I need to simulate the following system of equations in python: \begin{align*} &\frac{\partial S}{\partial t}=-I S-\beta A S+\rho_{d} I+\rho_{c} A-\left(\alpha_{d}+\alpha_{c}\right) S + d_s \frac{\partial ^2S}{\partial x^2}\\ &\frac{\partial I}{\partial t}=I S-\rho_{d} I+\alpha_{d} S-\sigma_{d c} I+\sigma_{c d} A +d_I\frac{\partial ^2 I}{\partial x^2}\\ &\frac{\partial A}{\partial t}=\beta A S-\rho_{c} A+\alpha_{c} S+\sigma_{d c} I-\sigma_{c d} A + d_A \frac{\partial ^2 A}{\partial x^2} \end{align*}
Where I can chose the parameter values.
I have seen that perhaps I could take the Fourier transform of the equations and try to create ODE's. However, I don't know how to Fourier transform the non-linear terms like $-\beta AS$?
To clarify, I would like to solve these numerically in Python:)
Thank you, Rik.