COMMUTATIVITY ASSOCIATED WITH EULER SECOND-ORDER DIFFERENTIAL EQUATION
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.
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.