Journal of Function Spaces
Volume 2018 (2018), Article ID 1216901, 8 pages
https://doi.org/10.1155/2018/1216901

Research Article

Stability of the Diffusion Equation with a Source

Soon-Mo Jung¹ and Seungwook Min²

¹Mathematics Section, College of Science and Technology, Hongik University, 30016 Sejong, Republic of Korea
²Division of Computer Science, Sangmyung University, 03016 Seoul, Republic of Korea

Correspondence should be addressed to Soon-Mo Jung; smjung@hongik.ac.kr

Received 13 May 2018; Accepted 11 July 2018; Published 1 August 2018

Academic Editor: Shanhe Wu

Copyright © 2018 Soon-Mo Jung and Seungwook Min. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 equation

is 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 source

for 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 by

where 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 that

for all , where we define

Then we have

Comparing both (12) and (14), we obtain

From the last system of equations, we get a Riccati equation

one of whose solutions has the form , where is a real constant: If we put in the Riccati equation (16), then we have

Comparing (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 conditions

and

Suppose is a function for which there exists a continuous function such that

for all and and

Assume moreover that the constant in (19) is chosen such that

for some integer . For any function satisfying the inequality

for all and , there exists a solution of the diffusion equation (4) such that and

for 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 that

for 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) that

Hence, 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 that

for all and

for 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

or

for 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 that

for any . By (42) and (48), we easily get

It 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 that

for all and

for 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 .

Table 1

Theorem 3

(33)

Table 2

Theorem 3

(45)

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.

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