The Best Uniform Rational Approximation: Applications to Solving Equations Involving Fractional powers of Elliptic Operators

In this paper we consider one particular mathematical problem of this large area of fractional powers of self-adjoined elliptic operators, defined either by Dunford-Taylor-like integrals or by the representation through the spectrum of the elliptic operator. Due to the mathematical modeling of various non-local phenomena using such operators recently a number of numerical methods for solving equations involving operators of fractional order were introduced, studied, and tested. Here we consider the discrete counterpart of such problems obtained from finite difference or finite element approximations of the corresponding elliptic problems. In this report we provide all necessary information regarding the best uniform rational approximation (BURA) $r{k,\alpha}(t) := Pk(t)/Qk(t)$ of $t^{\alpha}$ on $[\delta, 1]$ for various $\alpha$, $\delta$, and $k$. The results are presented in 160 tables containing the coefficients of $Pk(t)$ and $Qk(t)$, the zeros and the poles of $r{k,\alpha}(t)$, the extremal point of the error $t^\alpha - r{k,\alpha}(t)$, the representation of $r{k,\alpha}(t)$ in terms of partial fractions, etc. Moreover, we provide links to the files with the data that characterize $r_{k,\alpha}(t)$ which are available with enough significant digits so one can use them in his/her own computations.