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## Homework Statement

Solve:

[tex] \frac{\partial u}{\partial t} = k\frac{\partial^2 u}{\partial x^2}, 0<x<\pi, t>0 [/tex]

with initial condition

[tex] u(x,0)=f(x)=\left\{\begin{array}{cc} 1,& 0\leq x< \pi/2 \\ 0, &\pi/2 \leq x < \pi \end{array}\right [/tex]

and with non-homogeneous boundary conditions

[tex] u(0,t) = 1, u(\pi,t)=0[/tex]

## The Attempt at a Solution

I've tried this one a couple different ways, I tried separation of variables, and fourier series. I can get a solution (or infinitely many) to the equation, but I can't seem to make them fit the boundary/initial conditions both.

My solution for the fourier series method is:

[tex]u(x,0)=\frac{A_0}{2} + \sum^{\infty}_{n=1} e^{-kn^2t}(A_n \cos(nx) + B_n \sin(nx) [/tex]

Any ideas how to make this (or any other solution) match both the boundary and initial conditions? Or is the problem incosistent?