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Analytical approach calculating eigenfrequencies and modes of stator cores in AC machines

Paweł Witczak

Archives of Electrical Engineering | 2012 | vol. 61 | No 1 March | 47-56 | DOI: 10.2478/v10171-012-0004-4

Keywords natural vibration AC machines differential equations

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Abstract

The paper describes analytical approach solving the problem of dynamic analysis of two-dimensional fields of vibrational displacements and rotations caused by magnetic forces acting on stator of AC machine. Final set of three differential equations converted into algebraic ones is given and it is confronted with numerical solutions obtained by finite element method.

Authors and Affiliations

Paweł Witczak

Optimization of periodic maintenance for wind turbines based on stochastic degradation model

Hongsheng Su Xuping Duan Dantong Wang

Archives of Electrical Engineering | 2021 | vol. 70 | No 3 | 585-599 | DOI: 10.24425/aee.2021.137575

Keywords availability maintenance optimization stochastic differential equation update process wind turbine

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Abstract

The degradation process of wind turbines is greatly affected by external factors. Wind turbine maintenance costs are high. The regular maintenance of wind turbines can easily lead to over and insufficient maintenance. To solve the above problems, a stochastic degradation model (SDE, stochastic differential equation) is proposed to simulate the change of the state of the wind turbine. First, the average degradation trend is obtained by analyzing the properties of the stochastic degradation model. Then the average degradation model is used to describe the predictive degradation model. Then analyze the change trend between the actual degradation state and the predicted state of the wind turbine. Secondly, according to the update process theory, the effect of maintenance on the state of wind turbines is comprehensively analyzed to obtain the availability. Then based on the average degradation process, the optimal maintenance period of the wind turbine is obtained. The optimal maintenance time of wind turbines is obtained by optimizing the maintenance cycle through availability constraints. Finally, an onshore wind turbine is used as an example to verification. Based on the historical fault data of wind turbines, the optimized maintenance decision is obtained by analyzing the reliability and maintenance cost of wind turbines under periodic and non-equal cycle conditions. The research results show that maintenance based on this model can effectively improve the performance of wind turbines and reduce maintenance costs.

Bibliography

[1] Tchakoua P., Wamkeue R., Ouhrouche M. et al., Wind turbine condition monitoring: state-of-the-art review, new trends, and future challenges, Energies, vol. 7, no. 4, pp. 2595–2630 (2014).
[2] Su C., Hu Z.Y., Reliability assessment for Chinese domestic wind turbines based on data mining techniques, Wind Energy, vol. 21, no. 3, pp. 198–209 (2018).
[3] Zhao Hongshan, Zhang Jianping, Gao Duo et al., A condition based opportunistic maintenance strategy for wind turbine, Proceedings of the CSEE, vol. 35, no. 15, pp. 3851–3858 (2015).
[4] ChengYujing, Optimization maintenance research of wind turbines pitch system based on opportunistic maintenance strategy, Shanghai, Shang Hai Dianji University (2013).
[5] Li Hui, Yang Chao, Li Xuewei et al., Conditions characteristic parameters mining and outlier identification for electric pitch system of wind turbine, Proceedings of the CSEE, vol. 34, no. 12, pp. 1922–1930 (2014).
[6] Besnard F., Bertling L., An approach for condition-based maintenance optimization applied to wind turbine blades, IEEE Transactions on Sustainable Energy, vol. 1, no. 2, pp. 77–83 (2010).
[7] Liu Lujie, FuYang,Ma Shiwei et al., Maintenance strategy for offshore wind turbine based on condition monitoring and prediction, Power System Technology, vol. 39, no. 11, pp. 3292–3297 (2015).
[8] Suprasad V., Amari Leland Mclaughlin, Hoang Pham, Cost-effective condition-based maintenance using Markov decision processes, Reliability and Maintainability Symposium, pp. 464–469 (2006).
[9] Zhao Hongshan, Zhang Jianping, Gao Duo et al., A condition based opportunistic maintenance strategy for wind turbine under imperfect maintenance, Proceedings of the CSEE, vol. 36, no. 3, pp. 3851–3858 (2016).
[10] Li Dazi, Feng Yuanyuan, Liu Zhan et al., Reliability modeling and maintenance strategy optimization for wind power generation sets, Power System Technology, vol. 35, no. 9, pp. 122–127 (2011).
[11] Fu Yang, Xu Weixin, Liu Lujie et al., Optimization of preventive opportunistic maintenance strategy for offshore wind turbine considering weather conditions, Proceedings of the CSEE, vol. 38, no. 20, pp. 5947–5956 (2018).
[12] Tian Z., Jin T., Wu B. et al., Condition based maintenance optimization for wind power generation systems under continuous monitoring, Renewable Energy, vol. 36, no. 5, pp. 1502–1509 (2011).
[13] Yildirim M., Gebraeel N., Sun X., Integrated Predictive Analytics and Optimization for Opportunistic Maintenance and Operations in Wind Farms, IEEE Transactions on Power Systems, pp. 4319–4328 (2017).
[14] Elwany A.H., Gebraeel N.Z., Sensor-driven prognostic models for equipment replacement and spare parts inventory, IIE Transactions, vol. 40, no. 7, pp. 629–639 (2008).
[15] Liu Haiqing, Lin Weijian, Li Yuancheng, Ultra-short-term wind power prediction based on copula function and bivariate EMD decomposition algorithm, Archives of Electrical Engineering, vol. 69, no. 2, pp. 271–286 (2020).
[16] Wang Shaohua, Zhang Yaohui et al., Optimal condition-based maintenance decision-making method of multi-component system based on simulation, Acta Armamentarii, vol. 38, no. 3, pp. 568–575 (2017).
[17] Liu Junqiang, Xie Jianwei et al., Residual lifetime prediction for aeroengines based on wiener process with random effect, Acta Aeronautica et Astronautica Sinica, vol. 36, no. 2, pp. 564–574 (2015).
[18] Palmer T.N., A nonlinear dynamical perspective on model error; A proposal for non-local stochasticdynamic parametrization in weather and climate prediction models, Quarterly Journal of the Royal Meteorological Society, vol. 127, no. 572, pp. 279–304 (2010).
[19] Gong Guanglu, Qian Minping, Application of stochastic process tutorial and its stochastic models in algorithms and intelligent computing, Beijing, Tsinghua University Press (2004).
[20] Rausand M., System Reliability Theory: Models, Statistical Methods, and Applications, 2nd Edition, Statistical methods in reliability theory and practice, E. Horwood (2004).
[21] Su Hongsheng, Control strategy on preventive maintenance of repairable device, Journal of Zhejiang University (Engineering Science), vol. 44, no. 7, pp. 1308–1314 (2010).

