Simple difference schemes for multidimensional fractional Laplacian and fractional gradient

Abstract The fractional Laplacian and fractional gradient are operators which play fundamental role in modeling of anomalous diffusion in d -dimensional space with the fractional exponent $$\alpha \in (1,2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . The principal-value integrals are split into singular and regular parts where we avoid using any weight function for the approximation in the singularity neighborhood. The resulting approximation coefficients are calculated from optimal value of the singular domain radius which is only a function of the exponent $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> and a given grid topology. Various difference schemes are presented for the regular rectangular grids with mesh size $$h&gt;0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>h</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , and also for the hexagonal and the dodecahedral ones. This technique enables to evaluate the fractional operators with the approximation error $$\textrm{O}(h^{4-\alpha })$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>O</mml:mtext> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>h</mml:mi> <mml:mrow> <mml:mn>4</mml:mn> <mml:mo>-</mml:mo> <mml:mi>α</mml:mi> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> which is verified using testing functions with known analytical expression of their fractional Laplacian and fractional gradient. Resulting formulas can be also used for the numeric solution of the fractional partial differential equations.