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.

No abstract

No abstract

A positive definite integral quadratic form is said to be almost (primitively) universal if it (primitively) represents all but at most finitely many positive integers. In general, almost primitive universality is a stronger property than almost universality. The two main results of this paper are: 1) every primitively universal form nontrivially represents zero over every ring Zp of p-adic integers, and 2) every almost universal form in five or more variables is almost primitively universal.