Parallelization of a Numerical Algorithm for Solving the Cauchy Problem for a Nonlinear Differential Equation of Fractional Variable Order Using OpenMP Technology

Abstract

The article presents a software implementation of a parallel efficient and fast computational algorithm for solving the Cauchy problem for a nonlinear differential equation of a fractional variable order. The computational algorithm is based on a non-local explicit finite-difference scheme, taking into account the approximation of the Gerasimov-Caputo fractional derivative VO included in the main differential equation. The algorithms for parallelization of the non-local explicit finite difference scheme were implemented as functions of the user library of the C programming language using the OpenMP technology. The OpenMP technology allows implementing parallel algorithms for working with the CPU computing node using its multithreading. The C language was chosen because of its versatility and lack of strict restrictions on memory handling. Further in the paper, the efficiency of the parallel algorithm is investigated. Efficiency is understood as the optimal ratio in coordinates: acceleration of calculations – the amount of RAM memory occupied, in comparison with the sequential version of the algorithm. The average computation time is analyzed in terms of: running time, acceleration, efficiency and cost of the algorithm. These algorithms were run on two different computing systems: a gaming laptop and a computing server. For a non-local explicit scheme, a significant performance increase of 3-5 times is shown for various methods of software implementation.

Keywords

• fractional derivatives
• heredity
• memory effect
• finite difference schemes
• parallel computing
• openmp
• дробные производные
• эредитарность
• эффект памяти
• явные конечно-разностные схемы
• параллельные вычисления
• openmp