Authors and Affiliations

Hongsheng Su

1

Xuping Duan

1

e-mail:

ORCID:

Dantong Wang

1

Lanzhou Jiaotong University, China

Automated Linearization of a System of Nonlinear Ordinary Differential Equations

Aigerim Mazakova Sholpan Jomartova Waldemar Wójcik Talgat Mazakov Gulzat Ziyatbekova

International Journal of Electronics and Telecommunications | 2023 | vol. 69 | No 4 | 655-600 | DOI: 10.24425/ijet.2023.147684

Keywords Ordinary differential equation Computer algebra stability controllability MatLab

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Abstract

This paper investigates the possibility of automatically linearizing nonlinear models. Constructing a linearised model for a nonlinear system is quite labor-intensive and practically unrealistic when the dimension is greater than 3. Therefore, it is important to automate the process of linearisation of the original nonlinear model. Based on the application of computer algebra, a constructive algorithm for the linearisation of a system of non-linear ordinary differential equations was developed. A software was developed on MatLab. The effectiveness of the proposed algorithm has been demonstrated on applied problems: an unmanned aerial vehicle dynamics model and a twolink robot model. The obtained linearized models were then used to test the stability of the original models. In order to account for possible inaccuracies in the measurements of the technical parameters of the model, an interval linearized model is adopted. For such a model, the procedure for constructing the corresponding interval characteristic polynomial and the corresponding Hurwitz matrix is automated. On the basis of the analysis of the properties of the main minors of the Hurwitz matrix, the stability of the studied system was analyzed.

Authors and Affiliations

Aigerim Mazakova

3

Sholpan Jomartova

3

Waldemar Wójcik

2

Talgat Mazakov

1

Gulzat Ziyatbekova

1

Institute of Information and Computational Technologies CS MES RK, Al-Farabi Kazakh National University, Kazakhstan
Lublin Technical University, Poland
Al-Farabi Kazakh NationalUniversity, Kazakhstan

Physics-guided neural networks (PGNNs) to solve differential equations for spatial analysis

Bartłomiej Borzyszkowski Karol Damaszke Jakub Romankiewicz Marcin Świniarski Marek Moszyński

Bulletin of the Polish Academy of Sciences Technical Sciences | 2021 | 69 | 6 | e139391 | DOI: 10.24425/bpasts.2021.139391

Keywords physics-guided neural networks spatial analysis differential equations machine learning

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Abstract

Numerous examples of physically unjustified neural networks, despite satisfactory performance, generate contradictions with logic and lead to many inaccuracies in the final applications. One of the methods to justify the typical black-box model already at the training stage involves extending its cost function by a relationship directly inspired by the physical formula. This publication explains the concept of Physics-guided neural networks (PGNN), makes an overview of already proposed solutions in the field and describes possibilities of implementing physics-based loss functions for spatial analysis. Our approach shows that the model predictions are not only optimal but also scientifically consistent with domain specific equations. Furthermore, we present two applications of PGNNs and illustrate their advantages in theory by solving Poisson’s and Burger’s partial differential equations. The proposed formulas describe various real-world processes and have numerous applications in the area of applied mathematics. Eventually, the usage of scientific knowledge contained in the tailored cost functions shows that our methods guarantee physics-consistent results as well as better generalizability of the model compared to classical, artificial neural networks.

