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The study of generalising the central difference for integer order Laplacianto fractional order is discussed in this paper. Analysis shows that, incontrary to the conclusion of a previous study, difference stencils evaluatedthrough fast Fourier transform prevents the convergence of the solution offractional Laplacian. We propose a composite quadrature rule in order toefficiently evaluate the stencil coefficients with the required convergencerate in order to guarantee convergence of the solution. Furthermore, we proposethe use of generalised higher order lattice Boltzmann method to generatestencils which can approximate fractional Laplacian with higher orderconvergence speed and error isotropy. We also review the formulation of thelattice Boltzmann method and discuss the explicit sparse solution formulatedusing Smolyak's algorithm, as well as the method for the evaluation of theHermite polynomials for efficient generation of the higher order stencils.Numerical experiments are carried out to verify the error analysis andformulations.

On Generalisation of Isotropic Central Difference for Higher Order Approximation of Fractional Laplacian