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Stability of the Diffusion Equation with a Source.

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 [epsilon] [greater than or equal to] 0: If a function f satisfies the inequality [parallel]f(x + y) - f(x) - f(y) [parallel] [less than or equal to] [epsilon] for all x, then there exists an exact additive function F such that [parallel]f(x) - F(x)[parallel] [less than or equal to] [epsilon] for all x. In that case, the Cauchy additive functional equation, f(x + y) = f(x) + f(y), is said to have (satisfy) the Hyers-Ulam stability.

Assume that V is a normed space and I is an open interval of R. The nth order linear differential equation

[a.sub.n] (x) [y.sup.(n)] (x) + [a.sub.n-1] (x) [y.sup.(n-1)] (x) + ... + [a.sub.1] (x) y' (x) + [a.sub.0] (x) u(x) + h (x) = 0 (1)

is said to have (satisfy) the Hyers-Ulam stability provided the following condition is satisfied for all [epsilon] [greater than or equal to] 0: If a function u : I [right arrow] V satisfies the differential inequality

[parallel][a.sub.n] (x) [y.sup.(n)] (x) + [a.sub.n-1] (x) [y.sup.(n-1)] (x) + ... + [a.sub.1] (x) y' (x) + [a.sub.0] (x) u(x) + h (x) [parallel] [less than or equal to] [epsilon]. (2)

for all x [member of] I, then there exists a solution [u.sub.0] : I [right arrow] V to the differential equation (1) and a continuous function K such that [parallel]u(x) - [u.sub.0](x)[parallel] [less than or equal to] K([epsilon]) for any x [member of] I and [lim.sub.[epsilon][right arrow]0] K([epsilon]) = 0.

When the above statement is true even if we replace e and K([epsilon]) by [phi](x) and [PHI](x), where [phi], [PHI] : I [right arrow] [0, [infinity]) are functions not depending on u and u0 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 g, r : (a, b) [right arrow] R are continuous functions with [[integral].sup.b.sub.a] [absolute value of g(x)]dx < [infinity]. Suppose [epsilon] is an arbitrary positive real number. Obloza proved that there exists a constant [delta] > 0 such that [absolute value of y(x) - [y.sub.0](x)] [less than or equal to] [delta] for all x [member of] (a, b) whenever a differentiable function y : (a, b) [right arrow] R satisfies [absolute value of y'(x) + g(x)y(x) - r(x)] [less than or equal to] [epsilon] for all x [member of] (a, b) and a function [y.sub.0] : (a, b) [right arrow] R satisfies [y'.sub.0](x) + g(x) [y.sub.0](x) = r(x) for all x [member of] (a, b) and y([tau]) = [y.sub.0]([tau]) for some r [micro] (a, b). Since then, a number of mathematicians have dealt with this subject (see [3,12]).

Prastaro 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

[au.sub.x] (x, y) + [bu.sub.y] (x, y) + cu (x, y) + d = 0, (3)

where a, b [member of] R and c, d [member of] C are constants with R(c) [not equal to] 0. 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

[u.sub.t] (x, t) - k [DELTA] u (x, t) = f (x, t) (4)

for x [member of] [R.sup.n] \ {(0, ..., 0)} and t > 0, where k is a positive constant, [DELTA] = [[partial derivative].sup.2]/[partial derivative][x.sup.2.sub.1] + ... +[partial derivative][x.sup.2.sub.n], and n 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 u(x,t) is a solution to the diffusion equation (4) with n = 1 and a is a positive constant, then the dilated function w(x, t) := u([square root of a]x, at) satisfies the equality, [w.sub.t](x, t) - [kw.sub.xx](x, t) = af([square root of a]x, at), for all x > 0 and t > 0. When the source term f(x, t) satisfies the additional condition

af ([square root of a]x, at) = f (x, t), (5)

w(x, t) is also a solution to (4) with n = 1. 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

u(x, t) = 1/[(4kt).sup.[alpha]] - v ([absolute value of x]/[square root of 4kt]), (6)

where [alpha] is a real parameter which will be determined later and v is a twice continuously differentiable function. That is, u(x, t) depends on x and t primarily through the term [absolute value of x]/ [absolute value of 4kt]. We note that the intention of inclusion of the factor 1/[absolute value of 4k] in the above formula is to simplify our formulations later.

