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

Meerschaert, Mark M.; Mortensen, Jeff; Scheffler, Hans-Peter

Fract. Calc. Appl. Anal. 7, No. 1, 61-81 (2004).

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.

Reviewer: John Pryce (Swindon)

MSC:

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

Keywords:

Grünwald formula; vector fractional derivative; finite differences; fractional partial differential equation; diffusion in porous media; Fourier transform; convergence