Solving a PDE involving Hilbert transform
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While solving one research paper published in Physical Review Letters, I came across the following equation and I am unable to solve it.
∂f/∂t−(H(f)(∂f/∂x)=0
where [H(f)] is hilbert transform
and f=f(x,t) and initial condition is f(x,0)=cos(x)
In the paper it is given that the solution of the above mentioned equation is obtained with periodic boundary conditions using pseudospectral method given below,
F{H{f(x′)}}=i⋅sgn(k)F{f(x)}
where
F{f(x)} is Fourier transform of f(x)
sgn(k) is signum function of k
So I am thinking of application of Fourier transforms on both sides of the equation but I am unable proceed forward. Please solve the equation and can give the code for the same in Matlab
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