Parallel Implementation of Modeling of Fractional-Order State-Space Systems Using the Fixed-Step Euler Method

Stanisławski, Rafał; Kozioł, Kamil

doi:10.3390/e21100931

Entropy

2019

DOI: 10.3390/e21100931

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Parallel Implementation of Modeling of Fractional-Order State-Space Systems Using the Fixed-Step Euler Method

Rafał Stanisławski

Kamil Kozioł

Abstract: This paper presents new results in implementation of parallel computing in modeling of fractional-order state-space systems. The methods considered in the paper are based on the Euler fixed-step discretization scheme and the Grünwald-Letnikov definition of the fractional-order derivative. Two different parallelization approaches for modeling of fractional-order state-space systems are proposed, which are implemented both in Central Processing Unit (CPU)- and Graphical Processing Unit (GPU)-based hardware envir… Show more

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Cited by 3 publications

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References 30 publications

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“…Two different parallelization approaches are proposed. Simulation results illustrate the high efficiency of the introduced schemes [ 16 ].…”

mentioning

confidence: 99%

The Fractional View of Complexity

Lopes

Machado

2019

Entropy

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Fractal analysis and fractional differential equations have been proven as useful tools for describing the dynamics of complex phenomena characterized by long memory and spatial heterogeneity. There is a general agreement about the relation between the two perspectives, but the formal mathematical arguments supporting their relation are still being developed.The fractional derivative of real order appears as the degree of structural heterogeneity between homogeneous and inhomogeneous domains. A purely real derivative order would imply a system with no characteristic scale, where a given property would hold regardless of the scale of the observations. However, in real-world systems, physical cut-offs may prevent the invariance spreading over all scales and, therefore, complex-order derivatives could yield more realistic models.Information theory addresses the quantification and communication of information. Entropy and complexity are concepts that often emerge in relation to systems composed of many elements that interact with each other, which appear intrinsically difficult to model. This Special Issue focuses on the synergies of fractals or fractional calculus and information theory tools, such as entropy, when modeling complex phenomena in engineering, physics, life, and social sciences. It includes 16 manuscripts addressing novel issues and specific topics that illustrate the role of entropy-based techniques in fractality, fractionality, and complexity. In the follow-up the selected manuscripts are presented in alphabetic order.The manuscript "A Fractional-Order Partially Non-Linear Model of a Laboratory Prototype of Hydraulic Canal System", by Saddam Gharab, Vicente Feliu-Batlle and Raul Rivas-Perez, addresses the identification of the nonlinear dynamics of the main pool of a laboratory hydraulic canal installed in the University of Castilla La Mancha. A new dynamic model is developed by taking into account the measurement errors caused by the different parts of the experimental setup. Fractional and integer order plus time delay models are used to approximate the responses of the main pool of the canal in its different flow regimes [1].In the paper "Adaptive Synchronization Strategy between Two Autonomous Dissipative Chaotic Systems Using Fractional-Order Mittag-Leffler Stability", Licai Liu, Chuanhong Du, Xiefu Zhang, Jian Li and Shuaishuai Shi investigate two novel four-dimensional, continuous, fractional-order, autonomous, and dissipative chaotic system models with high complexity. Based on the fractional Mittag-Leffler stability theory, an adaptive, large-scale, and asymptotic synchronization control method is derived [2].In "An Entropy Formulation Based on the Generalized Liouville Fractional Derivative", Rui A. C. Ferreira and J. Tenreiro Machado present a new entropy formula based in the Liouville fractional derivative. The new concept is illustrated when applied to the Dow Jones Industrial Average time series. The Jensen-Shannon divergence is also generalized and its variation with the fractional ord...

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“…Two different parallelization approaches are proposed. Simulation results illustrate the high efficiency of the introduced schemes [ 16 ].…”

mentioning

confidence: 99%

The Fractional View of Complexity

Lopes

Machado

2019

Entropy

View full text Add to dashboard Cite

show abstract

Solution of nonlinear fractional-order models of nuclear reactor with parallel computing: Implementation on GPU platform

Keluskar,

Singhaniya,

Vyawahare

et al. 2024

Annals of Nuclear Energy

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Effects of OpenCL-Based Parallelization Methods on Explicit Numerical Methods to Solve the Heat Equation

Koics,

Kovács,

Hornyák

2024

Computers

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In recent years, the need for high-performance computing solutions has increased due to the growing complexity of computational tasks. The use of parallel processing techniques has become essential to address this demand. In this study, an Open Computing Language (OpenCL)-based parallelization algorithm is implemented for the Constant Neighbors (CNe) and CNe with Predictor–Corrector (CpC) numerical methods, which are recently developed explicit and stable numerical algorithms to solve the heat conduction equation. The CPU time and error rate performance of these two methods are compared with the sequential implementation and Euler’s explicit method. The results demonstrate that the parallel version’s CPU time remains nearly constant under the examined circumstances, regardless of the number of spatial mesh points. This leads to a remarkable speed advantage over the sequential version for larger data point counts. Furthermore, the impact of the number of timesteps on the crossover point where the parallel version becomes faster than the sequential one is investigated.

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Parallel Implementation of Modeling of Fractional-Order State-Space Systems Using the Fixed-Step Euler Method

Cited by 3 publications

References 30 publications

The Fractional View of Complexity

The Fractional View of Complexity

Solution of nonlinear fractional-order models of nuclear reactor with parallel computing: Implementation on GPU platform

Effects of OpenCL-Based Parallelization Methods on Explicit Numerical Methods to Solve the Heat Equation

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