Simple Harmonic Oscillator Equation and Its Hyers-Ulam Stability

Abstract

We solve the inhomogeneous simple harmonic oscillator equation and apply this result to obtain a partial solution to the Hyers-Ulam stability problem for the simple harmonic oscillator equation.

1. Introduction

Let 𝑋 be a normed space over a scalar field 𝕂 and let 𝐼 be an open interval, where 𝕂 denotes either ℝ or ℂ. Assume that 𝑎₀,𝑎₁,…,𝑎_𝑛∶𝐼→𝕂 are given continuous functions, 𝑔∶𝐼→𝑋 is a given continuous function, and 𝑦∶𝐼→𝑋 is an 𝑛 times continuously differentiable function satisfying the inequality:‖‖𝑎_𝑛(𝑡)𝑦^(𝑛)(𝑡)+𝑎_𝑛−1(𝑡)𝑦^(𝑛−1)(𝑡)+⋯+𝑎₁(𝑡)𝑦^(𝑡)+𝑎₀(𝑡)𝑦(𝑡)+𝑔(𝑡)‖‖≤𝜀,(1.1)

for all 𝑡∈𝐼 and for a given 𝜀>0. If there exists an 𝑛 times continuously differentiable function 𝑦₀∶𝐼→𝑋 satisfying𝑎_𝑛(𝑡)𝑦^0(𝑛)(𝑡)+𝑎_𝑛−1(𝑡)𝑦^0(𝑛−1)(𝑡)+⋯+𝑎₁(𝑡)𝑦^0(𝑡)+𝑎₀(𝑡)𝑦₀(𝑡)+𝑔(𝑡)=0,(1.2)

and ‖𝑦(𝑡)−𝑦₀(𝑡)‖≤𝐾(𝜀) for any 𝑡∈𝐼, where 𝐾(𝜀) is an expression of 𝜀 with lim_𝜀→0𝐾(𝜀)=0, then we say that the above differential equation has the Hyers-Ulam stability. For more detailed definitions of the Hyers-Ulam stability, we refer the reader to [1–7].

Obloza seems to be the first author who has investigated the Hyers-Ulam stability of linear differential equations (see [8, 9]). Here, we will introduce a result of Alsina and Ger (see [10]). If a differentiable function 𝑓∶𝐼→ℝ satisfies the inequality |𝑦′(𝑡)−𝑦(𝑡)|≤𝜀, where 𝐼 is an open subinterval of ℝ, then there exists a solution 𝑓₀∶𝐼→ℝ of the differential equation 𝑦^(𝑡)=𝑦(𝑡) such that |𝑓(𝑡)−𝑓₀(𝑡)|≤3𝜀 for any 𝑡∈𝐼. This result has been generalized by Takahasi et al. Indeed, it was proved in [11] that the Hyers-Ulam stability holds true for the Banach space valued differential equation 𝑦^(𝑡)=𝜆𝑦(𝑡) (see also [12, 13]).

Moreover, Miura et al. [14] investigated the Hyers-Ulam stability of 𝑛th order linear differential equation with complex coefficients. They [15] also proved the Hyers-Ulam stability of linear differential equations of first order, 𝑦^(𝑡)+𝑔(𝑡)𝑦(𝑡)=0, where 𝑔(𝑡) is a continuous function.

Jung also proved the Hyers-Ulam stability of various linear differential equations of first order [16–19]. Moreover, he could successfully apply the power series method to the study of the Hyers-Ulam stability of Legendre differential equation (see [20]). Subsequently, the authors [21] investigated the Hyers-Ulam stability problem for Bessel differential equation by applying the same method.

In Section 2 of this paper, by using the ideas from [20, 21], we investigate the general solution of the inhomogeneous simple harmonic oscillator equation of the form:𝑦^(𝑥)+𝜔²𝑦(𝑥)=^∞_𝑚₌₀𝑎_𝑚𝑥^𝑚,(1.3)

where 𝜔 is a given positive number. Section 3 will be devoted to a partial solution of the Hyers-Ulam stability problem for the simple harmonic oscillator equation (2.1) in a subclass of analytic functions.

2. Inhomogeneous Simple Harmonic Oscillator Equation

A function is called a simple harmonic oscillator function if it satisfies the simple harmonic oscillator equation:𝑦^(𝑥)+𝜔²𝑦(𝑥)=0.(2.1)

The simple harmonic oscillator equation plays a great role in physics and engineering. In particular, it describes quantum particles confined in potential wells in quantum mechanics and the Hyers-Ulam stability of solutions of this equation is very important.

