Mathematical Physics Vol 1

A.8 An Introduction to Fractional Vector Operators

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as a Fractional Laplacian. However commonly in the literature the fractional Laplacian is introduced by the following property h F ( − ∆ ) 1 2 f ( x ) i ( k )= | k | − α F f ( k ) , (A.95) where F is the Fourier transform, but it is well known that there are several operators verifying this condition. On the other hand, the relevance of vectorial calculus in the scientific fie1d is well known. We gather here some definitions of the extension of this operators to the fractional case. Let us consider a general vector F ( x )= F s ( x ) e s = F x e x + F y e y + F z e z , s ∈{ 1 , 2 , 3 } , (A.96) where e s are the orthogonal unit vectors. We will use D α s to denote any fractional differential operator with respect to the variable x s . Gradient of scalar function G Classical gradient = ∇ G grad G = ∂ G ∂ x e 1 + ∂ G ∂ y e 2 + ∂ G ∂ z e 3 Fbactional gradient. Definition 1 grad α f ( x )= e s D α s f ( x ) Fbactional gradient. Definition 2 grad α f ( x )= e s Γ ( 1 + α ) D α s f ( x ) Divergence of vector function F Classical divergence ∇ · F ( x ) div F ( x )= ∂ F x ∂ x + ∂ F y ∂ y + ∂ F z ∂ z Fbactional divergence. Definition 1 div α F ( x )= e s D α s F s ( x ) Fbactional divergence. Definition 2 div α F ( x )= e s Γ ( 1 + α ) D α s F s ( x ) Curl of vector function F Classical Curl curl F = e ℓ ε ℓ mn D m F n Fbactional Curl. Definition 1 curl α F = e ℓ ε ℓ mn D m F n Fbactional Curl. Definition 2 curl α F = e ℓ Γ ( α + 1 ) ε ℓ mn D m F n Fbactional Curl. Definition 3 curl α F = e ℓ Γ ( α + 1 ) ε ℓ mn D m I 1 − α n F n Cite this article Valério, D., Trujillo, J.J., Rivero, M. et al. Fractional calculus: A survey of useful formulas. Eur. Phys. J. Spec. Top. 222, 1827–1846 (2013). https://doi.org/10.1140/epjst/e2013 01967-y