Vector Grünwald formula for fractional derivatives. (English) Zbl 1084.65024

Fractional ordinary and partial differential equations, involving fractional derivatives, are used to model evolution phenomena such as diffusion in porous media. The authors consider a general multivariable fractional derivative defined in terms of the Fourier transform and show that for a certain function class it can be expressed as a mixture of directional derivatives. They derive a generalization of the Grünwald formula, which itself generalizes the classical one-sided difference approximation to derivatives, and prove convergence results as meshsize \(h\to0\). The proofs are not straightforward, and are well explained. However, though the formulae are intended to assist a numerical solution, they are only first-order accurate (error \(O(h)\)). It would be valuable to derive higher-order formulae.


65D25 Numerical differentiation
26B12 Calculus of vector functions
35Q35 PDEs in connection with fluid mechanics
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
26A33 Fractional derivatives and integrals
76S05 Flows in porous media; filtration; seepage
74M20 Impact in solid mechanics