Ulam type stability of second-order linear differential equations with constant coefficients having damping term by using the Aboodh transform
DOI:
https://doi.org/10.22199/issn.0717-6279-5260Keywords:
Hyers-Ulam stability, Mittag-Leffler-Hyers-Ulam stability, linear differential equation, Aboodh transformAbstract
The main aim of this paper is to investigate various types of Ulam stability and Mittag-Leffler stability of linear differential equations of second order with constant coefficients having damping term using the Aboodh transform method. We also obtain the Hyers-Ulam stability constants of these differential equations using the Aboodh transform and some examples to illustrate our main results are given.
References
S. M. Ulam, Problem in Modern Mathematics. New York: Willey, 1960.
D. H. Hyers, ”On the Stability of a Linear functional equation”, Proceedings of the National Academy of Sciences, vol. 27, pp. 222-224, 1941. https://doi.org/10.1073/pnas.27.4.222
T. Aoki, ”On the stability of the linear transformation in Banach spaces”, Journal of the Mathematical Society of Japan, vol. 2, pp. 64-66, 1950. https://doi.org/10.2969/jmsj/00210064
D. G. Bourgin, ”Classes of transformations and bordering transformations”, Bulletin of the American Mathematical Society, vol. 57, pp. 223-237, 1951. https://doi.org/10.1090/s0002-9904-1951-09511-7
J. M. Rassias, ”On approximately of approximately linear mappings by linear mappings”, Journal of Functional Analysis, vol. 46, pp. 126-130, 1982. https://doi.org/10.1016/0022-1236(82)90048-9
T. M. Rassias, ”On the stability of the linear mappings in Banach Spaces”, Proceedings of the American Mathematical Society, vol. 72, pp. 297-300, 1978. https://doi.org/10.2307/2042795
H. Vaezi, ”Hyers-Ulam stability of weighted composition operators on disc algebra”, International Journal of Mathematics and Computation, vol. 10, pp. 150-154, 2011.
T. Miura, ”On the Hyers-Ulam stability of a differentiable map”, Scientiae Mathematicae Japonicae, vol. 55, pp. 17-24, 2002.
I. A. Rus, ”Ulam Stabilities of ordinary differential equations in a Banach space”, Carpathian Journal of Mathematics, vol. 26, pp. 103-107, 2010.
Q. H. Alqifiary and S. Jung, ”Laplace transform and generalized Hyers-Ulam stability of differential equations”, Electronic Journal of Differential Equations, 2014, no. 80, 2014. [On line]. Available: https://bit.ly/3Tatk5u
A. Buakird and S. Saejung, ”Ulam stability with respect to a directed graph for some fixed point equations”, Carpathian Journal of Mathematics, vol. 35, pp. 23-30, 2019.
R. Fukutaka and M. Onitsuka, ”Best constant in Hyers-Ulam stability of first-order homogeneous linear differential equations with a periodic coefficient”, Journal of Mathematical Analysis and Applications, vol. 473, pp. 1432-1446, 2019. https://doi.org/10.1016/j.jmaa.2019.01.030
J. Huang, S. Jung and Y. Li, ”On Hyers-Ulam stability of nonlinear differential equations”, Bulletin of the Korean Mathematical Society, vol. 52, pp. 685-697, 2015.
T. Li, A. Zada and S. Faisal, ”Hyers-Ulam stability of nth order linear differential equations”, Journal of Nonlinear Science and Applications, vol. 9, no. 5, pp. 2070-2075, 2016. https://doi.org/10.22436/jnsa.009.05.12
Y. Li and Y. Shen, ”Hyers-Ulam stability of linear differential equations of second order”, Applied Mathematics Letters, vol. 23, pp. 306-309, 2010. https://doi.org/10.1016/j.aml.2009.09.020
R. Murali, A. Bodaghi and A. P. Selvan, ”Stability for the third order linear ordinary differential equation”, International Journal of Computer Mathematics, vol. 30, pp. 87-92, 2019.
