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An approximation property of simple harmonic functions
Journal of Inequalities and Applications volume 2013, Article number: 3 (2013)
In this paper, applying the power series method, we approximate analytic functions by simple harmonic functions in a neighborhood of zero.
Differential equations have been studied for more than 300 years since the seventeenth century when the concepts of differentiation and integration were formulated by Newton and Leibniz. By use of differential equations, we can explain many natural phenomena: gravity, projectiles, wave, vibration, nuclear physics, and so on.
Let us consider a closed system which can be explained by the first-order linear differential equation, namely, . The past, present, and future of this system are completely determined if we know the general solution and an initial condition of that differential equation. So, we can say that this system is ‘predictable.’ Sometimes, because of the disturbances (or noises) of the outside, the system may not be determined by but can only be explained by an inequality like . Then it is impossible to predict the exact future of the disturbed system.
Even though the system is not predictable exactly because of outside disturbances, we say the differential equation has the Hyers-Ulam stability if the ‘real’ future of the system follows the solution of with a bounded error. But if the error bound is ‘too big,’ we say that the differential equation does not have the Hyers-Ulam stability. Resonance is the case.
There is another way to explain the Hyers-Ulam stability. Usually the experiment (or the observed) data do not exactly coincide with theoretical expectations. We may express natural phenomena by use of equations, but because of the errors due to measurement or observance, the actual experiment data can almost always be a little bit off the expectations. If we used inequalities instead of equalities to explain natural phenomena, then these errors could be absorbed into the solutions of inequalities, i.e., those errors would no longer be errors. Considering this point of view, the Hyers-Ulam stability (of differential equations) is fundamental. (Hyers-Ulam stability is not same as the concept of the stability of differential equations which has been studied by many mathematicians for a long time.)
We will now introduce the concept of Hyers-Ulam stability of differential equations. Let X be a normed space over a scalar field and let be an open interval, where denotes either ℝ or ℂ. Assume that and are continuous functions and that is an n times continuously differentiable function satisfying the inequality
for all and for a given . If there exists an n times continuously differentiable function satisfying
and for all , where is an expression of ε with , 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 [1–6].
Hyers-Ulam stability of functional equations has been studied for more than 70 years. But the history of the Hyers-Ulam stability of differential equations is less than 20 years. For example, Obłoza seems to be the first author who investigated the Hyers-Ulam stability of linear differential equations in 1993 (see [7, 8]). Thereafter, Alsina and Ger published their paper , which handles the Hyers-Ulam stability of the linear differential equation : If a differentiable function is a solution of the inequality for any , then there exists a constant c such that for all . We know that the general solution of the linear differential equation is , where c is a constant.
Therefore, we say that the differential equation has the Hyers-Ulam stability. If we can get a similar result with a control function in place of ε, we say that the differential equation has the Hyers-Ulam-Rassias stability.
In 2001 and 2002, Miura et al.  expanded Alsina and Ger’s result by proving that the differential equation has the Hyers-Ulam stability. The author wrote a paper with Miura and Takahasi which expanded the result of Hyers-Ulam stability of that differential equation. To be more precise, we may choose a constant c such that the solution of the inequality is not too far away from in the sense of upper norm (see ).
In Section 3 of this paper, we apply the power series method to prove the Hyers-Ulam stability of the simple harmonic oscillator equation
This paper is an extension and an improvement of the previous paper . We denote by the set of all nonnegative integers.
2 Inhomogeneous simple harmonic oscillator equation
Let ω be a positive constant. A function is called a simple harmonic oscillator function if it satisfies the simple harmonic oscillator equation (1.1), which plays an important role in quantum mechanics.
Theorem 2.1 Let L be a real number with . Assume that the radius of convergence of power series is , and a sequence satisfies the recurrence relation
for any . If
then every solution of the inhomogeneous simple harmonic oscillator equation
can be expressed by
for all , where is a simple harmonic oscillator function and the value of is given by (2.4).
Proof We can use induction on m and prove that if a sequence satisfies the relation (2.1), then
for all . We omit the proof.
