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It is well known that in the case of integer order differentiation the formula $\partial_{x}f(x,u(x))=\partial_{u}f\cdot \partial_{x}u+\partial_{x}f\cdot u$ holds. If we define fractional derivative via Fourier transform F as $D_{x}^{\alpha}u(x)=F^{-1}[{(i\xi)^{\alpha}\widehat{u}(\xi)}]$, where ^ denotes the Fourier transform, is there a similarly formula for $D_{x}^{\alpha}f(x,u(x))$?

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The fractional derivative is of order 0<\alpha<1 –  Milos Sep 19 '13 at 8:06
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dx.doi.org/10.1137/0501026 –  Carlo Beenakker Sep 19 '13 at 20:44

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