# Solution to a PDE (heat equation) with one initial condition

1. Sep 24, 2012

### frenchkiki

1. The problem statement, all variables and given/known data

By trial and error, find a solution of the diffusion equation du/dt = d^2u / dx^2 with
the initial condition u(x, 0) = x^2.

2. Relevant equations

3. The attempt at a solution

Given the initial condition, I tried finding a solution at the steady state (du/dt=0) which gives me d^2u / dx^2 = 0, from I found u(x,0)=c_1 + c_2 x = x^2 and now I am stuck. Thanks in advance!

2. Sep 30, 2012

### Dustinsfl

$u_t = u_{xx}$ with $u(x,0) = x^2$.

The basic solution is of the form $u(x,t) = \varphi(x)\psi(t)$.
You would use the method of separation of variables.
$$\begin{cases} \varphi = A\cos\lambda x + B\frac{\sin\lambda x}{\lambda}\\ \psi = C\exp(-\lambda^2t) \end{cases}$$
The above is basic and nothing ground breaking.
Right now your solution is of the form
$$a_0 + \sum_{n = 1}^{\infty}\left[\left(A_n\cos\lambda_n x + B_n\frac{\sin\lambda_n x}{\lambda}\right)\exp(-\lambda^2t)\right]$$
When you have non-homogeneous boundary conditions, you would solve the steady state solution. What are your BC?
I would then use the orthogonality and Fourier series to finish the problem solving for the Fourier coefficients.