On the Stability of One-Dimensional Wave Equation

Abstract

We prove the generalized Hyers-Ulam stability of the one-dimensional wave equation, 𝑢_𝑡𝑡=𝑐²𝑢_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 𝐺₁ be a group and let 𝐺₂ be a metric group with the metric 𝑑(⋅,⋅). Given 𝜀>0, does there exist a 𝛿>0 such that if a function ℎ∶𝐺₁→𝐺₂ satisfies the inequality 𝑑(ℎ(𝑥𝑦),ℎ(𝑥)ℎ(𝑦))<𝛿, for all 𝑥,𝑦∈𝐺₁, then there exists a homomorphism 𝐻∶𝐺₁→𝐺₂ with 𝑑(ℎ(𝑥),𝐻(𝑥))<𝜀, for all 𝑥∈𝐺₁?

The case of approximately additive functions was solved by Hyers [2] under the assumption that 𝐺₁ and 𝐺₂ 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: 𝑓𝑥+𝑦−𝑓(𝑥)−𝑓𝑦≤𝜀‖𝑥‖^𝑝+𝑦^𝑝 (1) 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 𝑢_𝑡𝑡(𝑥,𝑡)−𝑐²𝑢_𝑥𝑥(𝑥,𝑡)=0 (2) is called the (one-dimensional) wave equation, where 𝑢_𝑡𝑡(𝑥,𝑡) and 𝑢_𝑥𝑥(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 𝑢_𝑡𝑡(𝑥,𝑡)−𝑐²𝑢_𝑥𝑥(𝑥,𝑡)≤𝜑(𝑥,𝑡)(𝑥,𝑡∈ℝ), (3) there exist a solution 𝑢₀∶ℝ×ℝ→ℂ of the (one-dimensional) wave equation (2) and a function Φ∶ℝ×ℝ→[0,∞) such that 𝑢(𝑥,𝑡)−𝑢₀(𝑥,𝑡)≤Φ(𝑥,𝑡)(𝑥,𝑡∈ℝ), (4) where Φ(𝑥,𝑡) is independent of 𝑢(𝑥,𝑡) and 𝑢₀(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 ^𝑏₀^𝑎₀𝜑𝜇+𝜈2,𝜇−𝜈2𝑐𝑑𝜇𝑑𝜈 (5) exists for all 𝑎,𝑏∈ℝ. If a twice continuously differentiable function 𝑢∶ℝ×ℝ→ℂ satisfies the inequality 𝑢_𝑡𝑡(𝑥,𝑡)−𝑐²𝑢_𝑥𝑥(𝑥,𝑡)≤𝜑(𝑥,𝑡) (6) for all 𝑥,𝑡∈ℝ, then there exists a solution 𝑢₀∶ℝ×ℝ→ℂ of the wave equation (2) which satisfies 𝑢(𝑥,𝑡)−𝑢₀(≤1𝑥,𝑡)4𝑐²₀^{𝑥−𝑐𝑡}₀^{𝑥+𝑐𝑡}𝜑𝜇+𝜈2,𝜇−𝜈2𝑐𝑑𝜇𝑑𝜈 (7) for all 𝑥,𝑡∈ℝ.

