  In many recent applications when new materials and technologies are developed
it is important to describe and simulate new nonlinear and nonlocal diffusion
transport processes. A general class of such models deals with nonlocal
fractional power elliptic operators. In order to solve these problems
numerically it is proposed (Petr N. Vabishchevich, Journal of Computational
Physics. 2015, Vol. 282, No.1, pp.289--302) to consider equivalent local
nonstationary initial value pseudo-parabolic problems. Previously such problems
were solved by using the standard implicit backward and symmetrical Euler
methods. In this paper we use the one-parameter family of three-level finite
difference schemes for solving the initial value problem for the first order
nonstationary pseudo-parabolic problem. The fourth-order approximation scheme
is developed by selecting the optimal value of the weight parameter. The
results of the theoretical analysis are supplemented by results of extensive
computational experiments.
