I am trying to solve a convection-diffusion problem for multi-stacked layer system. I obtained a system of second-order PDEs with variable coefficients. The independent variables are radius (r, {0 - infinity}) and time (t, {0 - infinity}).
I have used Laplace transformation to obtain a system of ODEs in the time - Laplace domain so that the independent variables are (s, r). I wonder if Fourier transformation can be used to simplify the problem in space - Fourier domain, if the domain of (r) is {0 - infinity}. The ODEs in Laplace domain are as follow.
$$y_1''(r) + \frac{C_1}{r}y_1'(r)= C_2 y_1(r) + C_3 y_2(r)$$ $$y_2''(r) + \frac{C_4}{r}y_2'(r)= C_5 y_2(r) + C_6 y_1(r)$$
Where $r$ is the radius and the coefficients $C_1 - C_6$ include the time variable $s$.