Computational Physics Lectures: Partial differential equations

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Solution for the One-dimensional Diffusion Equation

The coefficient

$A_n$ is in turn determined from the initial condition. We require

$\begin{equation*} u(x,0)=g(x)=\sum_{n=1}^{\infty} A_n \sin(n\pi x/L). \end{equation*}$ The coefficient

$A_n$ is the Fourier coefficients for the function

$g(x)$ . Because of this,

$A_n$ is given by (from the theory on Fourier series)

$\begin{equation*} A_n=\frac{2}{L}\int_0^L g(x)\sin(n\pi x/L) \mathrm{d}x. \end{equation*}$ Different

$g(x)$ functions will obviously result in different results for

$A_n$ .