Proof. Let us define a function 𝑣∶ℝ×ℝ→ℂ by 𝑣(𝑤,𝑧)∶=𝑢𝑤+𝑧2,𝑤−𝑧.2𝑐 (8) If we set 𝑤=𝑥+𝑐𝑡 and 𝑧=𝑥−𝑐𝑡, then we have 𝑢(𝑥,𝑡)=𝑣(𝑤,𝑧) and 𝑢_𝑡(𝑥,𝑡)=𝑣_𝑤(𝑤,𝑧)𝜕𝑤𝜕𝑡+𝑣_𝑧(𝑤,𝑧)𝜕𝑧𝜕𝑡=𝑐𝑣_𝑤(𝑤,𝑧)−𝑐𝑣_𝑧(),𝑢𝑤,𝑧_𝑡𝑡(𝑥,𝑡)=𝑐𝑣_𝑤𝑤(𝑤,𝑧)𝜕𝑤𝜕𝑡+𝑐𝑣_𝑤𝑧(𝑤,𝑧)𝜕𝑧𝜕𝑡−𝑐𝑣_𝑧𝑤(𝑤,𝑧)𝜕𝑤𝜕𝑡−𝑐𝑣_𝑧𝑧(𝑤,𝑧)𝜕𝑧𝜕𝑡=𝑐²𝑣_𝑤𝑤(𝑤,𝑧)−2𝑐²𝑣_𝑤𝑧(𝑤,𝑧)+𝑐²𝑣_𝑧𝑧(),𝑢𝑤,𝑧_𝑥(𝑥,𝑡)=𝑣_𝑤(𝑤,𝑧)𝜕𝑤𝜕𝑥+𝑣_𝑧(𝑤,𝑧)𝜕𝑧𝜕𝑥=𝑣_𝑤(𝑤,𝑧)+𝑣_𝑧(),𝑢𝑤,𝑧_𝑥𝑥(𝑥,𝑡)=𝑣_𝑤𝑤(𝑤,𝑧)𝜕𝑤𝜕𝑥+𝑣_𝑤𝑧(𝑤,𝑧)𝜕𝑧𝜕𝑥+𝑣_𝑧𝑤(𝑤,𝑧)𝜕𝑤𝜕𝑥+𝑣_𝑧𝑧(𝑤,𝑧)𝜕𝑧𝜕𝑥=𝑣_𝑤𝑤(𝑤,𝑧)+2𝑣_𝑤𝑧(𝑤,𝑧)+𝑣_𝑧𝑧(𝑤,𝑧), (9) for all 𝑥,𝑡∈ℝ. Hence, we have 𝑢_𝑡𝑡(𝑥,𝑡)−𝑐²𝑢_𝑥𝑥(𝑥,𝑡)=−4𝑐²𝑣_𝑤𝑧(𝑤,𝑧), (10) for any 𝑥,𝑡∈ℝ. Thus, it follows from inequality (6) that 𝑣_𝑤𝑧(≤1𝑤,𝑧)4𝑐²𝜑𝑤+𝑧2,𝑤−𝑧,2𝑐 (11) for any 𝑤,𝑧∈ℝ.
Therefore, we get −14𝑐^2𝑧₀^𝑤₀𝜑𝜇+𝜈2,𝜇−𝜈≤2𝑐𝑑𝜇𝑑𝜈_𝑧0𝑤0𝑣_𝑤𝑧≤1𝜇,𝜈𝑑𝜇𝑑𝜈4𝑐^2𝑧₀^𝑤₀𝜑𝜇+𝜈2,𝜇−𝜈2𝑐𝑑𝜇𝑑𝜈 (12) or equivalently |𝑣()()()(≤1𝑤,𝑧−𝑣𝑤,0−𝑣0,𝑧+𝑣0,0)|4𝑐^2𝑧₀^𝑤₀𝜑𝜇+𝜈2,𝜇−𝜈,2𝑐𝑑𝜇𝑑𝜈 (13) for all 𝑤,𝑧∈ℝ.
On account of (8), we get 𝑣(𝑤,𝑧)=𝑢𝑤+𝑧2,𝑤−𝑧()𝑤2𝑐,𝑣𝑤,0=𝑢2,𝑤,𝑣()𝑧2𝑐0,𝑧=𝑢2𝑧,−()().2𝑐,𝑣0,0=𝑢0,0 (14) Hence, it follows from (13) and the last equalities that 𝑢𝑤+𝑧2,𝑤−𝑧𝑤2𝑐−𝑢2,𝑤𝑧2𝑐−𝑢2𝑧,−(≤12𝑐+𝑢0,0)4𝑐^2𝑧₀^𝑤₀𝜑𝜇+𝜈2,𝜇−𝜈,2𝑐𝑑𝜇𝑑𝜈 (15) for all 𝑤,𝑧∈ℝ.
If we set 𝑤=𝑥+𝑐𝑡 and 𝑧=𝑥−𝑐𝑡 in the last inequality, then we obtain 𝑢(𝑥,𝑡)−𝑢₀(≤1𝑥,𝑡)4𝑐²₀^{𝑥−𝑐𝑡}₀^{𝑥+𝑐𝑡}𝜑𝜇+𝜈2,𝜇−𝜈,2𝑐𝑑𝜇𝑑𝜈 (16) for all 𝑥,𝑡∈ℝ, where we set 𝑢₀()𝑥𝑥,𝑡∶=𝑢2+𝑐2𝑥𝑡,+𝑡2𝑐2𝑥+𝑢2−𝑐2𝑥𝑡,−+𝑡2𝑐2−𝑢(0,0). (17)