R. Murali and A. P. Selvan, ”On the generalized Hyers-Ulam stability of linear ordinary differential equations of higher order”, International Journal of Pure and Applied Mathematics, vol. 117, no. 12, pp. 317-326, 2017. [On line]. Available: https://bit.ly/3zWuAT9
R. Murali and A. P. Selvan, ”Hyers-Ulam-Rassias stability for the linear ordinary differential equation of third order”, Kragujevac Journal of Mathematics, vol. 42, pp. 579-590, 2018. https://doi.org/10.5937/kgjmath1804579m
R. Murali and A. P. Selvan, ”Ulam stability of third order linear differential equations”, International Journal of Pure and Applied Mathematics , vol. 120, no. 9, pp. 217-225, 2018. [On line]. Available: https://bit.ly/3t9LqKi
R. Murali and A. P. Selvan, ”Hyers-Ulam stability of nth order linear differential equation”, Proyecciones (Antofagasta), vol. 38, no. 3, pp. 553-566, 2019. https://doi.org/10.22199/issn.0717-6279-2019-03-0035
R. Murali and A. P. Selvan, ”Hyers-Ulam stability of a free and forced vibrations”, Kragujevac Journal of Mathematics, vol. 44, no. 2, pp. 299-312, 2020. https://doi.org/10.46793/kgjmat2002.299m
M. Onitsuka, ”Hyers-Ulam stability of first order linear differential equations of Caratheodory type and its application”, Applied Mathematics Letters, vol. 90, pp. 61-68, 2019. https://doi.org/10.1016/j.aml.2018.10.013
M. Onitsuka and T. Shoji, ”Hyers-Ulam stability of first order homogeneous linear differential equations with a real valued coefficients”, Applied Mathematics Letters, vol. 63, pp. 102-108, 2017. https://doi.org/10.1016/j.aml.2016.07.020
R. Murali, A. P. Selvan and C. Park, ”Ulam stability of linear differential equations using Fourier transform”, AIMS Mathematics, vol. 5, no. 2, pp. 766-780, 2019. https://doi.org/10.3934/math.2020052
R. Murali and A. P. Selvan, ”Fourier transforms and Ulam stabilities of linear differential equations”, in Frontiers in Functional Equations and Analytic Inequalities, G.A. Ansatassiou and J.M. Rassias, Eds., Springer, 2019, pp. 195-217. https://doi.org/10.1007/978-3-030-28950-8_12
J. M. Rassias, R. Murali and A. P. Selvan, ”Mittag-Leffler-Hyers-Ulam stability of linear differential equations using Fourier transforms”, Journal of Computational Analysis and Applications, vol. 29, pp. 68-85, 2021.
A. A. Alshikh and M. M. Abdelrahim Mahgob, ”A comparative study between Laplace transform and two new integrals ELzaki” transform and Aboodh transform”, Pure and Applied Mathematics Journal, vol. 5, no. 5, pp. 145-150, 2016. https://doi.org/10.11648/j.pamj.20160505.11
K. S. Aboodh, ”Solving porous medium equation using Aboodh transform homotopy perturbation method”, American Journal of Applied Mathematics, vol. 4, no. 6, pp. 271-276, 2016. https://doi.org/10.11648/j.ajam.20160406.12
S. Aggarwal, N. Sharma and R. Chauhan, ”Solution of linear Volterra integro-differential equations of second kind using Mahgoub transform”, International Journal of Latest Technology in Engineering, Management & Applied Science, vol. 7, no. 5, pp. 173-176, 2018.
B. O. Osu and V. U. Sampson, ”Application of Aboodh transform to the solution of stochastic differential equation”, Journal of Advanced Research in Applied Mathematics and Statistics, vol. 3, no. 4, pp. 12-18, 2018. https://doi.org/10.24321/2455.7021.201802
C. Alsina and R. Ger, ”On some inequalities and stability results related to the exponential function”, Journal of Inequalities and Applications, vol. 2, pp. 373-380, 1998. https://doi.org/10.1155/s102558349800023x
D. S. Cimpean and D. Popa, ”On the stability of the linear differential equation of higher order with constant coefficients”, Applied Mathematics and Computation, vol. 217, pp. 4141-4146, 2010. https://doi.org/10.1016/j.amc.2010.09.062
B. Davies, Integral Transforms and Their Applications. New York: Springer, 2001.
S. M. Jung, ”Hyers-Ulam stability of linear differential equations of first order”, Applied Mathematics Letters, vol. 17, pp. 1135-1140, 2004. https://doi.org/10.1016/j.aml.2003.11.004
S. M. Jung, ”Hyers-Ulam stability of linear differential equations of first order, III”, Journal of Mathematical Analysis and Applications, vol. 311, pp. 139-146, 2005. https://doi.org/10.1016/j.jmaa.2005.02.025
S. M. Jung, ”Hyers-Ulam stability of linear differential equations of first order, II”, Applied Mathematics Letters, vol. 19, pp. 854-858, 2006. https://doi.org/10.1016/j.aml.2005.11.004
Y. Li and Y. Shen, ”Hyers-Ulam stability of linear differential equations of second order”, Applied Mathematics Letters, vol. 23, pp. 306-309, 2010.