Bibliography

R. Vinuesa et al., “The role of artificial intelligence in achieving the Sustainable Development Goals,” Nat. Commun., vol. 11, no. 1, pp. 1‒10, 2020.
M. Grochowski, A. Kwasigroch, and A. Mikołajczyk, “Selected technical issues of deep neural networks for image classification purposes,” Bull. Pol. Acad. Sci. Tech. Sci., vol. 67, no. 2, pp. 363–376, 2019.
T. Poggio and Q. Liao, “Theory II: Deep learning and optimization,” Bull. Pol. Acad. Sci. Tech. Sci., vol. 66, no. 6, pp. 775–787, 2018.
A. Lüdeling, M. Walter, E. Kroymann, and P. Adolphs, “Multilevel error annotation in learner corpora,” Proc. Corpus Linguistics Conf., vol. 1, pp. 14–17, 2005.
A. Mikołajczyk and M. Grochowski, “Data augmentation for improving deep learning in image classification problem,” Proc. Int. Interdiscipl. PhD Workshop (IIPhDW), 2018, pp. 117–122.
A.W. Moore and M.S. Lee, “Efficient algorithms for minimizing cross validation error,” Proc. 11th Int’l Conf. Machine Learning, 1994, pp. 190–198.
J.M. Benitez, J.L. Castro, and I. Requena, “Are artificial neural networks black boxes?,” IEEE Trans. Neural Networks, vol. 8, pp. 1156– 1164, 1997.
T. Hagendorff, “The ethics of AI ethics: An evaluation of guidelines,” Minds Mach., vol. 30, pp. 99–120, 2020.
T. Miller, P. Howe, and L. Sonenberg, “Explainable AI: Beware of inmates running the asylum,” Proc. IJCAI Workshop Explainable AI (XAI), 2017, pp. 36–42.
A. Rai, “Explainable AI: from black box to glass box,” Journal of the Academy of Marketing Science, vol. 48, pp. 137–141, 2020.
G. Shortley and G. Weller, “Numerical solution of Laplace’s equation,” J. Appl. Phys., vol. 9, no. 1, pp. 334–336, 1938.
R.C. Mittal and P. Singhal, “Numerical solution of Burger’s equation,” Commun. Numer. Methods Eng., vol. 9, no. 5, pp. 397–406, 1993.
R. French, “Subcognition and the limits of the Turing test,” Mind, vol. 99, no. 393, pp. 53–65, 1990.
J. McCarthy, “What is artificial intelligence?,” 1998.
I. Rojek, M. Macko, D. Mikołajewski, M. Saga, and T. Burczyński, “Modern methods in the field of machine modelling and simulation as a research and practical issue related to industry 4.0,” Bull. Pol. Acad. Sci. Tech. Sci., vol. 69, no. 2, p. e136717, 2021.
A. Karpatne, W. Watkins, J. Read, and V. Kumar, “Physicsguided neural networks (PGNN): An application in lake temperature modeling,” 2017, [Online], Available: http://arxiv.orgabs/1710.11431.
J. Willard et al., “Integrating Physics-Based Modeling with Machine Learning: A Survey,” 2020, [Online], Available: http://arxiv.org/abs/2003.04919.
X. Jia et al., “Physics guided RNNs for modeling dynamical systems: A case study in simulating lake temperature profiles,” Proc. SIAM Int. Conf. Data Mining, pp. 558–566, 2019.
A. Daw et al., “Physics-Guided Architecture (PGA) of neural networks for quantifying uncertainty in lake temperature modeling,” Proc. SIAM Int. Conf. Data Mining, pp. 532–540, 2020.
Y. Yang and P. Perdikaris, “Physics-informed deep generative models,” 2018, [Online], Available: http://arxiv.org/abs/1812. 03511.
R. Singh, V. Shah, B. Pokuri, and S. Sarkar, “Physics-aware deep generative models for creating synthetic microstructures,” 2018, [Online], Available: http://arxiv.org/abs/1811.09669.
L. Wang, Q. Zhou, and S. Jin, “Physics-guided deep learning for power system state estimation,” J. Mod. Power Syst. Clean Energy, vol. 8, no. 4, pp. 607–615, 2020.
N. Muralidhar et al., “Physics-guided design and learning of neural networks for predicting drag force on particle suspensions in moving fluids,” 2019, [Online], Available: http://arxiv.org/abs/1911.04240.
J. Park and J. Park, “Physics-induced graph neural network: An application to wind-farm power estimation,” Energy, vol. 187, p. 115883, 2019.
R. Wang, K. Kashinath, M. Mustafa, A. Albert, and R. Yu, “Towards physics-informed deep learning for turbulent flow prediction,” Proc. 26th SIGKDD Int. Conf. Knowledge Discovery and Data Mining, 2020, pp. 1457–1466.
T. Yang et al., “Evaluation and machine learning improvement of global hydrological model-based flood simulations,” Environ. Res. Lett., vol. 14, no. 11, p. 114027, 2019.
Y. Zhang et al., “Pgnn: Physics-guided neural network for fourier ptychographic microscopy,” 2019, [Online], Available: http://arxiv.org/ abs/1909.08869.
M.G. Poirot et al., “Physics-informed deep learning for dualenergy computed tomography image processing,” Sci. Rep., vol. 9, no. 1, 2019.
F. Sahli Costabal, Y. Yang, P. Perdikaris, D. E. Hurtado, E. Kuhl, “Physics-informed neural networks for cardiac activation mapping,” Front. Phys., vol. 8, p. 42, 2020.
M. Raissi, P. Perdikaris, and G.E. Karniadakis, “Physics informed deep learning (part I): Data-driven solutions of nOnlinear partial dif- ferential equations,” 2017, [Online], Available: http://arxiv.org/abs/1711.10561.
M. Raissi, P. Perdikaris, and G.E. Karniadakis, “Physics informed deep learning (part II): Data-driven solutions of nOnlinear partial differential equations,” 2017, [Online], Available: http://arxiv.org/abs/1711.10566.
Z. Fang and J. Zhan, “A physics-informed neural network framework for PDEs on 3D surfaces: Time independent problems,” IEEE Access, vol. 8, pp. 26328–26335, 2019.
B. Paprocki, A. Pregowska, and J. Szczepanski, “Optimizing information processing in brain-inspired neural networks,” Bull. Pol. Acad. Sci. Tech. Sci., vol. 68, no. 2, pp. 225–233, 2020.
M. Pfeiffer and T. Pfeil, “Deep learning with spiking neurons: opportunities and challenges,” Front. Neurosci., vol. 12, pp. 774, 2018.
Z. Bing et al., “A survey of robotics control based on learninginspired spiking neural networks,” Front. Neurorob., vol. 12, pp. 35, 2018.
B. Borzyszkowski, “Neuromorphic Computing in High Energy Physics,” 2020, doi: 10.5281/zenodo.3755310.
J. George, C. Soci, M. Miscuglio, and V. Sorger, “Symmetry perception with spiking neural networks,” Sci. Rep., vol. 11, no. 1. pp. 1–14, 2021.
K. Janocha and W.M. Czarnecki, “On loss functions for deep neural networks in classification,” 2017, [Online], Available: http://arxiv. org/abs/1702.05659.
L. Bottou, “Stochastic gradient descent tricks,” in Neural networks: Tricks of the trade, Berlin, Heidelberg: Springer 2012, pp. 421–436.
T. Dockhorn, “A discussion on solving partial differential equations using neural networks,” 2019, [Online], Available: https://arxiv.org/abs/1904.07200.
A. Blumer, A. Ehrenfeucht, D. Haussler, and M.K. Warmuth, “Occam’s razor,” Inf. Process. Lett., vol. 24, no. 6, pp. 377–380, 1987.
A. Marreiros, J. Daunizeau, S. Kiebel, and K. Friston, “Population dynamics: variance and the sigmoid activation function,” Neuroimage, vol. 42, no. 1, pp. 147–157, 2008.
S.K. Kumar, “On weight initialization in deep neural networks.” 2017, [Online]. Available: http://arxiv.org/abs/1704.08863.
N. Nawi, M. Ransing, and R. Ransing, “An improved learning algorithm based on the Broyden-fletcher-goldfarb-shanno (BFGS) method for back propagation neural networks,” Proc. 6th Int. Conf. Intell. Syst. Design Appl., 2006, vol. 1, pp. 152–157.
P. Constantin and C. Foias, “Navier-stokes equations,” Chicago: University of Chicago Press, 1988.
G.A. Anastassiou, “Multivariate hyperbolic tangent neural network approximation,” Comput. Math. Appl., vol. 61, no. 4, pp. 809–821, 2011.
B. Hanin and D. Rolnick, “How to start training: The effect of initialization and architecture,” Proc. Adv. Neural Inf. Process. Syst., 2018, pp. 571–581.
R. Ghanem and P. Spanos, “Stochastic finite elements: a spectral approach,” New York: Springer, 1991.

