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Stability of Partial Differential Equations by Mahgoub Transform Method

Year 2022, , 1267 - 1273, 31.12.2022
https://doi.org/10.16984/saufenbilder.1142084

Abstract

The stability theory is an important research area in the qualitative analysis of partial differential equations. The Hyers-Ulam stability for a partial differential equation has a very close exact solution to the approximate solution of the differential equation and the error is very small which can be estimated. This study examines Hyers-Ulam and Hyers-Ulam Rassias stability of second order partial differential equations. We present a new method for research of the Hyers-Ulam stability of partial differential equations with the help of the Mahgoub transform. The Mahgoub transform method is practical as a fundamental tool to demonstrate the original result on this study. Finally, we give an example to illustrate main results. Our findings make a contribution to the topic and complete those in the relevant literature.

References

  • [1] S. M. Ulam, “Problems in Modern Mathematics,’’ Science Editions, John Wiley & Sons, Inc., New York, 1964.
  • [2] D. H. Hyers, “On the stability of the linear Functional equation,’’ Proceedings of the National Academy of Sciences, U.S.A., vol. 27, pp. 222-224, 1941.
  • [3] E. Biçer, C. Tunç, “New Theorems for Hyers-Ulam stability of Lienard equation with variable time lags,’’ International Journal of Mathematics and Computer Science, vol. 3, no. 2, pp. 231-242, 2018.
  • [4] E. Biçer, C. Tunç, "On the Hyers-Ulam stability of certain partial differential equations of second order," Nonlinear Dynamics and Systems Theory, vol. 17, no.2, pp. 150-157, 2017.
  • [5] D. H. Hyers, T. M. Rassias, “Approximate homomorphisms,’’ Aequationes Mathematicae, vol. 44, pp. 125-153, 1992.
  • [6] D. H. Hyers, G. Isac, TM. Rassias, “Stability of Functional Equations in Several Variables,’’ Progress in Nonlinear Differential Equations and their Applications, vol. 34, Boston, 1998.
  • [7] S. M. Jung, "Hyers–Ulam stability of linear partial differential equations of first order," Applied Mathematics Letters, vol. 22, no.1, pp. 70-74, 2009.
  • [8] S. M. Jung, J. Brzdek, “Hyers-Ulam stability of the delay equation y′(t)=λy(t-τ),’’ Abstract and Appllied Analysis, vol. 2010, pp. 1-10, 2010.
  • [9] N. Lungu, D. Popa, “Hyers-Ulam stability of a first order partial differential equation,’’ Journal of Mathematical Analysis and Appllications, vol. 385, pp. 86-91, 2012.
  • [10] J. Huang, Y. Li, “Hyers-Ulam Stability of Linear Functional Differential Equation,’’ Journal of Mathematical Analysis and Appllications, pp. 1192-1200, 2015.
  • [11] M. Obłoza, “Connections between Hyers and Lyapunov stability of the ordinary differential equations,’’ Rocznik Naukowo Dydaktyczny Wsp W Krakowie, vol. 14, pp. 141-146, 1997.
  • [12] D. Otrocol, V. Ilea, “Ulam stability for a delay differential equation,’’ Central European. Journal of Mathematics, vol. 11, no. 7, pp. 1296-1303, 2013.
  • [13] TM. Rassias, “On the stability of the linear mapping in Banach spaces,’’ Proceedings of the American Mathematical Society, vol. 72, no. 2, pp. 297-300, 1978.
  • [14] T. M. Rassias, “On the Stability of Functional Equations and a Problem of Ulam,’’ Acta Applicandae. Mathematicae, vol. 62, pp. 23-130, 2000.
  • [15] S. E. Takahasi, T. Miura, S. Miyajima, “On the Hyers-Ulam stability of the Banach space-valued differential equation y′=λy,’’ Bulletin of the Korean Mathematical Society, vol. 39, pp. 309-315, 2002.
  • [16] C. Tunç, E. Biçer, “Hyers-Ulam-Rassias stability for a first order functional differential equation,’’ Journal of Mathematical and Fundamental Sciences, vol. 47, no. 2, pp. 143-153, 2015.
  • [17] S. M. Jung, P. S. Arumugam, R. Murali, “Mahgoub Transform and Hyers Ulam stability of first order linear differential equations,’’ Journal of Mathematical Inequalities, vol.15, no. 3, pp. 1201-1218, 2021.
  • [18] S. Aggarwal, N. Sarma, N. Chauan, “Solution of linear Volterra integro-differential equations of second kind using Mahgoub transform,” International Journal of Latest Techonology in Engineering Management and Appllied Science, vol. 7, no. 5, pp. 173-176, 2018.
Year 2022, , 1267 - 1273, 31.12.2022
https://doi.org/10.16984/saufenbilder.1142084

