The Scientific World Journal

Volume 2013 (2013), Article ID 978754, 3 pages

http://dx.doi.org/10.1155/2013/978754

Research Article

On the Stability of One-Dimensional Wave Equation

Soon-Mo Jung

Mathematics Section, College of Science and Technology, Hongik University, Sejong 339-701, Republic of Korea

Received 5 August 2013; Accepted 16 September 2013

Academic Editors: K. Ammari, I. Canak, and M. M. Cavalcanti

Copyright © 2013 Soon-Mo Jung. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We prove the generalized Hyers-Ulam stability of the one-dimensional wave equation, , in a class of twice continuously differentiable functions.

1. Introduction

In 1940, Ulam [1] gave a wide ranging talk before the mathematics club of the University of Wisconsin in which he discussed a number of important unsolved problems. Among those was the question concerning the stability of group homomorphisms:

Let be a group and let be a metric group with the metric . Given , does there exist a such that if a function satisfies the inequality , for all , then there exists a homomorphism with , for all ?

The case of approximately additive functions was solved by Hyers [2] under the assumption that and are Banach spaces. Indeed, he proved that each solution of the inequality , for all and , can be approximated by an exact solution, say an additive function. In this case, the Cauchy additive functional equation, , is said to have the Hyers-Ulam stability.

Rassias [3] attempted to weaken the condition for the bound of the norm of the Cauchy difference as follows:

and proved Hyers’ theorem. That is, Rassias proved the generalized Hyers-Ulam stability (or Hyers-Ulam-Rassias stability) of the Cauchy additive functional equation. Since then, the stability of several functional equations has been extensively investigated [4–9].

The terminologies, the generalized Hyers-Ulam stability, and the Hyers-Ulam stability can also be applied to the case of other functional equations, differential equations, and various integral equations.

Given a real number , the partial differential equation

is called the (one-dimensional) wave equation, where and denote the second time derivative and the second space derivative of , respectively.

Let be a function. If, for each twice continuously differentiable function satisfying

there exist a solution of the (one-dimensional) wave equation (2) and a function such that

where is independent of and , then we say that the wave equation (2) has the generalized Hyers-Ulam stability (or the Hyers-Ulam-Rassias stability).

In this paper, using an idea from [10], we prove the generalized Hyers-Ulam stability of the (one-dimensional) wave equation (2).

2. Generalized Hyers-Ulam Stability

In the following theorem, using the d’Alembert method (method of characteristic coordinates), we prove the generalized Hyers-Ulam stability of the (one-dimensional) wave equation (2).

Theorem 1. Let a function be given such that the double integral

exists for all . If a twice continuously differentiable function satisfies the inequality

for all , then there exists a solution of the wave equation (2) which satisfies

for all .

Proof. Let us define a function by

If we set and , then we have and

for all . Hence, we have

for any . Thus, it follows from inequality (6) that

for any .

Therefore, we get

or equivalently

for all .

On account of (8), we get

Hence, it follows from (13) and the last equalities that

for all .

If we set and in the last inequality, then we obtain

for all , where we set

By some tedious calculations, we get

for all . Hence, we know that

for any ; that is, is a solution of the wave equation (2).

Corollary 2. Given a constant , let a function be given as

If a twice continuously differentiable function satisfies inequality (6), for all , then there exists a solution of the wave equation (2) which satisfies

for all .

Proof. Since

for all , in view of Theorem 1, we conclude that the statement of this corollary is true.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

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).

References

S. M. Ulam, A Collection of Mathematical Problems, Interscience, New York, NY, USA, 1960.

D. H. Hyers, “On the stability of the linear functional equation,” Proceedings of the National Academy of Sciences of USA, vol. 27, pp. 222–224, 1941.

T. M. Rassias, “On the stability of the linear mapping in Banach spaces,” Proceedings of the American Mathematical Society, vol. 72, pp. 297–300, 1978.

G. L. Forti, “Hyers-Ulam stability of functional equations in several variables,” Aequationes Mathematicae, vol. 50, no. 1-2, pp. 143–190, 1995.

P. Găvrută, “A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings,” Journal of Mathematical Analysis and Applications, vol. 184, no. 3, pp. 431–436, 1994.

D. H. Hyers, G. Isac, and T. M. Rassias, Stability of Functional Equations of Several Variables, Birkhauser, Boston, Mass, USA, 1998.

D. H. Hyers and T. M. Rassias, “Approximate homomorphisms,” Aequationes Mathematicae, vol. 44, no. 2-3, pp. 125–153, 1992.

S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, vol. 48 of Springer Optimization and Its Applications, Springer, New York, NY, USA, 2011.

T. M. Rassias, “On the stability of functional equations and a problem of Ulam,” Acta Applicandae Mathematicae, vol. 62, no. 1, pp. 23–130, 2000.

B. Hegyi and S.-M. Jung, “On the stability of Laplace's equation,” Applied Mathematics Letters, vol. 26, pp. 549–552, 2013.