Authors and Affiliations

Bartłomiej Borzyszkowski

1

e-mail:

ORCID:

Karol Damaszke

1

Jakub Romankiewicz

1

Marcin Świniarski

1

Marek Moszyński

1

Faculty of Electronics, Telecommunications and Informatics, Gdansk University of Technology, ul. G. Narutowicza 11/12, 80-233 Gdańsk, Poland

Steady-state modelling method for matrix-reactance frequency converter with boost topology

Igor Ye. Korotyeyev Beata Zięba

Archives of Electrical Engineering | 2011 | vol. 60 | No 2 June | 137-148 | DOI: 10.2478/v10171-011-0013-8

Keywords AC/AC converters matrix-reactance frequency converter Galerkin's method extension of differential equations

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Abstract

This paper presents a method intended for calculation of steady-state processes in AC/AC three-phase converters that are described by nonstationary periodical differential equations. The method is based on the extension of nonstationary differential equations and the use of Galerkin's method. The results of calculations are presented in the form of a double Fourier series. As an example, a three-phase matrix-reactance frequency converter (MRFC) with boost topology is considered and the results of computation are compared with a numerical method.

Authors and Affiliations

Igor Ye. Korotyeyev

Beata Zięba

Analysis of three-phases matrix reactance frequency converter with two pulsations of control signal

Igor Ye. Korotyeyev Beata Zięba

Archives of Electrical Engineering | 2012 | vol. 61 | No 3 September | 359-371 | DOI: 10.2478/v10171-012-0029-8

Keywords matrix-reactance frequency converter non-stationary periodic differential Galerkin's method extension of differential equations

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Abstract

This paper presents a method of calculation of steady-state processes in threephases matrix-reactance frequency converters (MRFC's), in which voltages and currents are transformed by control signals with two pulsations. A solution of nonstationary differentia equations with periodic coefficients that describe this system is obtained by using Galerkin's method and an extension of equations of one variable of time to equations of two variables of time. The results of calculations are presented in an example of three-phases MRFC with buck-boost topology and compared with a numerical metod embedded in the program Mathematica.

Authors and Affiliations

Igor Ye. Korotyeyev

Beata Zięba

Modelling of a non-linear coil with loss in iron using the Runge-Kutta methods

Joanna Kolańska-Płuska Barbara Grochowicz

Archives of Electrical Engineering | 2016 | vol. 65 | No 3 September | 527-539 | DOI: 10.1515/aee-2016-0038

Keywords non-linear coil Runge-Kutta methods stiff non-linear differential equations

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Abstract

This work presents a study on dynamics of a circuit with a non-linear coil, where loss in iron is also taken into account. A coil model is derived using a state space description. The work also includes the development of an application in C# for coil dynamics examination, where the implicit RADAU IIA method of various orders is applied for the purpose of solving non-linear differential equations modelling the non-linear coil with loss in iron.

