Abstract
We prove the generalized Hyers-Ulam stability of the wave equation with a source, , for a class of real-valued functions with continuous second partial derivatives in and .
1. Introduction
The stability problem for functional equations or (partial) differential equations started with the question of Ulam [1]: Under what conditions does there exist an additive function near an approximately additive function? In 1941, Hyers [2] answered the question of Ulam in the affirmative for the Banach space cases. Indeed, Hyers’ theorem states that the following statement is true for all : if a function satisfies the inequality for all , then there exists an exact additive function such that for all . In that case, the Cauchy additive functional equation, , is said to have (satisfy) the Hyers-Ulam stability.
Assume that is a normed space and is an open interval of . The th-order linear differential equationis said to have (satisfy) the Hyers-Ulam stability provided the following statement is true for all : if a function satisfies the differential inequality for all , then there exists a solution to the differential equation (1) and a continuous function such that for any and .
When the above statement is true even if we replace and by and , where are functions not depending on and explicitly, the corresponding differential equation (1) is said to have (satisfy) the generalized Hyers-Ulam stability. (This type of stability is sometimes called the Hyers-Ulam-Rassias stability.)
These terminologies will also be applied for other differential equations and partial differential equations. For more detailed definitions, we refer the reader to [1–9].
To the best of our knowledge, Obłoza was the first author who investigated the Hyers-Ulam stability of differential equations (see [10, 11]): assume that are continuous functions with and is an arbitrary positive real number. Obłoza’s theorem states that there exists a constant such that for all whenever a differentiable function satisfies the inequality for all and a function satisfies for all and for some . Since then, a number of mathematicians have dealt with this subject (see [3, 12, 13]).
Prástaro and Rassias are the first authors who investigated the Hyers-Ulam stability of partial differential equations (see [14]). Thereafter, the first author [15], together with Lee, proved the Hyers-Ulam stability of the first-order linear partial differential equation of the form, , where and are constants with . As a further step, the first author proved the generalized Hyers-Ulam stability of the wave equation without source (see [16, 17]).
One of typical examples of hyperbolic partial differential equations is the wave equation with a spatial variable and a time variable ,where is a constant, whose solution is a scalar function describing the propagation of a wave at a speed in the spatial direction.
In this paper, applying ideas from [16, 18], we investigate the generalized Hyers-Ulam stability of the wave equation (3) with a source, where and with . The main advantages of this present paper over the previous papers [16, 17] are that this paper deals with the wave equation with a source and it describes the behavior of approximate solutions of wave equation in the vicinity of origin while the previous one [17] can only deal with domains excluding the vicinity of origin. (Roughly speaking, a solution to a perturbed equation is called an approximate solution.)
2. Main Results
We know that if we introduce the characteristic coordinatesthen the wave equation, , is transformed into , which seems to be handled easily.
Given real constants and with , we define We note that the map , where and , is a one-to-one correspondence from onto (see Figure 1).
Theorem 1. Assume that and are continuous functions with the propertiesIf a function has continuous second partial derivatives and satisfies the inequalityfor all , then there exists a function with continuous second partial derivatives such that is a solution to the wave equation (3) andfor all , where is the interior of the parallelogram having the points as its vertices.
Proof. We introduce the characteristic coordinates (4) and we setfor all and corresponding with the relations in (4).
By the chain rule, we getand hence,for all and corresponding with the relations in (4).
It then follows from (7) and (10) that or or for any .
Considering the conditions in (6) and Figure 2, we can integrate each term of the last inequality from to with respect to the first variable and then we integrate each term of the resulting inequality from to with respect to the second variable to obtain for any .
If we define the function by then we havefor all . Moreover, we getWe now set and, analogously to (11), we compute the partial derivatives:In view of (10), (12), (19), and (20), we get for all , that is, is a solution to wave equation (3).
We compute the Jacobian determinant By (10) and (18), we obtain for all (see Figure 2).
Remark 2. In general, it is somewhat tedious to estimate the upper bound of inequality (8). However, in view of (10) and (18), we can compute the upper bound less tediously: for all .
When in Theorem 1, . In that case, by Theorem 1 and Remark 2, we have the following corollary.
Corollary 3. Assume that and are continuous functions satisfying the conditions If a function has continuous second partial derivatives and satisfies the inequality for all , then there exists a function with continuous second partial derivatives such that is a solution to the wave equation (3) and for all .
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.
Authors’ Contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Acknowledgments
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (no. 2016R1D1A1B03931061).