Research Article | Open Access

# On the Stability of One-Dimensional Wave Equation

**Academic Editor:**K. Ammari

#### 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

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#### Copyright

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.