Authors and Affiliations

Joanna Kolańska-Płuska

Barbara Grochowicz

Linear Dynamic System Identification in the Frequency Domain Using Fractional Derivatives

Tomasz Janiczek Janusz Janiczek

Metrology and Measurement Systems | 2010 | No 2 | DOI: 10.2478/v10178-010-0024-6

Keywords fractional differential equations fractional differential systems Fourier transform method identification methods

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Abstract

This paper presents a study of the Fourier transform method for parameter identification of a linear dynamic system in the frequency domain using fractional differential equations. Fundamental definitions of fractional differential equations are briefly outlined. The Fourier transform method of identification and their algorithms are generalized so that they include fractional derivatives and integrals.

Authors and Affiliations

Tomasz Janiczek

Janusz Janiczek

Explicit commutativity conditions for second-order linear time-varying systems with non-zero initial conditions

Mehmet Emir Koksal

Archives of Control Sciences | 2019 | vol. 29 | No 3 | 529–548 | DOI: 10.24425/acs.2019.130204

Keywords commutativity linear systems analogue control robust control differential equations non-zero initial conditions

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Abstract

Although the explicit commutativitiy conditions for second-order linear time-varying systems have been appeared in some literature, these are all for initially relaxed systems. This paper presents explicit necessary and sufficient commutativity conditions for commutativity of second-order linear time-varying systems with non-zero initial conditions. It has appeared interesting that the second requirement for the commutativity of non-relaxed systems plays an important role on the commutativity conditions when non-zero initial conditions exist. Another highlight is that the commutativity of switched systems is considered and spoiling of commutativity at the switching instants is illustrated for the first time. The simulation results support the theory developed in the paper.

Authors and Affiliations

Mehmet Emir Koksal

The mathematical characteristic of the Laplace contour filters used in digital image processing. The third order filters

Ireneusz Winnicki Janusz Jasinski Slawomir Pietrek Krzysztof Kroszczynski

Advances in Geodesy and Geoinformation | 2022 | vol. 71 | No 2 | e23 | DOI: 10.24425/gac.2022.141176

Keywords transfer function finite element method finite difference methods matrices of the third-order Laplace filters modified differential equations

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Abstract

The Laplace operator is a differential operator which is used to detect edges of objects in digital images. This paper presents the properties of the most commonly used third-order 3x3 pixels Laplace contour filters including the difference schemes used to derive them. The authors focused on the mathematical properties of the Laplace filters. The basic reasons of the differences of the properties were studied and indicated using their transfer functions and modified differential equations. The relations between the transfer function for the differential Laplace operator and its difference operators were described and presented graphically. The impact of the corner elements of the masks on the results was discussed. This is a theoretical work. The basic research conducted here refers to a few practical examples which are illustrations of the derived conclusions.We are aware that unambiguous and even categorical final statements as well as indication of areas of the results application always require numerous experiments and frequent dissemination of the results. Therefore, we present only a concise procedure of determination of the mathematical properties of the Laplace contour filters matrices. In the next paper we shall present the spectral characteristic of the fifth order filters of the Laplace type.

Authors and Affiliations

Ireneusz Winnicki

1

e-mail:

ORCID:

Janusz Jasinski

1

e-mail:

ORCID:

Slawomir Pietrek

1

e-mail:

ORCID:

Krzysztof Kroszczynski

1

e-mail:

ORCID:

Military University of Technology, Warsaw, Poland

Fixed terminal time fractional optimal control problem for discrete time singular system

Tirumalasetty Chiranjeevi Ramesh Devarapalli Naladi Ram Babu Kiran Babu Vakkapatla R. Gowri Sankara Rao Fausto Pedro Garcìa Màrquez

Archives of Control Sciences | 2022 | vol. 32 | No 3 | 489-506 | DOI: 10.24425/acs.2022.142846

Keywords fractional order differential equation discrete time singular system fractional derivative linear quadratic optimal control problem

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Abstract

This paper presents the formulation and numerical simulation for linear quadratic optimal control problem (LQOCP) of free terminal state and fixed terminal time fractional order discrete time singular system (FODSS). System dynamics is expressed in terms of Riemann-Liouville fractional derivative (RLFD), and performance index (PI) in terms of state and costate. Because of its complexity, finding analytical and numerical solutions to singular system (SS) is difficult. As a result, we use coordinate transformation to convert FODSS to its corresponding fractional order discrete time nonsingular system (FODNSS). After that, we obtain the necessary conditions by employing a Hamiltonian approach. The relevant conditions are solved using the general solution approach. For the analysis of formulation and solution algorithm, a numerical example is illustrated. Results are obtained for various �� values. According to state of the art, this is the first time that a formulation and numerical simulation of free terminal state and fixed terminal time optimal control problem (OCP) of FODSS is presented.

