Abstract
We will prove the generalized Hyers-Ulam stability of the (inhomogeneous) diffusion equation with a source, , for a class of scalar functions with continuous second partial derivatives.
1. Introduction
The stability problem for functional equations or differential equations started with the well-known question of Ulam [1]: Under what conditions does there exist an additive function near an approximately additive function? In 1941, Hyers [2] gave an affirmative answer to the question of Ulam 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 condition is satisfied 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 of these terminologies, refer to [1–9].
To the best of our knowledge, Obloza was the first author who investigated the Hyers-Ulam stability of differential equations (see [10, 11]): Assume that are continuous functions with . Suppose is an arbitrary positive real number. Obloza proved that there exists a constant such that for all whenever a differentiable function satisfies 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]).
Prástaro and Rassias seem to be the first authors who investigated the Hyers-Ulam stability of partial differential equations (see [13]). Thereafter, Jung and Lee [14] 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, Hegyi and Jung proved the generalized Hyers-Ulam stability of the diffusion equation on the restricted domain or with an initial condition (see [15, 16]).
In this paper, applying ideas from [15, 17], we investigate the generalized Hyers-Ulam stability of the (inhomogeneous) diffusion equation with a sourcefor and , where is a positive constant, , and is a positive integer.
The main advantages of this present paper over the previous works [15, 16] are that this paper deals with the inhomogeneous diffusion equation with a source and it describes the behavior of approximate solutions of diffusion equation in the vicinity of origin (roughly speaking, an approximate solution is a solution to a perturbed equation), while the previous works deal with domains not including the vicinity of origin or the homogeneous diffusion equation (without source term).
2. Preliminaries
If is a solution to the diffusion equation (4) with and is a positive constant, then the dilated function satisfies the equality, , for all and . When the source term satisfies the additional condition is also a solution to (4) with . This property is called the invariance under dilation. Hence, it is worth searching for approximate solutions to (4), which are scalar functions of the form where is a real parameter which will be determined later and is a twice continuously differentiable function. That is, depends on and primarily through the term . We note that the intention of inclusion of the factor in the above formula is to simplify our formulations later.
Throughout this paper, let be a fixed positive integer if there is no specification. Each point in is expressed as , where denotes the th coordinate of . Moreover, denotes the Euclidean distance of from the origin; i.e.,
Based on this argument, we define where is a positive integer and is a parameter.
The proof of the following lemma runs in the usual and routine way. Hence, we omit the proof.
Lemma 1. If a function belongs to and a twice continuously differentiable function is correspondingly given by then for all , , and for all obeying the relation .
Let us define the second-order differential operator bywhere and denote the set of all continuous real-valued functions and the set of all twice continuously differentiable real-valued functions defined on , respectively.
We now try to decompose the differential operator into differential operators and of first order such thatfor all , where we defineThen we have
Comparing both (12) and (14), we obtain From the last system of equations, we get a Riccati equationone of whose solutions has the form , where is a real constant: If we put in the Riccati equation (16), then we haveComparing (16) with (17) and considering that is a constant, we conclude that for all integers . (Even if is not defined for , we can also verify the truth of the formulas for and for by a direct calculation.)
Using this particular solution and in view of [18, § 1.2.1], the general solution of the Riccati equation (16) with is given by where is a nonnegative fixed real number, is a constant, and we set for the particular solution .
Lemma 2. Let be a positive integer. Then for all , where is defined in (19).
3. Main Results
Before starting with our main theorem, we modify the theorem ([19, Theorem 1]) into a more suitable form for practical applications and we will apply this modified version to the proof of our main theorem (cf. [20, Theorem 2.2]). Indeed, the hypotheses of the original theorem [19, Theorem 1] were formulated with instead of which impose a constraint on its usability. The proof of Theorem 3 exactly follows the lines of the proof of [19, Theorem 1]. Hence, we omit the proof.
Theorem 3 ([19, Theorem 1, Remark 3]). Assume that is a real Banach space and is an open interval for arbitrary constants with . Let and be continuous functions such that there exists a constant with the following properties: (i) exists for each . (ii) exists for any . Moreover, assume that is a function such that (iii) exists. If a continuously differentiable function satisfies the differential inequality for all , then there exists a unique continuously differentiable function such that for all and for all .
The following theorem is the main result of this paper which deals with the generalized Hyers-Ulam stability of the diffusion equation (4) when is an integer larger than .