In this section, we define 𝑐₀=𝑐₁=0 and for 𝑚≥1,𝑐_2𝑚=^𝑚−1_𝑖=0(−1)^{𝑚−𝑖−1}𝑎_2𝑖(2𝑖)!(2𝑚)!𝜔^{2𝑚−2𝑖−2}𝑐_2𝑚+1,=^𝑚−1_𝑖=0(−1)^{𝑚−𝑖−1}𝑎_2𝑖+1(2𝑖+1)!(2𝑚+1)!𝜔^{2𝑚−2𝑖−2},(2.2) where we refer to (1.3) for the 𝑎_𝑚. We can easily check that these 𝑐_𝑚 satisfy the following𝑎_𝑚=(𝑚+2)(𝑚+1)𝑐_𝑚+2+𝜔²𝑐_𝑚,(2.3) for any 𝑚∈{0,1,2,…}.

Lemma 2.1. (a) If the power series ∑^∞𝑚=0𝑎_𝑚𝑥^𝑚 converges for all 𝑥∈(−𝜌,𝜌) with 𝜌>1, then the power series ∑^∞𝑚=2𝑐_𝑚𝑥^𝑚 with 𝑐_𝑚 given in (2.2) satisfies the inequality |∑^∞𝑚=2𝑐_𝑚𝑥^𝑚|≤𝐶₁/(1−|𝑥|) for some positive constant 𝐶₁ and for any 𝑥∈(−1,1).
(b) If the power series ∑^∞𝑚=0𝑎_𝑚𝑥^𝑚 converges for all 𝑥∈(−𝜌,𝜌) with 𝜌≤1, then for any positive 𝜌₀<𝜌, the power series ∑^∞𝑚=2𝑐_𝑚𝑥^𝑚 with 𝑐_𝑚 given in (2.2) satisfies the inequality |∑^∞𝑚=2𝑐_𝑚𝑥^𝑚|≤𝐶₂ for any 𝑥∈[−𝜌₀,𝜌₀] and for some positive constant 𝐶₂ which depends on 𝜌₀. Since 𝜌₀ is arbitrarily close to 𝜌, this means that ∑^∞𝑚=2𝑐_𝑚𝑥^𝑚 is convergent for all 𝑥∈(−𝜌,𝜌).