Authors and Affiliations

Tirumalasetty Chiranjeevi

1

Ramesh Devarapalli

2

e-mail:

ORCID:

Naladi Ram Babu

3

Kiran Babu Vakkapatla

4

R. Gowri Sankara Rao

5

Fausto Pedro Garcìa Màrquez

6

Department of Electrical Engineering, Rajkiya Engineering College Sonbhadra, U.P., India
Department of EEE, Lendi Institute of Engineering and Technology, Vizianagaram-535005, India
Department of EEE, Aditya Engineering College, Surampalem, Andhra Pradesh, India
Lingayas Institute of Management and Technology Madalavarigudem, A.P., India
Department of EEE, MVGR College of Engineering Vizianagaram, A.P., India
Ingenium Research Group, University of Castilla-La Mancha, Spain

Theoretical Analysis and Experimental Verification of Top Orthogonal to Bottom Arrays of Conducting Strips on Piezoelectric Slab

Jurij Tasinkevych Ihor Trots Ryszard Tymkiewicz

Archives of Acoustics | 2020 | vol. 45 | No 3 | 433-444 | DOI: 10.24425/aoa.2020.134059

Keywords boundary value problem Fourier series Bloch series partial differential equations piezoelectric transducer

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Abstract

The purpose of this work is to present a theoretical analysis of top orthogonal to bottom arrays of conducting electrodes of infinitesimal thickness (conducting strips) residing on the opposite surfaces of piezoelectric slab. The components of electric field are expanded into double periodic Bloch series with corresponding amplitudes represented by Legendre polynomials, in the proposed semi-analytical model of the considered two-dimensional (2D) array of strips. The boundary and edge conditions are satisfied directly by field representation, as a result. The method results in a small system of linear equations for unknown expansion coefficients to be solved numerically. A simple numerical example is given to illustrate the method. Also a test transducer was designed and a pilot experiment was carried out to illustrate the acoustic-wave generating capabilities of the proposed arrangement of top orthogonal to bottom arrays of conducting strips.

Authors and Affiliations

Jurij Tasinkevych

Ihor Trots

Ryszard Tymkiewicz

Solution of nonlinear stiff differential equations for a three-phase no-load transformer using a Runge–Kutta implicit method

Bernard Baron Joanna Kolańska-Płuska Marian Łukaniszyn Dariusz Spałek Tomasz Kraszewski

Archives of Electrical Engineering | 2022 | vol. 71 | No 4 | 1081-1106 | DOI: 10.24425/aee.2022.142126

Keywords circuit model of a three-phase transformer Runge–Kutta implicit methods stiff nonlinear ordinary differential equations

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Abstract

The paper presents an approach to differential equation solutions for the stiff problem. The method of using the classic transformer model to study nonlinear steady states and to determine the current pulses appearing when the transformer is turned on is given. Moreover, the stiffness of nonlinear ordinary differential state equations has to be considered. This paper compares Runge–Kutta implicit methods for the solution of this stiff problem.

Authors and Affiliations

Bernard Baron

1

e-mail:

ORCID:

Joanna Kolańska-Płuska

1

e-mail:

ORCID:

Marian Łukaniszyn

1

e-mail:

ORCID:

Dariusz Spałek

2

e-mail:

ORCID:

Tomasz Kraszewski

3

e-mail:

ORCID:

Faculty of Electrical Engineering, Automatic Control and Informatics, Opole University of Technology, Prószkowska 76, 45-758 Opole, Poland
Institute of Electrotechnics and Informatics, Silesian University of Technology, 10 Akademicka St., 44-100 Gliwice, Poland
Research and Development Center GLOKOR Sp. z o.o., Górnych Wałów 27A St., 44-100 Gliwice, Poland

Existence, uniqueness and parameter perturbation analysis results of a fractional integro-differential boundary problem

Nan Zhang Lingling Zhang Mercy Ngungu Adejimi Adeniji Emmanuel Addai

Bulletin of the Polish Academy of Sciences Technical Sciences | 2023 | 71 | 4 | e145938 | DOI: 10.24425/bpasts.2023.145938

Keywords existence and uniqueness stability analysis parameter perturbation fractional differential equation integro-differential boundary condition

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Abstract

In the formulation, the existence, uniqueness and stability of solutions and parameter perturbation analysis to Riemann-Liouville fractional differential equations with integro-differential boundary conditions are discussed by the properties of Green’s function and cone theory. First, some theorems have been established from standard fixed point theorems in a proper Banach space to guarantee the existence and uniqueness of positive solution. Moreover, we discuss the Hyers-Ulam stability and parameter perturbation analysis, which examines the stability of solutions in the presence of small changes in the equation main parameters, that is, the derivative order η, the integral order β of the boundary condition, the boundary parameter ξ , and the boundary value τ. As an application, we present a concrete example to demonstrate the accuracy and usefulness of the proposed work. By using numerical simulation, we obtain the figure of unique solution and change trend figure of the unique solution with small disturbances to occur in different kinds of parameters.

Authors and Affiliations

Nan Zhang

1

Lingling Zhang

2

e-mail:

ORCID:

Mercy Ngungu

3

Adejimi Adeniji

4

Emmanuel Addai

2

College of Mathematics, Taiyuan University of Technology, 030024, TaiYuan, Shanxi, ChinaCollege of Mathematics, Taiyuan University of Technology, 030024, TaiYuan, Shanxi, China
College of Mathematics, Taiyuan University of Technology, 030024, TaiYuan, Shanxi, China
Human Sciences Research Council (HSRC), South Africa
Tshwane university of Technology, South Africa

The implicit numerical method for the one-dimensional anomalous subdiffusion equation with a nonlinear source term

Marek Błasik

Bulletin of the Polish Academy of Sciences Technical Sciences | 2021 | 69 | 6 | e138240 | DOI: 10.24425/bpasts.2021.138240

