Parallelization of a numerical algorithm for solving the Сauchy problem for a nonlinear differential equation of fractional variable order using OpenMP technology

D. A. Tvyordiy   ; R. I. Parovik   ; A. R. Hayotov   ; A. K. Boltaev

Vestnik KRAUNC. Fiziko-matematičeskie nauki, Tome 43 (2023) no. 2, pp. 87-110

Résumé

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.

Détail

Keywords: fractional derivatives, heredity, memory effect, finite difference schemes, parallel computing, OpenMP.

Bibliographie

[1] Acioli P. S., Xavier F. A., Moreira D. M., “Mathematical Model Using Fractional Derivatives Applied to the Dispersion of Pollutants in the Planetary Boundary Layer”, Boundary-Layer Meteorology, 170:2 (2019), 285–304 DOI: 10.1007/s10546-018-0403-1 | DOI

[2] Aslam M., Farman M., Ahmad H., Gia T. N., Ahmad A., Askar S., “Fractal fractional derivative on chemistry kinetics hires problem”, AIMS Mathematics, 7:1 (2021), 1155–1184 DOI: 10.3934/math.2022068 | DOI | MR

[3] Jamil B., Anwar M. S., Rasheed A., Irfan M., “MHD Maxwell flow modeled by fractional derivatives with chemical reaction and thermal radiation”, Chinese Journal of Physics, 67 (2020), 512–533 DOI: 10.1016/j.cjph.2020.08.012 | DOI | MR

[4] Fellah M., Fellah Z. E. A., Mitri F., Ogam E., Depollier C., “Transient ultrasound propagation in porous media using Biot theory and fractional calculus: Application to human cancellous bone”, The Journal of the Acoustical Society of America, 133:4 (2013), 1867–1881 DOI: 10.1121/1.47927213 | DOI

[5] Chen W., “An Intuitive Study of Fractional Derivative Modeling and Fractional Quantum in Soft Matter”, Journal of Vibration and Control, 14:9–10 (2008), 1651–1657 DOI: 10.1177/1077546307087398 | DOI | Zbl

[6] Garrappa R., “Numerical Solution of Fractional Differential Equations: A Survey and a Software Tutorial”, Mathematics, 6:2:16 (2018), 1–23 DOI: 10.3390/math6020016 | DOI

[7] Bohaienko V., “Parallel algorithms for modelling two-dimensional non-equilibrium salt transfer processes on the base of fractional derivative model”, Fractional Calculus and Applied Analysis, 21:3 (2018), 654–671 DOI: 10.1515/fca-2018-0035 | DOI | MR | Zbl

[8] Bogaenko V. A., Bulavatskiy V. M., Kryvonos I. G., “On Mathematical modeling of Fractional-Differential Dynamics of Flushing Process for Saline Soils with Parallel Algorithms Usage”, Journal of Automation and Information Sciences, 48:10 (2016), 1–12 DOI: 10.1615/JAutomatInfScien.v48.i10.10 | DOI

[9] Gerasimov A. N., “Generalization of linear deformation laws and their application to internal friction problems”, Applied Mathematics and Mechanics, 12 (1948), 529–539 | MR

[10] Caputo M., “Linear models of dissipation whose Q is almost frequency independent – II”, Geophysical Journal International, 13:5 (1967), 529–539 DOI: 10.1111/j.1365-246X.1967.tb02303.x | DOI

[11] Sanders J., Kandrot E., CUDA by Example: An Introduction to General-Purpose GPU Programming, Addison-Wesley Professional, London, 2010, 311 pp.

[12] Bogaenko V. O., “Parallel finite-difference algorithms for three-dimensional space-fractional diffusion equation with phi–Caputo derivatives”, Computational and Applied Mathematics, 39:163 (2020), 1–20 DOI: 10.1007/s40314-020-01191-x

[13] Machado J. T., Kiryakova V., Mainardi F., “Recent history of fractional calculus”, Communications in nonlinear science and numerical simulation, 16:3 (2011), 1140–1153 DOI: 10.1016/j.cnsns.2010.05.027 | DOI | MR | Zbl

[14] Patnaik S., Hollkamp J. P., Semperlotti F., “Applications of variable-order fractional operators: a review”, Proceedings of the Royal Society A, 476:2234 (2020), 20190498 DOI: 10.1098/rspa.2019.0498 | DOI | MR

[15] Ortigueira M. D., Valerio D., Machado J. T., “Variable order fractional systems”, Communications in Nonlinear Science and Numerical Simulation, 71 (2019), 231–243 DOI: 10.1016/j.cnsns.2018.12.003 | DOI | MR | Zbl

[16] Tvoyrdyj D. A., “The Cauchy problem for the Riccati equation with variable power memory and non-constant coeffcients”, Vestnik KRAUNC. Fiz.-mat. nauki, 23:3 (2018), 148–157 (In Russian) DOI: 10.18454/2079-6641-2018-23-3-148-157

[17] Borzunov S. V., Kurgalin S. D., Flegel A. V., Workshop on parallel programming: tutorial, BHV, St. Petersburg, 2017, 236 pp. (In Russian)

