Results on Ulam-type stability of linear differential equation with integral transform. (English) Zbl 1539.34064
Summary: The main theme of this study is to implement the Sumudu integral transform technique to solve the stability problem of linear differential equations. Another important aspect of this paper is to investigate the Ulam-Hyers and Ulam-Hyers-JRassias stability of linear differential equations by using Sumudu transform method. Further, the results are extended to the Mittag-Leffler-Ulam-Hyers and Mittag-Leffler-Ulam-Hyers-JRassias stability of these differential equations. As an application point of view, the Sumudu transform is used to find Ulam stabilities of differential equations arising in field-controlled DC servo motor with position control.
© 2023 John Wiley & Sons Ltd.
© 2023 John Wiley & Sons Ltd.
MSC:
| 34H15 | Stabilization of solutions to ordinary differential equations |
| 34H05 | Control problems involving ordinary differential equations |
| 44A15 | Special integral transforms (Legendre, Hilbert, etc.) |
Keywords:
DC servo motor; linear differential equation; Sumudu transform; Ulam-Hyers and Ulam-Hyers-Jrassias stabilityReferences:
| [1] | S. M.Ulam, Problems in modern mathematics, John Wiley & Sons, New York, 1964. · Zbl 0137.24201 |
| [2] | D. H.Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci.27 (1941), no. 4, 222-224. · JFM 67.0424.01 |
| [3] | T.Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan2 (1950), no. 1‐2, 64-66. · Zbl 0040.35501 |
| [4] | T. M.Rassias, On the stability of the linear mapping in Banach spaces, Proc. Am. Math. Soc.72 (1978), no. 2, 297-300. · Zbl 0398.47040 |
| [5] | L.Cădariu, L.Găvruţa, and P.Găvruţa, Fixed points and generalized Hyers‐Ulam stability, Abstract Appl. Anal.2012 (2012), 712743. · Zbl 1252.39030 |
| [6] | P.Găvruţa, S. M.Jung, and Y.Li, Hyers‐Ulam stability for second‐order linear differential equations with boundary conditions, Electron. J. Differ. Equ.2011 (2011), no. 80, 1-5. · Zbl 1230.34020 |
| [7] | V.Kalvandi, N.Eghbali, and J. M.Rassias, Mittag‐Leffler‐Hyers‐Ulam stability of fractional differential equations of second order, J. Math. Extens.13 (2019), no. 1, 29-43. · Zbl 1462.34013 |
| [8] | A.Ponmana Selvan, S.Sabarinathan, and A.Selvam, Approximate solution of the special type differential equation of higher order using Taylor’s series, J. Math. Comput. Sci.27 (2022), 131-141. |
| [9] | A.Selvam, S.Sabarinathan, S.Noeiaghdam, and V.Govindan, Fractional Fourier transform and Ulam stability of fractional differential equation with fractional Caputo‐type derivative, J. Funct. Spaces2022 (2022), 3777566. · Zbl 1506.34020 |
| [10] | M.Sivashankar, S.Sabarinathan, V.Govindan, U.Fernandez‐Gamiz, and S.Noeiaghdam, Stability analysis of COVID‐19 outbreak using Caputo‐Fabrizio fractional differential equation, Am. Inst. Math. Sci. (AIMS) Math.8 (2023), no. 2, 2720-2735. |
| [11] | G. K.Watugala, Sumudu transform: a new integral transform to solve differential equations and control engineering problems, Int. J. Math. Educ. Sci. Technol.24 (1993), no. 1, 35-43. · Zbl 0768.44003 |
| [12] | S.Weerakoon, Application of Sumudu transform to partial differential equations, Int. J. Math. Educ. Sci. Technol.25 (1994), no. 2, 277-283. · Zbl 0812.35004 |
| [13] | S.Weerakoon, Complex inversion formula for Sumudu transform, Int. J. Math. Educ. Sci. Technol.29 (1998), no. 4, 618-621. · Zbl 1018.44004 |
| [14] | G. K.Watugala, The Sumudu transform for functions of two variables, Math. Eng. Ind.8 (2002), no. 4, 293-302. · Zbl 1025.44003 |
| [15] | M. A.Asiru, Further properties of the Sumudu transform and its applications, Int. J. Math. Educ. Sci. Technol.33 (2002), no. 3, 441-449. · Zbl 1013.44001 |
| [16] | F. B. M.Belgacem and A. A.Karaballi, Sumudu transform fundamental properties investigations and applications, J. Appl. Math. Stoch. Anal.2006 (2006), 91083. · Zbl 1115.44001 |
| [17] | H.Eltayeb and A.Kılıçman, A note on the Sumudu transforms and differential equations, Appl. Math. Sci.4 (2010), no. 22, 1089-1098. · Zbl 1207.34014 |
| [18] | A.Kılıçman and H. E.Gadain, On the applications of Laplace and Sumudu transforms, J. Franklin Inst.347 (2010), no. 5, 848-862. · Zbl 1286.35185 |
