# On the Stability of One-Dimensional Wave Equation

Authors: Soon-Mo Jung
Journal: The Scientific World Journal
 (2013)

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## Abstract

We prove the generalized Hyers-Ulam stability of the one-dimensional wave equation,utt=c2uxx, in a class of twice continuously differentiable functions.


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## Body

## 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: LetG1 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:
(1)∥f(x+y)-f(x)-f(y)∥≤ε(∥x∥p+∥y∥p)
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 numberc>0, the partial differential equation
(2)utt(x,t)-c2uxx(x,t)=0
is called the (one-dimensional) wave equation, where utt(x,t) and uxx(x,t) denote the second time derivative and the second space derivative of u(x,t), respectively.Letφ:ℝ×ℝ→[0,∞) be a function. If, for each twice continuously differentiable function u:ℝ×ℝ→ℂ satisfying
(3)|utt(x,t)-c2uxx(x,t)|≤φ(x,t)(x,t∈ℝ),
there exist a solution u0:ℝ×ℝ→ℂ of the (one-dimensional) wave equation (2) and a function Φ:ℝ×ℝ→[0,∞) such that
(4)|u(x,t)-u0(x,t)|≤Φ(x,t)(x,t∈ℝ),
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 functionφ:ℝ×ℝ→[0,∞) be given such that the double integral
(5)∫0b‍∫0a‍φ(μ+ν2,μ-ν2c)dμdν
exists for all a,b∈ℝ. If a twice continuously differentiable function u:ℝ×ℝ→ℂ satisfies the inequality
(6)|utt(x,t)-c2uxx(x,t)|≤φ(x,t)
for all x,t∈ℝ, then there exists a solution u0:ℝ×ℝ→ℂ of the wave equation (2) which satisfies
(7)|u(x,t)-u0(x,t)|≤14c2|∫0x-ct‍∫0x+ct‍φ(μ+ν2,μ-ν2c)dμdν|
for all x,t∈ℝ.Proof.
Let us define a functionv:ℝ×ℝ→ℂ by
(8)v(w,z):=u(w+z2,w-z2c).
If we set w=x+ct and z=x-ct, then we have u(x,t)=v(w,z) and
(9)ut(x,t)=vw(w,z)∂w∂t+vz(w,z)∂z∂t=cvw(w,z)-cvz(w,z),utt(x,t)=cvww(w,z)∂w∂t+cvwz(w,z)∂z∂t-cvzw(w,z)∂w∂t-cvzz(w,z)∂z∂t=c2vww(w,z)-2c2vwz(w,z)+c2vzz(w,z),ux(x,t)=vw(w,z)∂w∂x+vz(w,z)∂z∂x=vw(w,z)+vz(w,z),uxx(x,t)=vww(w,z)∂w∂x+vwz(w,z)∂z∂x+vzw(w,z)∂w∂x+vzz(w,z)∂z∂x=vww(w,z)+2vwz(w,z)+vzz(w,z),
for all x,t∈ℝ. Hence, we have
(10)utt(x,t)-c2uxx(x,t)=-4c2vwz(w,z),
for any x,t∈ℝ. Thus, it follows from inequality (6) that
(11)|vwz(w,z)|≤14c2φ(w+z2,w-z2c),
for any w,z∈ℝ.
Therefore, we get(12)-14c2∫0z‍∫0w‍φ(μ+ν2,μ-ν2c)dμdν≤∫0z‍∫0w‍vwz(μ,ν)dμdν≤14c2∫0z‍∫0w‍φ(μ+ν2,μ-ν2c)dμdν
or equivalently
(13)|v(w,z)-v(w,0)-v(0,z)+v(0,0)|≤14c2|∫0z‍∫0w‍φ(μ+ν2,μ-ν2c)dμdν|,
for all w,z∈ℝ.
On account of (8), we get
(14)v(w,z)=u(w+z2,w-z2c),v(w,0)=u(w2,w2c),v(0,z)=u(z2,-z2c),v(0,0)=u(0,0).
Hence, it follows from (13) and the last equalities that
(15)|u(w+z2,w-z2c)-u(w2,w2c)-u(z2,-z2c)+u(0,0)|≤14c2|∫0z‍∫0w‍φ(μ+ν2,μ-ν2c)dμdν|,
for all w,z∈ℝ.
If we setw=x+ct and z=x-ct in the last inequality, then we obtain
(16)|u(x,t)-u0(x,t)|≤14c2|∫0x-ct‍∫0x+ct‍φ(μ+ν2,μ-ν2c)dμdν|,
for all x,t∈ℝ, where we set
(17)u0(x,t):=u(x2+c2t,x2c+t2)+u(x2-c2t,-x2c+t2)-u(0,0).
By some tedious calculations, we get(18)∂∂tu0(x,t)=c2ux(x2+ct2,x2c+t2)+12ut(x2+ct2,x2c+t2)-c2ux(x2-ct2,-x2c+t2)+12ut(x2-ct2,-x2c+t2),∂2∂t2u0(x,t)=c24uxx(x2+ct2,x2c+t2)+c2uxt(x2+ct2,x2c+t2)+14utt(x2+ct2,x2c+t2)+c24uxx(x2-ct2,-x2c+t2)-c2uxt(x2-ct2,-x2c+t2)+14utt(x2-ct2,-x2c+t2),∂∂xu0(x,t)=12ux(x2+ct2,x2c+t2)+12cut(x2+ct2,x2c+t2)+12ux(x2-ct2,-x2c+t2)-12cut(x2-ct2,-x2c+t2),∂2∂x2u0(x,t)=14uxx(x2+ct2,x2c+t2)+12cuxt(x2+ct2,x2c+t2)+14c2utt(x2+ct2,x2c+t2)+14uxx(x2-ct2,-x2c+t2)-12cuxt(x2-ct2,-x2c+t2)+14c2utt(x2-ct2,-x2c+t2),
for all x,t∈ℝ. Hence, we know that
(19)∂2∂t2u0(x,t)-c2∂2∂x2u0(x,t)=0,
for any x,t∈ℝ; that is, u0(x,t) is a solution of the wave equation (2).Corollary 2.
Given a constantα>0, let a function φ:ℝ×ℝ→[0,∞) be given as
(20)φ(x,t)=αe-x2-c2t2.
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
(21)|u(x,t)-u0(x,t)|≤απ8c2|erf(x-ct2)erf(x+ct2)|,
for all x,t∈ℝ.Proof.
Since(22)|∫0b‍∫0a‍φ(μ+ν2,μ-ν2c)dμdν|=|∫0b‍∫0a‍αe-μ2/2-ν2/2dμdν|=α|∫0b‍e-ν2/2dν||∫0a‍e-μ2/2dμ|=2απ4|(2π∫0b/2‍e-ν2dν)(2π∫0a/2‍e-μ2dμ)|=απ2|erf(b2)erf(a2)|<∞,
for all a,b∈ℝ, in view of Theorem 1, we conclude that the statement of this corollary is true.



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*Source: 978754-2013-10-27.xml*