Abstract- In this paper, a parallel Crank-Nicolson finite difference method (C-N-FDM) for time-fractional parabolic equation on a distributed system using MPI is investigated. The fractional derivative is described in the Caputos sense. The resultant large system of equations is studied using preconditioned conjugate gradient method (PCG), with the implementation of cluster computing on it. The proposed approach fulfills the suitability for the implementation on Linux PC cluster through the minimization of inter-process communication. To examine the efficiency and accuracy of the proposed method, numerical test experiment using different number of nodes of the Linux PC cluster is studied. The performance metrics clearly show the benefit of using the proposed approach on the Linux PC cluster in terms of execution time reduction and speedup with respect to the sequential running in a single PC.

Recently, many numerical techniques were presented to solve the fractional anomalous sub‐diffusion equations, and the results were excellent. In this paper, we study a simple numerical technique to solve two important types of fractional anomalous sub‐diffusion equations that appear strongly in chemical reactions and spiny neuronal dendrites, which are the two‐dimensional fractional Cable equation and the two‐dimensional fractional reaction sub‐diffusion equation. The proposed technique is a simple one which is an extension of the weighted average finite difference technique. The stability analysis of the proposed method is studied by means of John von Neumann stability analysis technique. An accurate stability criterion which is valid for different discretization schemes of the fractional derivative and arbitrary weight factor is introduced. Four numerical examples are presented (two for the Cable equation and two for the reaction sub‐diffusion equation) to demonstrate the effectiveness and the accuracy of the presented method.

An analysis of the molecular dynamics of ethanol solvated by water molecules in the absence and presence of a Pt surface has been performed using DL_POLY_2.19 code. The structure and diffusion properties of an ethanol–water system have been studied at various temperatures from 250 to 600 K. We have measured the self-diffusion coefficients of the 50:50% ethanol–water solution; in the absence of a Pt surface our results show an excellent agreement–within an error of 7.4% – with the experimental data. An increase in the self-diffusion coefficients with the inclusion of a Pt surface has been observed. The estimation of the diffusion coefficients of both water and ethanol in the presence of a Pt surface shows that they obey the Arrhenius equation; the calculated activation energies of diffusion of ethanol and water are 2.47 and 2.98 Kcal/mole, respectively. The radial distribution function graphs and density profiles have been built; their correlations with the self-diffusion coefficients of both ethanol and water molecules are also illustrated

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