Keywords fractional derivatives and integrals integro-differential equations numerical methods finite difference methods

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Abstract

In the paper, the numerical method of solving the one-dimensional subdiffusion equation with the source term is presented. In the approach used, the key role is played by transforming of the partial differential equation into an equivalent integro-differential equation. As a result of the discretization of the integro-differential equation obtained an implicit numerical scheme which is the generalized Crank-Nicolson method. The implicit numerical schemes based on the finite difference method, such as the Carnk-Nicolson method or the Laasonen method, as a rule are unconditionally stable, which is their undoubted advantage. The discretization of the integro-differential equation is performed in two stages. First, the left-sided Riemann-Liouville integrals are approximated in such a way that the integrands are linear functions between successive grid nodes with respect to the time variable. This allows us to find the discrete values of the integral kernel of the left-sided Riemann-Liouville integral and assign them to the appropriate nodes. In the second step, second order derivative with respect to the spatial variable is approximated by the difference quotient. The obtained numerical scheme is verified on three examples for which closed analytical solutions are known.

Bibliography

T. Kosztołowicz, K. Dworecki, and S. Mrówczyński, “How to measure subdiffusion parameters,” Phys. Rev. Lett., vol. 94, p. 170602, 2005, doi: 10.1016/j.tins.2004.10.007.
T. Kosztołowicz, K. Dworecki, and S. Mrówczyński, “Measuring subdiffusion parameters,” Phys. Rev. E, vol. 71, p. 041105, 2005.
E. Weeks, J. Urbach, and L. Swinney, “Anomalous diffusion in asymmetric random walks with a quasi-geostrophic flow example,” Physica D, vol. 97, pp. 291–310, 1996.
T. Solomon, E. Weeks, and H. Swinney, “Observations of anomalous diffusion and Lévy flights in a 2-dimensional rotating flow,” Phys. Rev. Lett., vol. 71, pp. 3975–3979, 1993.
N.E. Humphries, et al., “Environmental context explains Lévy and Brownian movement patterns of marine predators,” Nature, vol. 465, pp. 1066–1069, 2010.
U. Siedlecka, “Heat conduction in a finite medium using the fractional single-phase-lag model,” Bull. Pol. Acad. Sci. Tech. Sci., vol. 67, pp. 402–407, 2019.
R. Metzler and J. Klafter, “The random walk:s guide to anomalous diffusion: a fractional dynamics approach,” Phys. Rep., vol. 339, pp. 1–77, 2000.
R. Metzler and J. Klafter, “The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics,” J. Phys. A: Math. Gen., vol. 37, pp. 161–208, 2004.
M. Aslefallah, S. Abbasbandy, and E. Shivanian, “Numerical solution of a modified anomalous diffusion equation with nonlinear source term through meshless singular boundary method,” Eng. Anal. Boundary Elem., vol. 107, pp. 198–207, 2019.
Y. Li and D. Wang, “Improved efficient difference method for the modified anomalous sub-diffusion equation with a nonlinear source term,” Int. J. Comput. Math., vol. 94, pp. 821–840, 2017.
X. Cao, X. Cao, and L. Wen, “The implicit midpoint method for the modified anomalous sub-diffusion equation with a nonlinear source term,” J. Comput. Appl. Math., vol. 318, pp. 199–210, 2017.
A. Kilbas, H. Srivastava, and J. Trujillo, Theory and Applications of Fractional Differential Equations. Amsterdam: Elsevier, 2006.
E.D. Rainville, Special Functions. New York: The Macmillan Company, 1960.
J.-L. Liu and H.MSrivastava, “Classes of meromorphically multivalent functions associated with the generalized hypergeometric function,” Math. Comput. Modell., vol. 39, pp. 21–34, 2004.
Y.L. Luke, “Inequalities for generalized hypergeometric functions,” J. Approximation Theory, vol. 5, pp. 41–65, 1972.
M. Włodarczyk and A. Zawadzki, “The application of hypergeometric functions to computing fractional order derivatives of sinusoidal functions,” Bull. Pol. Acad. Sci. Tech. Sci., vol. 64, pp. 243–248, 2016.
M. Błasik, “A generalized Crank-Nicolson method for the solution of the subdiffusion equation,” 23rd International Conference on Methods & Models in Automation & Robotics (MMAR), pp. 726–729, 2018.
M. Błasik, “Zagadnienie stefana niecałkowitego rzędu,” Ph.D. dissertation, Politechnika Częstochowska, 2013.
M. Błasik and M. Klimek, “Numerical solution of the one phase 1d fractional stefan problem using the front fixing method,” Math. Methods Appl. Sci., vol. 38, no. 15, pp. 3214–3228, 2015.
K. Diethelm, The Analysis of Fractional Differential Equations. Berlin: Springer-Verlag, 2010.

