I have an equation of the form $(1-0.1 \Delta)f=1$ with boundary condition $f=0$. I need to solve it for $f$ by pseudospectral method in python. Apparently, it should be $f=\frac{1}{1-0.1\Delta}$. I know that by fourier and inverse fourier transform i can get laplace operator but i am confused that the fourier tranform works in a way, that if you have a function $f$, you take its fourier tranform $f_{k}$ then multiply it with $-k^2$ which will give you $\Delta f_k$ and then using inverse fourier transform we get $\Delta f$. But i am supposed to find $f$ from the equation then if dont have $f$ initially how can i compute its fourier ransform and how can i get the inverse operator $\frac{1}{1-0.1 \Delta}$ using fourier tranform theory?


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