Due to (2.2), there exists an such that for any integer . Moreover, for any real number δ with , we can choose an integer such that and
for all integers . We now define a real constant by
If , it then follows from (2.5) that
If , it then follows from (2.5) and (2.6) that
If and , then (2.6) implies that
In view of (2.4), (2.7), and (2.8), for any sufficiently large integer m, there exists a real constant such that
Hence, it holds that
Similarly, we have
which implies that the radius of convergence of power series is at least . Moreover, we notice that the radius of convergence of a general solution of the simple harmonic oscillator equation (1.1) is .
Since is an ordinary point, we can substitute for and use the formal multiplication of power series and consider (2.1) to get
for all . That is, is a particular solution of the inhomogeneous simple harmonic oscillator equation (2.3) and hence every solution of (2.3) can be expressed by
where is a simple harmonic oscillator function. □
3 Approximate simple harmonic oscillator equation
Let be a positive constant. We denote by the set of all functions with the following properties:
is expressible by a power series whose radius of convergence is at least ;
There exists a constant such that for any , where for all .
If we define
for all and , then is a vector space over complex numbers. We remark that the set is large enough to be a vector space.
Now, we prove the main theorem of this paper.
Theorem 3.1 Let L be a real number with . Assume that is an arbitrary function belonging to whose power series coefficients satisfy either
where ω is a positive number. If y satisfies the differential inequality
for all and for some , then there exists a solution of the simple harmonic oscillator equation (1.1) such that
for all , where C is determined by (3.3) and K is defined in (b).
Proof We will prove this theorem under the first condition of (3.1). Let be the radius of convergence of the power series which is a particular solution of the inhomogeneous simple harmonic oscillator equation (2.3), where is defined in (2.4). It follows from (b) and the first condition of (3.1) that
i.e., it holds that . In view of the proof of Theorem 2.1, it holds that .
Similarly as in the proof of Theorem 2.1, it follows from (2.4), (2.7), (2.8), (2.9), and (2.10) that there exists a real number such that
for any .
By the same argument presented in the proof of Theorem 2.1 with substitution of (instead of ) for , it follows from (b) that
for all . In view of (b), there exists a constant such that
for all .
Moreover, by using (3.2), (3.4), and (3.5), we get
for any . (That is, the radius of convergence of power series is at least .)
Since satisfies the recurrence relation (2.1), according to Theorem 2.1 and (3.4), can be written as
for all , where is a solution of the simple harmonic oscillator equation (1.1).
By the same argument as in the proof of Theorem 2.1 with substitution of for , since satisfies the relation (2.1), it is easy to see that is a particular solution of (3.4). Hence, it follows from (3.3), (3.6), and (3.7) that
for all . □
We remark that Theorem 3.1 is true whether the ‘radius’ of the domain interval of y is larger than one or not, while [, Theorem 3.1] holds only when the function y has the domain whose ‘radius’ is not larger than one.
4 An example
Let ε, ω, and be positive real numbers satisfying
Moreover, we set .
We will show that if for all , then with the constant . We know that the sine function can be expressed by the power series
where we set
for every . Hence, y satisfies the condition (a).
If we define
for any , then we have
Furthermore, by considering and by a long and tedious calculation, we get
for all . Therefore, we conclude that
for all , i.e., (b) is satisfied with and , and hence .
Now, we see that
where . Furthermore, it follows from (4.1) that
for all .
We remark that we can take zeros as the values of and in (2.4) without changing the validity of Theorem 2.1. If we set , , and , then it follows from (2.10) that
for any δ with and for all . Similarly, we have
for all δ satisfying and for all . Hence, we get
for all δ satisfying and for all , i.e., it holds that
for all .
According to Theorem 3.1, there exists a solution of the homogeneous differential equation (1.1) such that
for all .
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This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2011-0004919).
The author declares that he has no competing interests.
The author declares that this paper is his original paper.
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Cite this article
Jung, SM. An approximation property of simple harmonic functions. J Inequal Appl 2013, 3 (2013). https://doi.org/10.1186/1029-242X-2013-3
- Differential Equation
- Power Series
- Recurrence Relation
- Differentiable Function
- Natural Phenomenon