine series and the equation becomes an system of equations for A_n(t).

3. I can not understand what you write.
Yes Solution by Fourier series.
but before sulotion I need to find way to u(x,t)= \sum_{n=0}^\infty A_n(t) cos(nx)??? and I need to write u(x,t)=v(x,t)+w(x,t)