Abstract-We consider an approximation of one-dimensional fractional diffusion equation. We claim and show that the finite difference approximation obtained from the Grünwald-Letnikov formulation, often claimed to be of first order accuracy, is in fact a second order approximation of the fractional derivative at a point away from the grid points. We use this fact to device a second order accurate finite difference approximation for the fractional diffusion equation. The proposed method is also shown to be unconditionally stable. By this approach, we treat three cases of difference approximations in a unified setting. The results obtained are justified by numerical examples.Index Terms-Fractional derivative, diffusion equation, grunwald approximation, crank-nicolson method.

We propose a generalized theory to construct higher order Grünwald type approximations for fractional derivatives. We use this generalization to simplify the proofs of orders for existing approximation forms for the fractional derivative.  We also construct a set of higher order Grünwald type approximations for fractional derivatives in terms of a general real sequence and its generating function. From this, a second order approximation with shift is shown to be useful in approximating steady state problems and time dependent fractional diffusion problems. Stability and convergence for a Crank-Nicolson type scheme for this second order approximation are analyzed and are supported by numerical results.

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SUMMARYA new non-polar spherical co-ordinate system for the three-dimensional space is introduced. The co-ordinate system is composed of six local co-ordinate systems mapped from six faces of a cube on to the 2-sphere. Weakly orthogonal and orthogonal spherical harmonics are constructed in this coordinate system. The spherical harmonics are easily computable functions consisting of polynomials and square root of polynomials. Examples of finite Fourier series computations are given in terms of the new spherical harmonics to demonstrate their immediate applicability.