COMMUTATIVITY ASSOCIATED WITH EULER SECOND-ORDER DIFFERENTIAL EQUATION

Salisu Ibrahim

doi:10.17654/0974324322022

COMMUTATIVITY ASSOCIATED WITH EULER SECOND-ORDER DIFFERENTIAL EQUATION

Salisu Ibrahim Mathematics Education, Tishk International University-Erbil, Kurdistan Region, Iraq

Article ID: 2568

DOI: https://doi.org/10.17654/0974324322022

Keywords: commutativity; Euler differential equation; analogue system

Abstract

We study commutativity and the sensitivity of the second-order Euler differential equation. The necessary and sufficient conditions for commutativity of the second-order Euler differential equation are considered. Moreover, the stability, the robustness, and the effect due to disturbance on the second-order Euler linear time-varying system (LTVS) are investigated. An example is given to support the results. The results are well verified using the Matlab Simulink toolbox.

Published

2022-05-31

How to Cite

Ibrahim, S. (2022). COMMUTATIVITY ASSOCIATED WITH EULER SECOND-ORDER DIFFERENTIAL EQUATION. Advances in Differential Equations and Control Processes, 28, 29–36. https://doi.org/10.17654/0974324322022

Download Citation

Issue

Vol. 28 (2022)

Section

Article

This work is licensed under a Creative Commons Attribution 4.0 International License.

References

[1]S. Ibrahim, Numerical approximation method for solving differential equations, Eurasian Journal of Science and Engineering 6(2) (2020), 157-168.

[2]S. Ibrahim and A. Isah, Solving system of fractional order differential equations using Legendre operational matrix of derivatives, Eurasian Journal of Science and Engineering 7(1) (2021), 25-37.

[3]S. Ibrahim and M. E. Koksal, Decomposition of fourth-order linear time-varying, International Symposium on Innovative Approaches in Engineering and Natural Sciences, Samsun, Vol. 4, 2019, pp. 139-141.

[4]S. Ibrahim and M. E. Koksal, Commutativity of sixth-order time-varying linear systems, Circuits Syst. Signal Process, 2021. https://doi.org/10.1007/s00034-021-01709-6.

[5]S. Ibrahim and M. E. Koksal, Realization of a fourth-order linear time-varying differential system with nonzero initial conditions by cascaded two second-order commutative pairs, Circuits Syst. Signal Process, 2021. https://doi.org/10.1007/s00034-020-01617-1.

[6]A. Isah and S. Ibrahim, Shifted Genocchi polynomial operational matrix for solving fractional order system, Eurasian Journal of Science and Engineering 7(1) (2021), 25-37.

[7]M. Koksal, Commutativity of second-order time-varying systems, Internat. J. Control 36(3) (1982), 541-544.

[8]M. Koksal, General conditions for the commutativity of time-varying systems’ of second-order time-varying systems, International Conference on Telecommunication and Control, Halkidiki, Greece, 1987, pp. 223-225.

[9]M. Koksal, Commutativity of 4th order systems and Euler systems, Presented in National Congress of Electrical Engineers, Adana, Turkey, Paper no: BI-6, 1988.

[10]M. Koksal and M. E. Koksal, Commutativity of linear time-varying differential systems with non-zero initial conditions: a review and some new extensions, Math. Probl. Eng. 2011, Art. ID 678575, 1-25.

[11]E. Marshall, Commutativity of time-varying systems, Electro Letters 18 (1977), 539-540.

[12]A. Rababah and S. Ibrahim, Weighted -multi-degree reduction of Bézier curves, International Journal of Advanced Computer Science and Applications 7(2) (2016a), 540-545. https://thesai.org/Publications/ViewPaper?Volume=7&Issue=2&Code=ijacsa &SerialNo=70.

[13]A. Rababah and S. Ibrahim, Weighted degree reduction of Bézier curves with -continuity, International Journal of Advanced and Applied Science 3(3) (2016b), 13-18.

[14]A. Rababah and S. Ibrahim, Geometric degree reduction of Bézier curves, Springer Proceeding in Mathematics and Statistics, Book Chapter 8, 2018. https://www.springer.com/us/book/9789811320941.

[15]S. V. Saleh, Comments on commutativity of second-order time-varying systems, Internat. J. Control 37 (1983), 1195-1196.

[16]I. Salisu, Explicit Commutativity and Stability for the Heun’s Linear Time-Varying Differential Systems, Authorea, 2021. https://doi.org/10.22541/au.162566323.35099726/v1.

[17]I. Salisu and R. Abedallah, Decomposition of fourth-order Euler-type linear time-varying differential system into cascaded two second-order Euler commutative pairs, Complexity 2022 (2022), Article ID 3690019, 9 pp. https://doi.org/10.1155/2022/3690019.

[18]Salisu Ibrahim, Commutativity of high-order linear time-varying systems, Advances in Differential Equations and Control Processes 27 (2022a), 73-83. http://dx.doi.org/10.17654/0974324322013.

[19]Salisu Ibrahim, Discrete least square method for solving differential equations, Advances and Applications in Discrete Mathematics 30 (2022b), 87-102. http://dx.doi.org/10.17654/0974165822021.

Early fault signals of the rolling bearing in the rotor are weak and present the characteristics of non-periodic and non-stationary; it is more difficult to carry out fault diagnosis on it. In this regard, this paper proposes a weak rolling bearing fault diagnosis algorithm based on whale optimization algorithm, simplistic geometry mode decomposition, and maximum correlated kurtosis deconvolution (WOA-SGMD-MCKD). Firstly, the vibration signal of the rotor platform is obtained, and the Symmetric Geometric Mode Decomposition (SGMD) is used to reconstruct the vibration signal. To obtain the best decomposition effect of the SGMD and overcome modal aliasing, the Whale Optimization Algorithm (WOA) is used to optimize the embedding dimension. Secondly, for the reconstructed vibration signal, the Maximum Correlated Kurtosis Deconvolution (MCKD) is used to extract its impulse component, and the WOA is used to optimize the filter length and deconvolution period of the MCKD so that the frequency envelope spectrum of the vibration signal can be obtained, which can provide the basis for the fault diagnosis of rolling bearings. Finally, the effectiveness and feasibility of the algorithm proposed are verified by a non-periodic and non-stationary simulation platform and rotor maneuvering platform in this paper.

Numerical solution of a 3D mathematical model for the progression of tumor angiogenic factor in a tissue

In this work, the movement of tumor angiogenic factor in a three-dimensional tissue is obtained by the Method of Lines. This method transforms a partial differential equation into a system of ordinary differential equations together with the initial and boundary conditions. The more the number of lines is increased, the more the accuracy of the method increases. This method results in very accurate numerical solutions for linear and non-linear problems in contrast with other existing methods. We present Matlab-generated figures, which are the movement of tumor angiogenic factor in porous medium and explain the biological importance of this progression. The computer codes are also provided.

Mathematical modelling and controllability analysis of fractional order coal mill pulverizer model

This paper investigates the controllability of nonlinear dynamical systems and their applications, with a focus on fractional-order systems and coal mill models. A novel theorem is proposed, providing sufficient conditions for controllability, including constraints on the steering operator and nonlinear perturbation bounds. The theorem establishes the existence of a contraction mapping for the nonlinear operator, enabling effective control strategies for fractional systems. The methodology is demonstrated through rigorous proof and supported by an iterative algorithm for controller design. Additionally, the controllability of a coal mill system represented as a nonlinear differential system, is analyzed. The findings present new insights into the interplay of fractional dynamics and nonlinear systems, offering practical solutions for real-world control problems.