I need to solve following partial differential equation with Fourier transform numerically.
$ \frac{\partial T}{\partial t} = \nabla(c\nabla T) $
where T is temperature, c heat conductivity and t is time.
Now the problem is c itself has space dependence. Had it not been after Fourier transform equation would look like $ \frac{\partial \tilde T}{\partial t} = -k^2c\tilde T $
How should Fourier transform of first equation look like?
What I am doing is as follows:
Take Fourier transform of T. Multiply corresponding values of c(in real space) and T (in Fourier space). i.e. evaluate $g = k\cdot i \cdot c \cdot\tilde T$
Take $g$ back to real space. Now $g = c\nabla T$
Take $g$ back to Fourier space . Evaulate $f = k \cdot i \cdot \tilde g$
Take $f$ to real space. Now $f$ should be $\nabla c \nabla T$
But results of the above procedure are not matching with Finite Difference approach. What am I missing here? Using convolution theorem seems difficult. Is using convolution theorem the only option?
Thanks for any help in advance