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### Do different methods of calculating fractional derivatives have to be equal?

Do different methods of calculating fractional derivatives have to be equal? Or do they sometimes end up differently? An example would be nice, and if possible, an explanation as too why such ...
I want to solve this PDE: $$u_t-6uu_x+u_{xxx} = 0\,(1)$$ with the Inverse Scattering Method. This method is based on showing that the above equation can be expressed as $$L_t=LB-BL,\,(2)$$ where $L$ ...