By some tedious calculations, we get 𝜕𝑢𝜕𝑡⁰()=𝑐𝑥,𝑡2𝑢_𝑥𝑥2+𝑐𝑡2,𝑥+𝑡2𝑐2+12𝑢_𝑡𝑥2+𝑐𝑡2,𝑥+𝑡2𝑐2−𝑐2𝑢_𝑥𝑥2−𝑐𝑡2𝑥,−+𝑡2𝑐2+12𝑢_𝑡𝑥2−𝑐𝑡2𝑥,−+𝑡2𝑐2,𝜕²𝜕𝑡²𝑢₀()=𝑐𝑥,𝑡₂4𝑢_𝑥𝑥𝑥2+𝑐𝑡2,𝑥+𝑡2𝑐2+𝑐2𝑢_𝑥𝑡𝑥2+𝑐𝑡2,𝑥+𝑡2𝑐2+14𝑢_𝑡𝑡𝑥2+𝑐𝑡2,𝑥+𝑡2𝑐2+𝑐²4𝑢_𝑥𝑥𝑥2−𝑐𝑡2𝑥,−+𝑡2𝑐2−𝑐2𝑢_𝑥𝑡𝑥2−𝑐𝑡2𝑥,−+𝑡2𝑐2+14𝑢_𝑡𝑡𝑥2−𝑐𝑡2𝑥,−+𝑡2𝑐2,𝜕𝑢𝜕𝑥⁰()=1𝑥,𝑡2𝑢_𝑥𝑥2+𝑐𝑡2,𝑥+𝑡2𝑐2+1𝑢2𝑐^𝑡𝑥2+𝑐𝑡2,𝑥+𝑡2𝑐2+12𝑢_𝑥𝑥2−𝑐𝑡2𝑥,−+𝑡2𝑐2−1𝑢2𝑐^𝑡𝑥2−𝑐𝑡2𝑥,−+𝑡2𝑐2,𝜕²𝜕𝑥²𝑢₀()=1𝑥,𝑡4𝑢_𝑥𝑥𝑥2+𝑐𝑡2,𝑥+𝑡2𝑐2+1𝑢2𝑐^𝑥𝑡𝑥2+𝑐𝑡2,𝑥+𝑡2𝑐2+14𝑐²𝑢_𝑡𝑡𝑥2+𝑐𝑡2,𝑥+𝑡2𝑐2+14𝑢_𝑥𝑥𝑥2−𝑐𝑡2𝑥,−+𝑡2𝑐2−1𝑢2𝑐^𝑥𝑡𝑥2−𝑐𝑡2𝑥,−+𝑡2𝑐2+14𝑐²𝑢_𝑡𝑡𝑥2−𝑐𝑡2𝑥,−+𝑡2𝑐2, (18) for all 𝑥,𝑡∈ℝ. Hence, we know that 𝜕²𝜕𝑡²𝑢₀(𝑥,𝑡)−𝑐²𝜕²𝜕𝑥²𝑢₀(𝑥,𝑡)=0, (19) for any 𝑥,𝑡∈ℝ; that is, 𝑢₀(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 𝜑(𝑥,𝑡)=𝛼𝑒^{−𝑥2−𝑐2𝑡2}. (20) If a twice continuously differentiable function 𝑢∶ℝ×ℝ→ℂ satisfies inequality (6), for all 𝑥,𝑡∈ℝ, then there exists a solution 𝑢₀∶ℝ×ℝ→ℂ of the wave equation (2) which satisfies 𝑢(𝑥,𝑡)−𝑢₀(𝑥,𝑡)≤𝛼𝜋8𝑐²erf𝑥−𝑐𝑡√2erf𝑥+𝑐𝑡√2, (21) for all 𝑥,𝑡∈ℝ.

Proof. Since ^𝑏₀^𝑎₀𝜑𝜇+𝜈2,𝜇−𝜈=2𝑐𝑑𝜇𝑑𝜈_𝑏0𝑎0𝛼𝑒^{−𝜇2/2−𝜈2/2}𝑑𝜇𝑑𝜈=𝛼^𝑏₀𝑒^−𝜈2/2𝑑𝜈^𝑎₀𝑒^−𝜇2/2𝜋𝑑𝜇=2𝛼42√𝜋^√𝑏/2₀𝑒^−𝜈22𝑑𝜈√𝜋^√𝑎/2₀𝑒^−𝜇2=𝑑𝜇𝛼𝜋2𝑏erf√2𝑎erf√2<∞, (22) for all 𝑎,𝑏∈ℝ, 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).