| [19] | A. M.Abdel‐Hamed and E. A.Badran, An improved PID control scheme for DC servo motor using salp swarm algorithm, 2022 23rd International Middle East Power Systems Conference (MEPCON), IEEE, Cairo, Egypt, 2022, pp. 1-8. |
| [20] | A. S.Sadun, J.Jalani, and J. A.Sukor, A comparative study on the position control method of DC servo motor with position feedback by using Arduino, Proceedings of Engineering Technology International Conference (ETIC 2015), 10-11 August 2015, Bali, Indonesia, EDP Sciences, 2015, pp. 1-6. |
| [21] | H.Rezaei, S. M.Jung, and T. M.Rassias, Laplace transform and Hyers‐Ulam stability of linear differential equations, J. Math. Anal. Appl.403 (2013), no. 1, 244-251. · Zbl 1286.34077 |
| [22] | Q. H.Alqifiary and S. M.Jung, Laplace transform and generalized Hyers‐Ulam stability of linear differential equations, Electron. J. Differ. Equ.2014 (2014), no. 80, 1-11. · Zbl 1290.34059 |
| [23] | C.Dineshkumar, K.Sooppy Nisar, R.Udhayakumar, and V.Vijayakumar, A discussion on approximate controllability of Sobolev‐type Hilfer neutral fractional stochastic differential inclusions, Asian J. Control24 (2022), no. 5, 2378-2394. · Zbl 07887143 |
| [24] | V.Vijayakumar, Approximate controllability results for abstract neutral integro‐differential inclusions with infinite delay in Hilbert spaces, IMA J. Math. Control Inf.35 (2018), no. 1, 297-314. · Zbl 1402.93056 |
| [25] | V.Vijayakumar, Approximate controllability results for analytic resolvent integro‐differential inclusions in Hilbert spaces, Int. J. Control91 (2018), no. 1, 204-214. · Zbl 1393.34077 |
| [26] | V.Vijayakumar and R.Murugesu, Controllability for a class of second‐order evolution differential inclusions without compactness, Appl. Anal.98 (2019), no. 7, 1367-1385. · Zbl 1414.93038 |
| [27] | V.Vijayakumar, R.Udhayakumar, and C.Dineshkumar, Approximate controllability of second order nonlocal neutral differential evolution inclusions, IMA J. Math. Control Inf.38 (2021), no. 1, 192-210. · Zbl 1475.93017 |
| [28] | W. K.Williams, V.Vijayakumar, R.Udhayakumar, S. K.Panda, and K.Sooppy Nisar, Existence and controllability of nonlocal mixed Volterra‐Fredholm type fractional delay integro‐differential equations of order [( 1<r<2 \]\), Numer. Methods Partial Differ. Equ. (2020), 1-18. |
| [29] | M.Sivashankar, S.Sabarinathan, K.Sooppy Nisar, C.Ravichandran, and B. V.Senthil Kumar, Some properties and stability of Helmholtz model involved with nonlinear fractional difference equations and its relevance with quadcopter, Chaos, Solitons Fract.168 (2023), 113161. |
| [30] | V.Govindan, S.Noeiaghdam, U.Fernandez‐Gamiz, S. N.Sankeshwari, R.Arulprakasam, and B. Z.Li, Shehu integral transform and Hyers‐Ulam stability of [( {n}^{th} \]\) order linear differential equations, Sci. African18 (2022), e01427. |
| [31] | R.Murali, A.Ponmana Selvan, C.Park, and J. R.Lee, Aboodh transform and the stability of second order linear differential equations, Adv. Differ. Equ.2021 (2021), no. 1, 296. · Zbl 1494.34128 |
| [32] | S.Deepa, S.Bowmiya, A.Ganesh, V.Govindan, C.Park, and J. R.Lee, Mahgoub transform and Hyers‐Ulam stability of [( {n}^{th} \]\) order linear differential equations, Am. Inst. Math. Sci. (AIMS) Math.7 (2022), no. 4, 4992-5014. |
| [33] | S. M.Jung, A.Ponmana Selvan, and R.Murali, Mahgoub transform and Hyers‐Ulam stability of first‐order linear differential equations, J. Math. Inequa.15 (2021), no. 3, 1201-1218. · Zbl 1480.34009 |
| [34] | A.Raj Aruldass, P.Divyakumari, and C.Park, Hyers‐Ulam stability of second‐order differential equations using Mahgoub transform, Adv. Differ. Equ.2021 (2021), 23. · Zbl 1485.34181 |
| [35] | R.Murali, A.Ponmana Selvan, and C.Park, Ulam stability of linear differential equations using Fourier transform, Am. Inst. Math. Sci. (AIMS) Math.5 (2020), no. 2, 766-780. · Zbl 1484.34058 |
| [36] | J. M.Rassias, R.Murali, and A.Ponmana Selvan, Mittag‐Leffler‐Hyers‐Ulam stability of linear differential equations using Fourier transforms, J. Comput. Anal. Appl.29 (2021), no. 1, 68-85. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
