- Open Access
On the stability of the heat equation with an initial condition
© Jung; licensee Springer. 2013
- Received: 4 August 2013
- Accepted: 1 October 2013
- Published: 7 November 2013
In this paper, we prove the generalized Hyers-Ulam stability of the heat equation with an initial condition
in a class of twice continuously differentiable functions under certain conditions.
- Differential Equation
- Partial Differential Equation
- Brownian Motion
- Recent Result
- Normed Space
such that for any , where is an expression of ε only, then we say that the above differential equation has the Hyers-Ulam stability.
If the above statement is also true when we replace ε and by and , where are functions not depending on f and explicitly, then we say that the corresponding differential equation has the generalized Hyers-Ulam stability. (This type of stability is sometimes called the Hyers-Ulam-Rassias stability.)
We may apply these terminologies to other differential equations and partial differential equations. For more detailed definitions of the Hyers-Ulam stability and the generalized Hyers-Ulam stability, refer to [1–8].
Obłoza seems to be the first author who has investigated the Hyers-Ulam stability of linear differential equations (see [9, 10]). Here, we introduce the result of Alsina and Ger (see ): If a differentiable function is a solution of the differential inequality , where I is an open subinterval of ℝ, then there exists a solution of the differential equation such that for any . This result was generalized by Miura et al. (see [11, 12]).
where and are constants with . It seems that the first paper dealing with the Hyers-Ulam stability of partial differential equations was written by Prastaro and Rassias . For a recent result on this subject, refer to .
in the class of radially symmetric functions, where △ denotes the Laplace operator. The heat equation plays an important role in a number of fields of science. It is strongly related to the Brownian motion in probability theory. The heat equation is also connected with chemical diffusion, and it is sometimes called the diffusion equation.
for any .
for all .
for all and with , which proves the validity of inequality (2.4), and in view of (2.6), it is not difficult to prove that is a solution of the heat equation, i.e., .
for all and with , which shows the validity of inequality (2.5). □
The linearity of solutions of heat equation (1.1) implies that and are also solutions of the heat equation.
- (ii)As in the proof of [, Theorem 1 in Section 2.3], we can show that
for each .
- (iii)If a function satisfiesfor all and for some , then there exists a positive number A such that
i.e., satisfies condition (2.1).
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2013R1A1A2005557).
- Alsina C, Ger R: On some inequalities and stability results related to the exponential function. J. Inequal. Appl. 1998, 2: 373–380.MathSciNetGoogle Scholar
- Brillouet-Belluot N, Brzdek J, Cieplinski K: On some recent developments in Ulam’s type stability. Abstr. Appl. Anal. 2012., 2012: Article ID 716936 10.1155/2012/716936Google Scholar
- Czerwik S: Functional Equations and Inequalities in Several Variables. World Scientific, River Edge; 2002.View ArticleGoogle Scholar
- Hyers DH: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 1941, 27: 222–224. 10.1073/pnas.27.4.222MathSciNetView ArticleGoogle Scholar
- Hyers DH, Isac G, Rassias TM: Stability of Functional Equations in Several Variables. Birkhäuser, Boston; 1998.View ArticleGoogle Scholar
- Jung S-M Springer Optimization and Its Applications 48. In Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis. Springer, New York; 2011.View ArticleGoogle Scholar
- Rassias TM: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 1978, 72: 297–300. 10.1090/S0002-9939-1978-0507327-1View ArticleGoogle Scholar
- Ulam SM: Problems in Modern Mathematics. Wiley, New York; 1964.Google Scholar
- Obłoza M: Hyers stability of the linear differential equation. Rocznik Nauk.-Dydakt. Prace Mat. 1993, 13: 259–270.Google Scholar
- Obłoza M: Connections between Hyers and Lyapunov stability of the ordinary differential equations. Rocznik Nauk.-Dydakt. Prace Mat. 1997, 14: 141–146.Google Scholar
- Miura T, Jung S-M, Takahasi S-E: Hyers-Ulam-Rassias stability of the Banach space valued linear differential equations . J. Korean Math. Soc. 2004, 41: 995–1005. 10.4134/JKMS.2004.41.6.995MathSciNetView ArticleGoogle Scholar
- Takahasi S-E, Miura T, Miyajima S: On the Hyers-Ulam stability of the Banach space-valued differential equation . Bull. Korean Math. Soc. 2002, 39: 309–315. 10.4134/BKMS.2002.39.2.309MathSciNetView ArticleGoogle Scholar
- Jung S-M, Lee K-S: Hyers-Ulam stability of first order linear partial differential equations with constant coefficients. Math. Inequal. Appl. 2007, 10: 261–266.MathSciNetGoogle Scholar
- Prastaro A, Rassias TM: Ulam stability in geometry of PDE’s. Nonlinear Funct. Anal. Appl. 2003, 8: 259–278.MathSciNetGoogle Scholar
- Lungu N, Popa D: Hyers-Ulam stability of a first order partial differential equation. J. Math. Anal. Appl. 2012, 385: 86–91. 10.1016/j.jmaa.2011.06.025MathSciNetView ArticleGoogle Scholar
- Hegyi B, Jung S-M: On the stability of Laplace’s equation. Appl. Math. Lett. 2013, 26: 549–552. 10.1016/j.aml.2012.12.014MathSciNetView ArticleGoogle Scholar
- Hegyi, B, Jung, S-M: On the stability of heat equation. Abstr. Appl. Anal. (in press)Google Scholar
- Evans LC Graduate Studies in Mathematics 19. In Partial Differential Equations. American Mathematical Society, Providence; 1998.Google Scholar
- Jung S-M: Hyers-Ulam stability of linear differential equations of first order, II. Appl. Math. Lett. 2006, 19: 854–858. 10.1016/j.aml.2005.11.004MathSciNetView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.