[18] Kilbas A. A., Srivastava H. M., Trujillo J. J., Theory and Applications of Fractional Differential Equations, Elsevier Science Limited, Amsterdam, 2006, 523 pp. | MR | Zbl

[19] Tverdyi D. A., Parovik R. I., “Investigation of Finite-Difference Schemes for the Numerical Solution of a Fractional Nonlinear Equation”, Fractal and Fractional, 6(1):23 (2022), 1–27 DOI: 10.3390/fractalfract6010023

[20] Tvyordyj D. A., “Hereditary Riccati equation with fractional derivative of variable order”, Journal of Mathematical Sciences, 253:4 (2021), 564–572 DOI: 10.1007/s10958-021-05254-0 | DOI | Zbl

[21] Tverdyi D. A., Parovik R. I., “Application of the Fractional Riccati Equation for Mathematical Modeling of Dynamic Processes with Saturation and Memory Effect”, Fractal and Fractional, 6(3):163 (2022), 1–35 DOI: 10.3390/fractalfract6030163

[22] Daintith J., Wright E., A Dictionary of Computing, Oxford University Press, Oxford , 2008, 583 pp. DOI: 10.1093/acref/9780199234004.001.0001 | Zbl

[23] Miller R., Boxer L., Algorithms Sequential and Parallel: A Unified Approach. 3rd edition, Cengage Learning, Boston, 2013, 417 pp.

[24] Rauber T., Runger G., Parallel Programming for Multicore and Cluster Systems. 2nd edition, Springer, New York, 2013, 516 pp. | MR

[25] Al-hayanni M. A. N., Xia F., Rafiev A., Romanovsky A., Shafik R., Yakovlev A., “Amdahl's Law in the Context of Heterogeneous Many-core Systems - ASurvey”, IET Computers Digital Techniques, 14:4 (2020), 133–148 DOI: 10.1049/iet-cdt.2018.5220 | DOI

[26] Okrepilov V. V., Makarov V. L., Bakhtizin A. R., Kuzmina S. N., “Application of Supercomputer Technologies for Simulation of Socio-Economic Systems”, R-Economy, 1:2 (2015), 340–350 DOI: 10.15826/recon.2015.2.016 | DOI

[27] Il’in V. P., Skopin I. N., “About performance and intellectuality of supercomputer modeling”, Programming and Computer Software, 42:1 (2016), 5–16 DOI: 10.1134/S0361768816010047 | DOI

[28] Parovik R. I., “On the numerical solution of equations fractal oscillator with variable order fractional of time”, Vestnik KRAUNC. Fiz.-Mat. Nauki, 8:1 (2014), 60–65 (In Russian) DOI: 10.18454/2079-6641-2014-8-1-60-65 | Zbl

[29] Parovik R. I., “Mathematical models of oscillators with memory”, Oscillators-Recent Developments, 2019, 3–21 DOI: 10.5772/intechopen.81858

[30] Volterra V., Theory of functionals and of integral and integro-differential equations, Dover publications, New York, 1959, 226 pp. | MR | Zbl

[31] Uchaikin V. V., Fractional Derivatives for Physicists and Engineers, Background and Theory, v. I., Springer, Berlin, 2013, 373 pp. | MR | Zbl

[32] Tverdyi D. A., Makarov E. O., Parovik R. I., “Hereditary Mathematical Model of the Dynamics of Radon Accumulation in the Accumulation Chamber”, Mathematics, 11:4:850 (2023), 1–20 DOI: 10.3390/math11040850 | DOI

[33] Jeng S., Kilicman A., “Fractional Riccati Equation and Its Applications to Rough Heston Model Using Numerical Methods”, Symmetry, 12 (2020) DOI: 10.3390/sym12060959 | Zbl

[34] Sun H., et al., “Finite difference schemes for variable-order time fractional diffusion equation”, International Journal of Bifurcation and Chaos, 22:04 (2012), 1250085 DOI: 10.1142/S021812741250085X | DOI | MR

[35] Parovik R. I., “On a finite-difference scheme for an hereditary oscillatory equation”, Journal of Mathematical Sciences, 253:4 (2021), 547–557 DOI: 10.1007/s10958-021-05252-2 | DOI | Zbl

[36] Kalitkin N. N., Numerical methods. 2nd ed., BHV, St. Petersburg, 2011, 592 pp. (In Russian) | MR

[37] Brent R. P., “The parallel evaluation of general arithmetic expressions”, Journal of the Association for Computing Machinery, 21:2 (1974), 201–206 DOI: 10.1145/321812.321815 | DOI | MR | Zbl

[38] Corman T. H., Leiserson C. E., Rivet R. L., Stein C., Introduction to Algorithms, 3rd Edition, The MIT Press, Cambridge, 2009, 1292 pp. | MR

[39] Shao J., Mathematical Statistics. 2-ed, Springer, New York, 2003, 592 pp. | MR

[40] Gergel V. P., Vysokoproizvoditel'nyye vychisleniya dlya mnogoyadernykh mnogoprotsessornykh sistem [High performance computing for multi-core multiprocessor systems], Study guide, Moscow State University Publishing House, Moscow, 2010, 544 pp. (In Russian)