Proof. (a) Since the power series ∑^∞𝑚=0𝑎_𝑚𝑥^𝑚 is absolutely convergent on its interval of convergence, with 𝑥=1, ∑^∞𝑚=0𝑎_𝑚 converges absolutely, that is, ∑^∞𝑚=0|𝑎_𝑚|<𝑀₁ by some number 𝑀₁.
We know that ||𝑎_2𝑖||(2𝑖)!(2𝑚)!𝜔^{2𝑚−2𝑖−2}=||𝑎^2𝑖||≤2⎧⎪⎨⎪⎩𝑚(2𝑚||𝑎−1)(2𝑚𝜔−2)⋯(2𝑖𝜔+1)_2𝑖||2𝑚(||2𝑎𝑚−1)(for0<𝜔≤1)_2𝑖||2𝑚(2𝑚−1)𝜔^[𝜔]≤||𝑎(for𝜔>1)_2𝑖||2𝑚(2𝑚−1)max{1,𝜔^𝜔},(2.4) since for 𝜔>1, each factor of the form 𝜔/ℓ in the summand is either less than 1 if ℓ>[𝜔], or is bigger than or equal to 1 if ℓ≤[𝜔], where [𝜔] denotes the largest integer less than or equal to 𝜔. Thus, we obtain ||𝑐_2𝑚||≤^𝑚−1_𝑖=0||𝑎_2𝑖||(2𝑖)!(2𝑚)!𝜔^{2𝑚−2𝑖−2}≤^𝑚−1_𝑖=0||𝑎_2𝑖||2𝑚(2𝑚−1)max{1,𝜔^𝜔≤max{1,𝜔_𝜔}≤max𝑀₁}^∞_𝑖=0||𝑎_𝑖||,𝑀₁𝜔^𝜔≡𝐶₁.(2.5)
Similarly, we have ||𝑎_2𝑖+1||(2𝑖+1)!(2𝑚+1)!𝜔^{2𝑚−2𝑖−2}≤||𝑎^2𝑖+1||2𝑚(2𝑚−1)max{1,𝜔^𝜔},(2.6) and |𝑐_2𝑚+1|≤𝐶₁ for all 𝑚≥1.
Therefore, we get |||||_∞𝑚₌₂𝑐_𝑚𝑥^𝑚|||||≤_∞𝑚₌₂||𝑐_𝑚||||𝑥^𝑚||≤𝐶_1∞𝑚₌₂||𝑥^𝑚||≤𝐶¹1−|𝑥|,(2.7) for every 𝑥∈(−1,1).
(b) The power series ∑^∞𝑚=0𝑎_𝑚𝑥^𝑚 is absolutely convergent on its interval of convergence, and, therefore, for any given 𝜌₀<𝜌, the series ∑^∞𝑚=0|𝑎_𝑚𝑥^𝑚| is convergent on [−𝜌₀,𝜌₀] and ^∞_𝑚=0||𝑎_𝑚|||𝑥|^𝑚≤^∞_𝑚₌₀||𝑎_𝑚||𝜌^𝑚0≡𝑀₂(2.8) for any 𝑥∈[−𝜌₀,𝜌₀]. Now, it follows from (2.2), (2.4), (2.6), and (2.8) that |||||_∞𝑚₌₂𝑐_𝑚𝑥^𝑚|||||≤_∞𝑚₌₁||𝑐_2𝑚||𝜌^02𝑚+^∞_𝑚₌₁||𝑐_2𝑚+1||𝜌^02𝑚+1≤^∞_𝑚₌₁𝜌^{02𝑚𝑚−1}_𝑖=0||𝑎_2𝑖||(2𝑖)!(2𝑚)!𝜔^{2𝑚−2𝑖−2}+^∞_𝑚₌₁𝜌^{02𝑚+1𝑚−1}_𝑖=0||𝑎_2𝑖+1||(2𝑖+1)!(2𝑚+1)!𝜔^{2𝑚−2𝑖−2}≤^∞_{𝑚=1𝑚−1𝑖=0}||𝑎_2𝑖||𝜌^02𝑖max{1,𝜔^𝜔}+2𝑚(2𝑚−1)_∞_{𝑚=1𝑚−1𝑖=0}||𝑎_2𝑖+1||𝜌^02𝑖+1max{1,𝜔^𝜔}≤2𝑚(2𝑚−1)_∞𝑚₌₁𝑀₂max{1,𝜔^𝜔}2𝑚(2𝑚−1)+^∞_𝑚=1𝑀₂max{1,𝜔^𝜔}≤max𝑀₂2𝑚(2𝑚−1),𝑀₂𝜔^𝜔^∞_𝑚₌₁1≤3𝑚(2𝑚−1)2max𝑀²,𝑀₂𝜔^𝜔≡𝐶₂,(2.9) for any 𝑥∈[−𝜌₀,𝜌₀].

Lemma 2.2. Suppose that the power series ∑^∞𝑚=0𝑎_𝑚𝑥^𝑚 converges for all 𝑥∈(−𝜌,𝜌) with some positive 𝜌. Let 𝜌₁=min{1,𝜌}. Then, the power series ∑^∞𝑚=2𝑐_𝑚𝑥^𝑚 with 𝑐_𝑚 given in (2.2) is convergent for all 𝑥∈(−𝜌₁,𝜌₁). Further, for any positive 𝜌₀<𝜌₁, |∑^∞𝑚=2𝑐_𝑚𝑥^𝑚|≤𝐶 for any 𝑥∈[−𝜌₀,𝜌₀] and for some positive constant 𝐶 which depends on 𝜌₀.

Proof. The first statement follows from the latter statement. Therefore, let us prove the latter statement. If 𝜌≤1, then 𝜌₁=𝜌. By Lemma 2.1(b), for any positive 𝜌₀<𝜌=𝜌₁, |∑^∞𝑚=2𝑐_𝑚𝑥^𝑚|≤𝐶₂ for each 𝑥∈[−𝜌₀,𝜌₀] and for some positive constant 𝐶₂ which depends on 𝜌₀.
If 𝜌>1, then by Lemma 2.1(a), for any positive 𝜌₀<1=𝜌₁, we get |||||_∞𝑚₌₂𝑐_𝑚𝑥^𝑚|||||≤𝐶₁1−|𝑥|≤1−𝐶¹𝜌₀≤max𝐶¹1−𝜌₀,𝐶₂≡𝐶,(2.10) for all 𝑥∈[−𝜌₀,𝜌₀] and for some positive constant 𝐶 which depends on 𝜌₀.

