Positive Approximations of the Inverse of Fractional Powers of SPD M-Matrices

Abstract

This study is motivated by the recent development in the fractional calculus and its applications. During last few years, several different techniques are proposed to localize the nonlocal fractional diffusion operator. They are based on transformation of the original problem to a local elliptic or pseudoparabolic problem, or to an integral representation of the solution, thus increasing the dimension of the computational domain. More recently, an alternative approach aimed at reducing the computational complexity was developed. The linear algebraic system Aαu= f, 0 < α < 1 is considered, where A is a properly normalized (scalded) symmetric and positive definite matrix obtained from finite element or finite difference approximation of second order elliptic problems in Ω⊂ ℝd, d = 1, 2, 3. The method is based on best uniform rational approximations (BURA) of the function tβ−α for 0