The necessary and sufficient conditions of Hyers-Ulam stability for a class of parabolic equation

Volume 10, Issue 11, pp 5781--5788 http://dx.doi.org/10.22436/jnsa.010.11.16

Publication Date: November 15, 2017 Submission Date: May 03, 2017

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 1833 Downloads
• 2982 Views

Authors

Xiangkui Zhao - School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China. Xiaojun Wu - School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China. Zhihong Zhao - School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China.

Abstract

The aim of this paper is to consider the Hyers-Ulam stability of a class of parabolic equation \[\left\{\begin{array}{ll} \frac{\partial u}{\partial t}- a^{2}\Delta u+b\cdot\nabla u+cu=0,~~~(x,t)\in\mathbb{R}^{n}\times(0,+\infty),\\ u(x,0)=\varphi(x),~~~x\in\mathbb{R}^{n}.\end{array}\right.\] We conclude that (i) it is Hyers-Ulam stable on any finite interval; (ii) if $c\neq0 $, it is Hyers-Ulam stable on the semi-infinite interval; (iii) if $c=0$, it is not Hyers-Ulam stable on the semi-infinite interval by using Fourier transformation. Furthermore, our results can be applied to the mean square Hyers-Ulam stability of parabolic equations driven by an \(n\)-dimensional Brownian motion.

Share and Cite

Keywords

• Hyers-Ulam stability
• parabolic equation
• stochastic differential equations

MSC

•  35A24
•  35A30
•  35A35

References

• [1] M. R. Abdollahpour, R. Aghayari, M. T. Rassias, Hyers-Ulam stability of associated Laguerre differential equations in a subclass of analytic functions , J. Math. Anal. Appl., 437 (2016), 605–612.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [2] S. András, J. J. Kolumbán, On the Ulam-Hyers stability of first order differential systems with nonlocal initial conditions, Nonlinear Anal., 82 (2013), 1–11.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [3] S. András, A. R. Mészáros, Ulam-Hyers stability of dynamic equations on time scales via Picard operators, Appl. Math. Comput., 219 (2013), 4853–4864.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [4] S. András, A. R. Mészáros, Ulam-Hyers stability of elliptic partial differential equations in Sobolev spaces, Appl. Math. Comput., 229 (2014), 131–138.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [5] L. Debnath, Nonlinear partial differential equations for scientists and engineers, Second edition, Birkhäuser Boston, Inc., Boston, MA (2005)
  • View Article
  • MathSciNet


• [6] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U. S. A., 27 (1941), 222–224.
  • View Article
  • Google Scholar


• [7] F. John, Partial differential equations, Second edition, Applied mathematical sciences, Springer-Verlag, New York- Heidelberg (1975)


• [8] Y.-J. Li, Y. Shen , Hyers-Ulam stability of linear differential equations of second order, Appl. Math. Lett., 23 (2010), 306–309.
  • View Article
  • Google Scholar


• [9] N. Lungu, D. Popa, Hyers-Ulam stability of a first order partial differential equation, J. Math. Anal. Appl., 385 (2012), 86–91.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [10] M. Obłoza, Hyers stability of the linear differential equation, Rocznik Nauk.-Dydakt. Prace Mat., (1993), 259–270.
  • Google Scholar


• [11] M. Obłoza , Connections between Hyers and Lyapunov stability of the ordinary differential equations, Rocznik Nauk.- Dydakt. Prace Mat., (1997), 141–146.
  • MathSciNet
  • Google Scholar
  • MATH


• [12] B. Øksendal, Stochastic differential equations, An introduction with applications, Universitext, Springer-Verlag, Berlin (1985)


• [13] M. Onitsuka, T. Shoji , Hyers-Ulam stability of first-order homogeneous linear differential equations with a real-valued coefficient, Appl. Math. Lett., 63 (2017), 102–108.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [14] D. Popa, I. Raşa, On the Hyers-Ulam stability of the linear differential equation, J. Math. Anal. Appl., 381 (2011), 530–537.
  • View Article
  • MathSciNet
  • Google Scholar


• [15] H. Rezaei, S.-M. Jung, T. M. Rassias, Laplace transform and Hyers-Ulam stability of linear differential equations , J. Math. Anal. Appl., 403 (2013), 244–251.
  • View Article
  • MathSciNet
  • Google Scholar


• [16] S. M. Ulam, Problems in modern mathematics , Science Editions John Wiley & Sons, Inc., New York (1964)
  • Google Scholar


• [17] J.-R. Wang, X.-Z. Li , Ulam-Hyers stability of fractional Langevin equations, Appl. Math. Comput., 258 (2015), 72–83.
  • View Article
  • Google Scholar


• [18] J.-R. Wang, L.-L. Lv, Y. Zhou, New concepts and results in stability of fractional differential equations, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 2530–2538.
  • View Article
  • MathSciNet
  • Google Scholar


• [19] G.-W. Wang, M.-R. Zhou, L. Sun, Hyers-Ulam stability of linear differential equations of first order, Appl. Math. Lett., 21 (2008), 1024–1028.
  • View Article
  • Google Scholar