Using these definitions and the lemmas above, we will now show that ∑^∞𝑚=2𝑐_𝑚𝑥^𝑚 is a particular solution of the inhomogeneous simple harmonic oscillator equation (1.3).

Theorem 2.3. Assume that 𝜔 is a given positive number and the radius of convergence of the power series ∑^∞𝑚=0𝑎_𝑚𝑥^𝑚 is 𝜌>0. Let 𝜌₁=min{1,𝜌}. Then, every solution 𝑦∶(−𝜌₁,𝜌₁)→ℂ of the simple harmonic oscillator equation (1.3) can be expressed by 𝑦(𝑥)=𝑦_ℎ(𝑥)+^∞_𝑚₌₂𝑐_𝑚𝑥^𝑚,(2.11)where 𝑦_ℎ(𝑥) is a simple harmonic oscillator function and 𝑐_𝑚 are given by (2.2).

Proof. We show that ∑^∞𝑚=2𝑐_𝑚𝑥^𝑚 satisfies (1.3). By Lemma 2.2, the power series ∑^∞𝑚=2𝑐_𝑚𝑥^𝑚 is convergent for each 𝑥∈(−𝜌₁,𝜌₁).
Substituting ∑^∞𝑚=2𝑐_𝑚𝑥^𝑚 for 𝑦(𝑥) in (1.3) and collecting like powers together, it follows from (2.2) and (2.3) that (with 𝑐₀=𝑐₁=0) 𝑦^′(𝑥)+𝜔²𝑦(𝑥)=^∞_𝑚₌₀(𝑚+2)(𝑚+1)𝑐_𝑚+2+𝜔²𝑐_𝑚𝑥^𝑚=^∞_𝑚₌₀𝑎_𝑚𝑥^𝑚,(2.12) for all 𝑥∈(−𝜌₁,𝜌₁).
Therefore, every solution 𝑦∶(−𝜌₁,𝜌₁)→ℂ of the inhomogeneous simple harmonic oscillator equation (1.3) can be expressed by 𝑦(𝑥)=𝑦_ℎ(𝑥)+^∞_𝑚₌₂𝑐_𝑚𝑥^𝑚,(2.13)where 𝑦_ℎ(𝑥) is a simple harmonic oscillator function.

3. Partial Solution to Hyers-Ulam Stability Problem

In this section, we will investigate a property of the simple harmonic oscillator equation (2.1) concerning the Hyers-Ulam stability problem. That is, we will try to answer the question whether there exists a simple harmonic oscillator function near any approximate simple harmonic oscillator function.

Theorem 3.1. Let 𝑦∶(−𝜌,𝜌)→ℂ be a given analytic function which can be represented by a power series ∑^∞𝑚=0𝑏_𝑚𝑥^𝑚 whose radius of convergence is at least 𝜌>0. Suppose there exists a constant 𝜀>0 such that ||𝑦^(𝑥)+𝜔²𝑦(𝑥)||≤𝜀,(3.1) for all 𝑥∈(−𝜌,𝜌) and for some positive number 𝜔. Let 𝜌₁=min{1,𝜌}. Define 𝑎_𝑚=(𝑚+2)(𝑚+1)𝑏_𝑚+2+𝜔²𝑏_𝑚 for all 𝑚∈{0,1,2,…} and suppose further that ^∞_𝑚=0||𝑎_𝑚𝑥^𝑚||≤𝐾|||||^∞_𝑚₌₀𝑎_𝑚𝑥^𝑚|||||,(3.2) for all 𝑥∈(−𝜌,𝜌) and for some constant 𝐾. Then, there exists a simple harmonic oscillator function 𝑦_ℎ∶(−𝜌₁,𝜌₁)→ℂ such that ||𝑦(𝑥)−𝑦_ℎ(𝑥)||≤𝐶𝜀,(3.3) for all 𝑥∈[−𝜌₀,𝜌₀], where 𝜌₀<𝜌₁ is any positive number and 𝐶 is some constant which depends on 𝜌₀.

