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.

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

The case of approximately additive functions was solved by Hyers [2] under the assumption that G 1 and G 2 are Banach spaces. Indeed, he proved that each solution of the inequality ||f(x + y) − f(x) − f(y)|| ≤ ε, for all x and y, can be approximated by an exact solution, say an additive function. In this case, the Cauchy additive functional equation, f(x + y) = f(x) + f(y), 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 c > 0, the partial differential equation is called the (one-dimensional) wave equation, where u (x, t) and u (x, t) denote the second time derivative and the second space derivative of u(x, t), respectively.

Let φ : ℝ × ℝ → [0, ∞) be a function. If, for each twice continuously differentiable function u : ℝ × ℝ → ℂ satisfying there exist a solution u 0 : ℝ × ℝ → ℂ of the (one-dimensional) wave equation (2) and a function Φ : ℝ × ℝ → [0, ∞) such that where Φ(x, t) is independent of u(x, t) and u 0(x, t), 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 φ : ℝ × ℝ → [0, ∞) be given such that the double integral exists for all a, b ∈ ℝ. If a twice continuously differentiable function u : ℝ × ℝ → ℂ satisfies the inequality for all x, t ∈ ℝ, then there exists a solution u 0 : ℝ × ℝ → ℂ of the wave equation (2) which satisfies for all x, t ∈ ℝ.

Proof

Let us define a function v : ℝ × ℝ → ℂ by If we set w = x + ct and z = x − ct, then we have u(x, t) = v(w, z) and for all x, t ∈ ℝ. Hence, we have for any x, t ∈ ℝ. Thus, it follows from inequality (6) that for any w, z ∈ ℝ.

Therefore, we get or equivalently for all w, z ∈ ℝ.

On account of (8), we get Hence, it follows from (13) and the last equalities that for all w, z ∈ ℝ.

If we set w = x + ct and z = x − ct in the last inequality, then we obtain for all x, t ∈ ℝ, where we set

By some tedious calculations, we get for all x, t ∈ ℝ. Hence, we know that for any x, t ∈ ℝ; that is, u 0(x, t) is a solution of the wave equation (2).

Corollary 2

Given a constant α > 0, let a function φ : ℝ × ℝ → [0, ∞) be given as If a twice continuously differentiable function u : ℝ × ℝ → ℂ satisfies inequality (6), for all x, t ∈ ℝ, then there exists a solution u 0 : ℝ × ℝ → ℂ of the wave equation (2) which satisfies for all x, t ∈ ℝ.

Since for all a, b ∈ ℝ, in view of Theorem 1, we conclude that the statement of this corollary is true.