Research Article
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Stability of Partial Differential Equations by Mahgoub Transform Method

Year 2022, Volume 26, Issue 6, 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, Volume 26, Issue 6, 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.

Details

Primary Language English
Subjects Mathematics
Journal Section Research Articles
Authors

Harun BİÇER> (Primary Author)
BINGOL UNIVERSITY
0000-0002-9854-0595
Türkiye

Publication Date December 31, 2022
Submission Date July 7, 2022
Acceptance Date October 23, 2022
Published in Issue Year 2022, Volume 26, Issue 6

Cite

Bibtex @research article { saufenbilder1142084, journal = {Sakarya University Journal of Science}, eissn = {2147-835X}, address = {}, publisher = {Sakarya University}, year = {2022}, volume = {26}, number = {6}, pages = {1267 - 1273}, doi = {10.16984/saufenbilder.1142084}, title = {Stability of Partial Differential Equations by Mahgoub Transform Method}, key = {cite}, author = {Biçer, Harun} }
APA Biçer, H. (2022). Stability of Partial Differential Equations by Mahgoub Transform Method . Sakarya University Journal of Science , 26 (6) , 1267-1273 . DOI: 10.16984/saufenbilder.1142084
MLA Biçer, H. "Stability of Partial Differential Equations by Mahgoub Transform Method" . Sakarya University Journal of Science 26 (2022 ): 1267-1273 <https://dergipark.org.tr/en/pub/saufenbilder/issue/74051/1142084>
Chicago Biçer, H. "Stability of Partial Differential Equations by Mahgoub Transform Method". Sakarya University Journal of Science 26 (2022 ): 1267-1273
RIS TY - JOUR T1 - Stability of Partial Differential Equations by Mahgoub Transform Method AU - HarunBiçer Y1 - 2022 PY - 2022 N1 - doi: 10.16984/saufenbilder.1142084 DO - 10.16984/saufenbilder.1142084 T2 - Sakarya University Journal of Science JF - Journal JO - JOR SP - 1267 EP - 1273 VL - 26 IS - 6 SN - -2147-835X M3 - doi: 10.16984/saufenbilder.1142084 UR - https://doi.org/10.16984/saufenbilder.1142084 Y2 - 2022 ER -
EndNote %0 Sakarya University Journal of Science Stability of Partial Differential Equations by Mahgoub Transform Method %A Harun Biçer %T Stability of Partial Differential Equations by Mahgoub Transform Method %D 2022 %J Sakarya University Journal of Science %P -2147-835X %V 26 %N 6 %R doi: 10.16984/saufenbilder.1142084 %U 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
AMA Biçer H. Stability of Partial Differential Equations by Mahgoub Transform Method. SAUJS. 2022; 26(6): 1267-1273.
Vancouver Biçer H. Stability of Partial Differential Equations by Mahgoub Transform Method. Sakarya University Journal of Science. 2022; 26(6): 1267-1273.
IEEE H. Biçer , "Stability of Partial Differential Equations by Mahgoub Transform Method", Sakarya University Journal of Science, vol. 26, no. 6, pp. 1267-1273, Dec. 2022, doi:10.16984/saufenbilder.1142084

Sakarya University Journal of Science (SAUJS)