Abstract

References

  • [1] S. M. Ulam, “Problems in Modern Mathematics,’’ Science Editions, John Wiley & Sons, Inc., New York, 1964.
  • [2] D. H. Hyers, “On the stability of the linear Functional equation,’’ Proceedings of the National Academy of Sciences, U.S.A., vol. 27, pp. 222-224, 1941.
  • [3] E. Biçer, C. Tunç, “New Theorems for Hyers-Ulam stability of Lienard equation with variable time lags,’’ International Journal of Mathematics and Computer Science, vol. 3, no. 2, pp. 231-242, 2018.
  • [4] E. Biçer, C. Tunç, "On the Hyers-Ulam stability of certain partial differential equations of second order," Nonlinear Dynamics and Systems Theory, vol. 17, no.2, pp. 150-157, 2017.
  • [5] D. H. Hyers, T. M. Rassias, “Approximate homomorphisms,’’ Aequationes Mathematicae, vol. 44, pp. 125-153, 1992.
  • [6] D. H. Hyers, G. Isac, TM. Rassias, “Stability of Functional Equations in Several Variables,’’ Progress in Nonlinear Differential Equations and their Applications, vol. 34, Boston, 1998.
  • [7] S. M. Jung, "Hyers–Ulam stability of linear partial differential equations of first order," Applied Mathematics Letters, vol. 22, no.1, pp. 70-74, 2009.
  • [8] S. M. Jung, J. Brzdek, “Hyers-Ulam stability of the delay equation y′(t)=λy(t-τ),’’ Abstract and Appllied Analysis, vol. 2010, pp. 1-10, 2010.
  • [9] N. Lungu, D. Popa, “Hyers-Ulam stability of a first order partial differential equation,’’ Journal of Mathematical Analysis and Appllications, vol. 385, pp. 86-91, 2012.
  • [10] J. Huang, Y. Li, “Hyers-Ulam Stability of Linear Functional Differential Equation,’’ Journal of Mathematical Analysis and Appllications, pp. 1192-1200, 2015.
  • [11] M. Obłoza, “Connections between Hyers and Lyapunov stability of the ordinary differential equations,’’ Rocznik Naukowo Dydaktyczny Wsp W Krakowie, vol. 14, pp. 141-146, 1997.
  • [12] D. Otrocol, V. Ilea, “Ulam stability for a delay differential equation,’’ Central European. Journal of Mathematics, vol. 11, no. 7, pp. 1296-1303, 2013.
  • [13] TM. Rassias, “On the stability of the linear mapping in Banach spaces,’’ Proceedings of the American Mathematical Society, vol. 72, no. 2, pp. 297-300, 1978.
  • [14] T. M. Rassias, “On the Stability of Functional Equations and a Problem of Ulam,’’ Acta Applicandae. Mathematicae, vol. 62, pp. 23-130, 2000.
  • [15] S. E. Takahasi, T. Miura, S. Miyajima, “On the Hyers-Ulam stability of the Banach space-valued differential equation y′=λy,’’ Bulletin of the Korean Mathematical Society, vol. 39, pp. 309-315, 2002.
  • [16] C. Tunç, E. Biçer, “Hyers-Ulam-Rassias stability for a first order functional differential equation,’’ Journal of Mathematical and Fundamental Sciences, vol. 47, no. 2, pp. 143-153, 2015.
  • [17] S. M. Jung, P. S. Arumugam, R. Murali, “Mahgoub Transform and Hyers Ulam stability of first order linear differential equations,’’ Journal of Mathematical Inequalities, vol.15, no. 3, pp. 1201-1218, 2021.
  • [18] S. Aggarwal, N. Sarma, N. Chauan, “Solution of linear Volterra integro-differential equations of second kind using Mahgoub transform,” International Journal of Latest Techonology in Engineering Management and Appllied Science, vol. 7, no. 5, pp. 173-176, 2018.
There are 18 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Articles
Authors

Harun Biçer 0000-0002-9854-0595

Publication Date December 31, 2022
Submission Date July 7, 2022
Acceptance Date October 23, 2022
Published in Issue Year 2022

Cite

APA Biçer, H. (2022). Stability of Partial Differential Equations by Mahgoub Transform Method. Sakarya University Journal of Science, 26(6), 1267-1273. https://doi.org/10.16984/saufenbilder.1142084
AMA Biçer H. Stability of Partial Differential Equations by Mahgoub Transform Method. SAUJS. December 2022;26(6):1267-1273. doi:10.16984/saufenbilder.1142084
Chicago Biçer, Harun. “Stability of Partial Differential Equations by Mahgoub Transform Method”. Sakarya University Journal of Science 26, no. 6 (December 2022): 1267-73. https://doi.org/10.16984/saufenbilder.1142084.
EndNote Biçer H (December 1, 2022) Stability of Partial Differential Equations by Mahgoub Transform Method. Sakarya University Journal of Science 26 6 1267–1273.
IEEE H. Biçer, “Stability of Partial Differential Equations by Mahgoub Transform Method”, SAUJS, vol. 26, no. 6, pp. 1267–1273, 2022, doi: 10.16984/saufenbilder.1142084.
ISNAD Biçer, Harun. “Stability of Partial Differential Equations by Mahgoub Transform Method”. Sakarya University Journal of Science 26/6 (December 2022), 1267-1273. https://doi.org/10.16984/saufenbilder.1142084.
JAMA Biçer H. Stability of Partial Differential Equations by Mahgoub Transform Method. SAUJS. 2022;26:1267–1273.
MLA Biçer, Harun. “Stability of Partial Differential Equations by Mahgoub Transform Method”. Sakarya University Journal of Science, vol. 26, no. 6, 2022, pp. 1267-73, doi:10.16984/saufenbilder.1142084.
Vancouver Biçer H. Stability of Partial Differential Equations by Mahgoub Transform Method. SAUJS. 2022;26(6):1267-73.

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