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Thanks for your useful post. However, could you please enlighten some parts of your post? 1. In fact, I think the "Equation 2: Time-varying non-linear PDEs: Fick's first law of diffusion" is actually the "linear" Heat PDE and it is not...
Let that we want to numerically solve the following PDE \begin{equation}\label{sde} -r V(S,t)+r S \frac{\partial V(S,t)}{\partial S}+0.5 S^2 \text{sigma}^2 \frac{\partial ^2V(S,t)}{\partial S^2}+\frac{\partial V(S,t)}{\partial t}=0, \end{equation}...