The diffusion equation is:
$ \frac{\partial T}{\partial t} = \alpha \left( \frac{\partial^2 T}{\partial x^2} \right) $
An explicit finite difference approach can be used to solve this, forward in time and central differences in space. Approximating the diffusion equation at a node i, yields,
$\frac{T_i^n+1-T_i^n}{\Delta t} = \alpha\frac{T_{i+1}^n-2T_i^n+T_{i-1}^n}{\Delta x^2}$$\frac{T_i^{n+1}-T_i^n}{\Delta t} = \alpha\frac{T_{i+1}^n-2T_i^n+T_{i-1}^n}{\Delta x^2}$
which gives
$T_i^{n+1} = T_i^n(1-\omega)+\omega(0.5T_{i+1}^n+0.5T_{i-1}^n)$
where $\omega = \frac{2\alpha \Delta t}{\Delta x^2}$
The book states that for stability condition, the coefficients of the right-hand side terms must be positive which implies that
$\Delta t \lt \frac{\Delta x^2}{2\alpha}$
My question is why should the coefficients of the right-hand side terms be positive for stability?