Proof. We assumed that 𝑦(𝑥) can be represented by a power series and 𝑦^(𝑥)+𝜔²𝑦(𝑥)=^∞_𝑚₌₀𝑎_𝑚𝑥^𝑚(3.4) also satisfies ^∞_𝑚=0||𝑎_𝑚𝑥^𝑚||≤𝐾|||||^∞_𝑚₌₀𝑎_𝑚𝑥^𝑚|||||≤𝐾𝜀,(3.5) for all 𝑥∈(−𝜌,𝜌) from (3.1).
According to Theorem 2.3, 𝑦(𝑥) can be written as 𝑦_ℎ(𝑥)+∑^∞𝑚=2𝑐_𝑚𝑥^𝑚 for all 𝑥∈(−𝜌₁,𝜌₁), where 𝑦_ℎ is some simple harmonic oscillator function and 𝑐_𝑚 are given by (2.2). Then by Lemmas 2.1 and 2.2 and their proofs (replace 𝑀₁ and 𝑀₂ with 𝐾𝜀 in Lemma 2.1), ||𝑦(𝑥)−𝑦_ℎ(𝑥)||=|||||^∞_𝑚₌₂𝑐_𝑚𝑥^𝑚|||||≤𝐶𝜀(3.6) for all 𝑥∈[−𝜌₀,𝜌₀], where 𝜌₀<𝜌₁ is any positive number and 𝐶 is some constant which depends on 𝜌₀.
Actually from the proof of Lemma 2.1, with both 𝑀₁ and 𝑀₂ replaced by 𝐾𝜀, we find 𝐶₁=max{𝐾𝜀𝜔^𝜔,𝐾𝜀} and 𝐶₂=3/2𝐶₁. Further from the proof of Lemma 2.2, we have 𝐶𝜀=max𝐶¹1−𝜌₀,𝐶₂=max1𝐾−𝜀𝜌₀𝜔^𝜔,𝐾𝜀1−𝜌₀,32𝐾𝜀𝜔^𝜔,32𝐾𝜀,(3.7) we find 𝐶=max𝐾1−𝜌₀𝜔^𝜔,𝐾1−𝜌₀,32𝐾𝜔^𝜔,32𝐾,(3.8)which completes the proof of our theorem.

4. Example

In this section, we show that there certainly exist functions 𝑦(𝑥) which satisfy all the conditions given in Theorem 3.1. We introduce an example related to the simple harmonic oscillator equation (1.3) for 𝜔=1/4.

Let 𝑦_ℎ(𝑥) be a simple harmonic oscillator function for 𝜔=1/4 and let 𝑦∶(−1,1)→ℝ be an analytic function given by𝑦(𝑥)=𝑦_ℎ(𝑥)+𝜀^∞_𝑚₌₀𝑥^2𝑚4^𝑚+1,(4.1) where 𝜀 is a positive constant. (We can easily show that the radius of convergence of the power series ∑^∞𝑚=0𝑥^2𝑚/4^𝑚+1 is 2). Then, we have𝑦^(𝑥)+116𝑦(𝑥)=^∞_𝑚=0𝑎_𝑚𝑥^𝑚,(4.2)

where𝑎_𝑚=⎧⎪⎨⎪⎩4𝑚²+12𝑚+92^𝑚+60((ffoorr𝑚𝑚∈∈{{01,,32,,54,,……}})),_∞𝑚₌₀||𝑎_𝑚𝑥^𝑚||=|||||^∞_𝑚₌₀𝑎_𝑚𝑥^𝑚|||||(4.3)

for any 𝑥∈(−1,1). So we can here choose 𝐾=1.

Furthermore, we get|||𝑦_(𝑥)+116𝑦(𝑥)|||≤^∞_𝑚=08𝜀(𝑚+1)(2𝑚+1)4^𝑚+3|𝑥|^2𝑚+^∞_𝑚₌₀𝜀4^𝑚+3|𝑥|^2𝑚≤^∞_𝑚₌₀15𝜀32|𝑥2|^2𝑚𝑚+^∞_𝑚₌₀𝜀4³|𝑥|^2𝑚4^𝑚≤^∞_𝑚₌₀15𝜀3221^𝑚+^∞_𝑚₌₀𝜀4³14^𝑚<𝜀,(4.4)and it follows from (4.1) that||𝑦(𝑥)−𝑦_ℎ(𝑥)||=|||||𝜀^∞_𝑚₌₀𝑥^2𝑚4^𝑚+1|||||≤𝜀4^∞_𝑚=014^𝑚=𝐶𝜀,(4.5)

for all 𝑥∈(−1,1), where we set 𝐶=1/3.