On the Stability of One-Dimensional Wave Equation

We prove the generalized Hyers-Ulam stability of the one-dimensional wave equation, u tt = c 2 u xx, in a class of twice continuously differentiable functions.


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 1 be a group and let 2 be a metric group with the metric (⋅, ⋅). Given > 0, does there exist a > 0 such that if a function ℎ : 1 → 2 satisfies the inequality (ℎ( ), ℎ( )ℎ( )) < , for all , ∈ 1 , then there exists a homomorphism : 1 → 2 with (ℎ( ), ( )) < , for all ∈ 1 ?
The case of approximately additive functions was solved by Hyers [2] under the assumption that 1 and 2 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][5][6][7][8][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 > 0, 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 there exist a solution 0 : R × R → C of the (one-dimensional) wave equation (2) and a function Φ : where Φ( , ) is independent of ( , ) and 0 ( , ), 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).

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).
exists for all , ∈ R. If a twice continuously differentiable function : R × R → C satisfies the inequality for all , ∈ R, then there exists a solution 0 : R × R → C of the wave equation (2) which satisfies for all , ∈ R.
Proof. Let us define a function V : for all , ∈ R. Hence, we have for any , ∈ R. Thus, it follows from inequality (6) that for any , ∈ R. Therefore, we get or equivalently for all , ∈ R.
On account of (8), we get Hence, it follows from (13) and the last equalities that for all , ∈ R. If we set = + and = − in the last inequality, then we obtain for all , ∈ R, where we set 0 ( , ) := ( 2 + 2 , 2 + 2 ) for all , ∈ R. Hence, we know that for any , ∈ R; that is, 0 ( , ) is a solution of the wave equation (2).

Corollary 2.
Given a constant > 0, let a function : R × R → [0, ∞) be given as If a twice continuously differentiable function : R × R → C satisfies inequality (6), for all , ∈ R, then there exists a solution 0 : R × R → C of the wave equation ( for all , ∈ R.
Proof. Since for all , ∈ R, in view of Theorem 1, we conclude that the statement of this corollary is true.