Authors and Affiliations

Marek Błasik

1

Institute of Mathematics, Czestochowa University of Technology, al. Armii Krajowej 21, 42-201 Czestochowa, Poland

Finite-Difference Operators for 2D problems

T. Sobczyk

Bulletin of the Polish Academy of Sciences Technical Sciences | 2020 | 68 | No. 6 | 1535-1541 | DOI: 10.24425/bpasts.2020.135387

Keywords second-order partial differential equations 2D finite-difference operators finite-difference equations iterative algorithms

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Abstract

This paper presents the concept of using algorithms for reducing the dimensions of finite-difference equations of two-dimensional (2D) problems, for second-order partial differential equations. Solutions are predicted as two-variable functions over the rectangular domain, which are periodic with respect to each variable and which repeat outside the domain. Novel finite-difference operators, of both the first and second orders, are developed for such functions. These operators relate the value of derivatives at each point to the values of the function at all points distributed uniformly over the function domain. A specific feature of the novel operators follows from the arrangement of the function values as well as the values of derivatives, which are rectangular matrices instead of vectors. This significantly reduces the dimensions of the finite-difference operators to the numbers of points in each direction of the 2D area. The finite-difference equations are created exemplary elliptic equations. An original iterative algorithm is proposed for reducing the process of solving finite-difference equations to the multiplication of matrices.

Authors and Affiliations

T. Sobczyk

Varying coefficients in parallel shared-memory variational splitting solvers for non-stationary Maxwell equations

Marcin Łoś Maciej Woźniak Maciej Paszynski

Bulletin of the Polish Academy of Sciences Technical Sciences | Early Access | e149179 | DOI: 10.24425/bpasts.2024.149179

Keywords partial differential equations finite element method Maxwell equations variational splitting MRI scan data

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Abstract

Direction-splitting implicit solvers employ the regular structure of the computational domain augmented with the splitting of the partial differential operator to deliver linear computational cost solvers for time-dependent simulations. The finite difference community originallye mployed this method to deliver fast solvers for PDE-based formulations. Later, this method was generalized into so-called variationals plitting. The tensor product structure of basis functions over regular computational meshes allows us to employ the Kronecker product structureo f the matrix and obtain linear computational cost factorization for finite element method simulations. These solvers are traditionally usedf or fast simulations over the structures preserving the tensor product regularity. Their applications are limited to regular problems and regularm odel parameters. This paper presents a generalization of the method to deal with non-regular material data in the variational splitting method. Namely, we can vary the material data with test functions to obtain a linear computational cost solver over a tensor product grid with nonregularm aterial data. Furthermore, as described by the Maxwell equations, we show how to incorporate this method into finite element methods imulations of non-stationary electromagnetic wave propagation over the human head with material data based on the three-dimensional MRI scan.

Authors and Affiliations

Marcin Łoś

e-mail:

ORCID:

Maciej Woźniak

e-mail:

ORCID:

Maciej Paszynski

e-mail:

ORCID:

Comprehensive online estimation of object signals for a control system with an adaptive approach and incomplete measurements

Tadeusz Kwater Przemysław Hawro Paweł Krutys Marek Gołębiowski Grzegorz Drałus

Bulletin of the Polish Academy of Sciences Technical Sciences | Early Access | e150111 | DOI: 10.24425/bpasts.2024.150111

Keywords online estimation and control nonlinear differential equations adaptive gain modification measurements window random lackof measurements

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Abstract

This paper presents the novel estimation algorithm that generates all signals of an object described by nonlinear ordinary differential equations based only on easy-to-implement measurements. Unmeasured signals are estimated by using an adaptive approach. For this purpose, a filtering equation with a continuously modified gain vector is used. Its value is determined by an incremental method, and the amount of correction depends on the current difference between the generated signal and its measured counterpart. In addition, the study takes into account the aging process of measurements and their random absence. The application of the proposed approach can be realized for any objects with a suitable mathematical description. A biochemically polluted river with an appropriate transformation of the notation of partial differential equations was chosen as an object. The results of numerical experiments are promising, and the process of obtaining them involves little computational necessity, so the approach is aimed at the needs of control implemented online.

Authors and Affiliations

Tadeusz Kwater

e-mail:

ORCID:

Przemysław Hawro

e-mail:

ORCID:

Paweł Krutys

Marek Gołębiowski

Grzegorz Drałus

The mathematical characteristic of the fifth order Laplace contour filters used in digital image processing

Ireneusz Winnicki Slawomir Pietrek Janusz Jasinski Krzysztof Kroszczynski

Advances in Geodesy and Geoinformation | 2022 | vol. 71 | No 2 | e26 | DOI: 10.24425/agg.2022.141300

Keywords finite element method finite difference methods modified differential equations matrices of the fifth-order Laplace filters Taylor expansion of the transfer function

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Abstract

The Laplace operator is a differential operator which is used to detect edges of objects in digital images. This paper presents the properties of the most commonly used fifth-order pixels Laplace filters including the difference schemes used to derive them (finite difference method – FDM and finite element method – FEM). The results of the research concerning third-order pixels matrices of the convolution Laplace filters used for digital processing of images were presented in our previous paper: The mathematical characteristic of the Laplace contour filters used in digital image processing. The third order filters is presented byWinnicki et al. (2022). As previously, the authors focused on the mathematical properties of the Laplace filters: their transfer functions and modified differential equations (MDE). The relations between the transfer function for the differential Laplace operator and its difference operators are described and presented here in graphical form. The impact of the corner elements of the masks on the results is also discussed. A transfer function, is a function characterizing properties of the difference schemes applied to approximate differential operators. Since they are relations derived in both types of spaces (continuous and discrete), comparing them facilitates the assessment of the applied approximation method.

Authors and Affiliations

Ireneusz Winnicki

1

e-mail:

ORCID:

Slawomir Pietrek

1

e-mail:

ORCID:

Janusz Jasinski

1

e-mail:

ORCID:

Krzysztof Kroszczynski

1

e-mail:

ORCID:

Military University of Technology, Warsaw, Poland

1

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