Theorem 4. Let be an integer and assume that are functions satisfying the conditionsandSuppose is a function for which there exists a continuous function such thatfor all and andAssume moreover that the constant in (19) is chosen such thatfor some integer . For any function satisfying the inequalityfor all and , there exists a solution of the diffusion equation (4) such that andfor all and .
Proof. Since , it follows from Lemma 1 with , (25), and (28) that for all and , where is the twice continuously differentiable function given in the definition of and we set . In view of (11), (24), and the last inequality, we have for all . We here note that .
On account of Lemma 2, we further have for all , where is defined in (19). If we define a continuously differentiable function by , then it follows from the last inequality thatfor all , where is given in (19) with a positive real constant .
We can now apply Theorem 3 to our inequality (33) by considering the substitutions as we see in Table 1.
Our hypothesis that is an integer not less than implies that It then follows from (19) and (27) thatHence, we have which implies that the condition of Theorem 3 is satisfied.
Moreover, it follows from the last inequality that and, by (26), we get for all , which means that the condition of Theorem 3 is satisfied.
Similarly, it also follows from (23) that by which we conclude that the condition of Theorem 3 is satisfied.
According to Theorem 3 (or [19, Theorem 1]) and (33), there exists a unique continuously differentiable function such thatfor all andfor all . In particular, by [19, Theorem 1], is explicitly given by with some .
By (35), we have for any with . Therefore, since , it follows from (41) that orfor all .
We apply Theorem 3 to our inequality (45) by considering the substitutions as we see in Table 2.
First, in view of (19) and (35), we get and hence for all , by which we see that the condition of Theorem 3 is satisfied.
Further, it follows from the last inequality thatfor any . By (42) and (48), we easily getIt now follows from (26), (48), and (49) that which means that the condition of Theorem 3 is satisfied.
Analogously, it follows from (23) and (48) that by which we conclude that the condition of Theorem 3 is satisfied.
Due to Theorem 3, (45), and (48), there exists a unique continuously differentiable function such thatfor all andfor all . In particular, by [19, Theorem 1], is explicitly given by with some . We remark that is indeed a twice continuously differentiable function.
Now, let us define the twice continuously differentiable function by where we set . Then, and inequality (29) follows directly from (53). By Lemma 1 with and and in place of , , and , respectively, we further have for all , , and for all obeying the relation .
Finally, it follows from Lemma 2, (11), (13), (19), (40), and (52) that for all . Hence, by (25), we have for all and .
If we set in the proof of Theorem 4, i.e., if we set in (19), then we obtain the following corollary by letting in (27) and (29) of Theorem 4.
Corollary 5. Under the same hypotheses and conditions of Theorem 4 except condition (27), there exists a solution of the diffusion equation (4) such that and for all and .
We now introduce a concrete example for Theorem 4 and Corollary 5 in the following corollary.
Corollary 6. Let be an integer and let be positive real constants and let a function belong to . If the function satisfies the inequality for all and , then there exists a solution of the diffusion equation (4) such that and for all and .
Proof. First, we set Then, all the conditions and hypotheses (23), (24), (25), (26), and (28) are fulfilled.
Due to Corollary 5, there exists a solution of the diffusion equation (4) such that and for all and .
4. Discussions and Conclusions
The diffusion equation is sometimes called a heat equation or a continuity equation and it plays an important role in a number of fields of science. For example, the diffusion equation describes the conduction of heat, the signal transmission in communication systems, and diffusion models of chemical diffusion phenomena and it is also connected with Brownian motion in probability theory.
This paper was partially motivated by a previous work [21] in which the generalized Hyers-Ulam stability of the one-dimensional wave equation with a source, was investigated by using the method of characteristic coordinates. On the other hand, we prove in this paper the generalized Hyers-Ulam stability of the -dimensional diffusion equation with a source, , by applying a kind of method for decomposition of differential operators.
The main advantages of this present paper over the existing results [15, 16] are that this paper deals with the (inhomogeneous) diffusion equation with a source and it can describe the behavior of approximate solutions of (inhomogeneous) diffusion equation in the vicinity of origin, while the previous work [15] deals with the case of domain with and , as we see that the domain does not include the vicinity of origin and while the other existing result [16] deals with the generalized Hyers-Ulam stability of the homogeneous diffusion equation with an initial condition (but without source term).
Data Availability
The data used to support the findings of this study are included within the article.
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). This work was supported by 2018 Hongik University Research Fund.