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Implementing convective boundary conditions for 1-dimensional finite difference computation of heat equation

Let us assume that you divide the space into $n$ equal partitions, such that $\Delta x=\frac{L}{n}$. There are $n+1$ nodes, denoted by $x_i$, where $i=(0,1,....n)$. The target is to find $u_i=u(x_i)$ ...
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