T. Miura, S. Miyajima and S. E. Takahasi, ”A characterization of Hyers-Ulam stability of first order linear differential operators”, Journal of Mathematical Analysis and Applications, vol. 286, pp. 136-146, 2003. https://doi.org/10.1016/s0022-247x(03)00458-x
T. Miura, S. Miyajima and S. E. Takahasi, ”Hyers-Ulam stability of linear differential operator with constant coefficients”, Mathematische Nachrichten, vol. 258, pp. 90-96, 2003. https://doi.org/10.1002/mana.200310088
M. Obloza, “Hyers stability of the linear differential equation”, Rocznik Naukowo-Dydaktyczny. Prace Matematyczne, vol. 13, pp. 259–270, 1993.
M. Obloza, ”Connections between Hyers and Lyapunov stability of the ordinary differential equations”, Rocznik Naukowo-Dydaktyczny. Prace Matematyczne, vol. 14, pp. 141-146, 1997. [On line]. Available: https://bit.ly/3FWgq8i
D. Popa and I. Raşa, ”On the Hyers-Ulam stability of the linear differential equation”, Journal of Mathematical Analysis and Applications, vol. 381, pp. 530-537, 2011. https://doi.org/10.1016/j.jmaa.2011.02.051
D. Popa and I. Raşa, ”Hyers-Ulam stability of the linear differential operator with non-constant coefficients”, Applied Mathematics and Computation, vol. 219, pp. 1562-1568, 2012. https://doi.org/10.1016/j.amc.2012.07.056
I. A. Rus, ”Remarks on Ulam stability of the operatorial equations”, Fixed Point Theory, vol. 10, pp. 305-320, 2009. [On line]. Available: https://bit.ly/3zYINPa
I. A. Rus, ”Ulam stability of ordinary differential equations”, Studia Universitatis Babeș-Bolyai Mathematica, vol. 54, pp. 125-134, 2009. [On line]. Available: https://bit.ly/3hs7qxB
S. E. Takahasi, T. Miura and S. Miyajima, ”On the Hyers-Ulam stability of the Banach space-valued differential equation y0’= λy”, Bulletin of the Korean Mathematical Society, vol. 39, pp. 309-315, 2002. https://doi.org/10.4134/bkms.2002.39.2.309
S. E. Takahasi, H. Takagi, T. Miura and S. Miyajima, ”The Hyers-Ulam stability constants of first order linear differential operators”, Journal of Mathematical Analysis and Applications, vol. 296, pp. 403-409, 2004. https://doi.org/10.1016/j.jmaa.2003.12.044
G. Wang, M. Zhou and L. Sun, ”Hyers-Ulam stability of linear differential equations of first order”, Applied Mathematics Letters, vol. 21, pp. 1024-1028, 2008. https://doi.org/10.1016/j.aml.2007.10.020
H. Rezaei, S. M. Jung and T. M. Rassias, ”Laplace transform and Hyers-Ulam stability of linear differential equations”, Journal of Mathematical Analysis and Applications, vol. 403, pp. 244-251, 2013. https://doi.org/10.1016/j.jmaa.2013.02.034
J. Xue, ”Hyres-Ulam stability of linear differential equations of second order with constant coefficients”, Italian Journal of Pure and Applied Mathematics, vol. 32, pp. 419-424, 2014. [On line]. Available: https://bit.ly/3DSUIPR
A. Najatia, M. R. Abdollahpoura and C. Park, ”On the stability of linear differential equations of second order”, International Journal of Nonlinear Analysis and Applications, vol. 8, no. 2, pp. 65-70, 2017.
A. R. Aruldass, D. Pachaiyappan and C. Park, ”Hyers-Ulam stability of second-order differential equations using Mahgoub transform”, Advances in Difference Equations, 2021. https://doi.org/10.1186/s13662-020-03172-0
V. Kalvandi, N. Eghbali and J. M. Rassias, ”Mittag-Leffler-Hyers-Ulam Stability of Fractional Differential Equations of Second Order”, Journal of Mathematical Extension, vol. 13, no. 1, pp. 29-43, 2019.
A. Mohanapriya1, C. Park, A. Ganesh and V. Govindan, ”Mittag-Leffler-Hyers-Ulam stability of differential equation using Fourier transform”, Advances in Difference Equations, vol. 2020, no. 1, 2020. https://doi.org/10.1186/s13662-020-02854-z
R. Murali, A. P. Selvan, C. Park and J. R. Lee, ”Aboodh transform and the stability of second order linear differential equations”, Advanced in Difference Equations, 2021. https://doi.org/10.1186/s13662-021-03451-4
Published
How to Cite
Issue
Section
Copyright (c) 2022 Adil Mısır, Süleyman Öğrekçi, Yasemin Başcı
This work is licensed under a Creative Commons Attribution 4.0 International License.
-
Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- No additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.