Throughout this paper, let n be a fixed positive integer if there is no specification. Each point x in [R.sup.n] is expressed as x = ([x.sub.1], ..., [x.sub.i], ..., [x.sub.n]), where [x.sub.i] denotes the ith coordinate of x. Moreover, [absolute value of x] denotes the Euclidean distance of x from the origin; i.e.,

[absolute value of x] = [square root of [x.sup.2.sub.1] + ... + [x.sup.2.sub.1] + ... + [x.sup.2.sub.n]. (7)

Based on this argument, we define

[U.sup.n.sub.[alpha]] := {u : R \ {(0, ..., 0)} x (0, [infinity]) R]

there exists a twice continuously differentiable function v :

[mathematical expression not reproducible], (8)

where n is a positive integer and a 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 u belongs to [U.sup.n.sub.[alpha]] and a twice continuously differentiable function v : (0, [infinity]) [right arrow] R is

correspondingly given by

u(x, t) = v 1/[(4kt).sup.[alpha]] v ([absolute value of x]/[square root of 4kt), (9)

then

[mathematical expression not reproducible] (10)

for all r [member of] [R.sup.n] \ {(0, ..., 0)}, t > 0, and for all r > 0 obeying the relation r = [absolute value of x]/[square root of 4kt].

Let us define the second-order differential operator [L.sup.2.sub.[alpha]]: [C.sup.2](0, [infinity]) [right arrow] C(0, [infinity]) by

[L.sup.2.sub.[alpha]] v(r) := v" (r) + (2r + n - 1/r) v'(r) + 4[alpha]v(r), (11)

where C(0, [infinity]) and [C.sup.2](0, [infinity]) denote the set of all continuous real-valued functions and the set of all twice continuously differentiable real-valued functions defined on (0, [infinity]), respectively.

We now try to decompose the differential operator [L.sup.2.sub.[alpha]] into differential operators [L.sub.A(r)] and [L.sub.B(r)] of first order such that

[L.sup.2.sub.[alpha]]v(r) = ([L.sub.A(r)] [omicron] [L.sub.B(r)]) v (r) (12)

for all v [member of] [C.sup.2](0, [infinity]), where we define

[L.sub.A(r)] v (r) := v' (r) + A (r) v (r),

[L.sub.B(r)] v (r) := v (r) + B(r) v (r). (13)

Then we have

([L.sub.A(r)] [omicron] [L.sub.B(r)]) v (r) = v" (r) + (A (r) + B (r)) v' (r)

+(A (r) + B (r) + B'(r)) v (r). (14)

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

A (r) + B (r) = 2r + n - 1/r

A (r) B (r) + B'(r) = 4[alpha]. (15)

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

B'(r) + (2r + n - 1/r) B(r) - B [(r).sup.2] = 4[alpha], (16)

one of whose solutions has the form (n - 1)[beta]/r, where [beta] [not equal to] 0 is a real constant: If we put [B.sub.p](r) = (n - 1)[beta]/r in the Riccati equation (16), then we have

[mathematical expression not reproducible]. (17)

Comparing (16) with (17) and considering that 4a is a constant, we conclude that

[alpha] = n - 2/2,

[beta] = n - 2/n - 1

[B.sub.p](r) = n - 2/r (18)

for all integers n [greater than or equal to] 2. (Even if [beta] is not defined for n = 1, we can also verify the truth of the formulas for [alpha] and [B.sub.p](r) for n = 1 by a direct calculation.)

Using this particular solution [B.sub.p](r) and in view of [18, [section] 1.2.1], the general solution of the Riccati equation (16) with 4[alpha] = 2n - 4 is given by

[mathematical expression not reproducible], (19)

where [r.sub.0] is a nonnegative fixed real number, [C.sub.n] is a constant, and we set [C.sub.n] = [infinity] for the particular solution [B.sub.p](r) = (n - 2)/r.

Lemma 2. Let n be a positive integer. Then

[L.sup.2.sub.(n-2)/2] v(r) = ([L.sub.2r+(n-1)/r-B(r)] [omicron] [L.sub.B(r)]) v (r) (20)

for all v [member of] [C.sup.2](0, m), where B(r) 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 a instead of a0 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 X is a real Banach space and I = (a, b) is an open interval for arbitrary constants a, b [member of] R [union] {[+ or -][infinity]} with a < b. Let p : I [right arrow] R and q : I [right arrow] X be continuous functions such that there exists a constant a0 e [a, b) with the following properties:

(i) [mathematical expression not reproducible] P(s)ds exists for each tel.

(ii) [mathematical expression not reproducible] p(s)ds}dy exists for any tel.

Moreover, assume that [phi] : I [right arrow] [0, [infinity]) is a function such that

(iii) [mathematical expression not reproducible] p(s)ds]dy exists.

If a continuously differentiable function v : I [right arrow] X satisfies the differential inequality

[parallel] v'(t) + p(t) v(t) + q(t) [parallel] [less than or equal to] [phi] (t) (21)

for all t [member of] l, then there exists a unique continuously differentiable function [v.sub.0] : I [right arrow] X such that [v'.sub.0](t)+p(t)v0(t)+q(t) = 0 for all t [member of] l and

[mathematical expression not reproducible]. (22)

The following theorem is the main result of this paper which deals with the generalized Hyers-Ulam stability of the diffusion equation (4) when n is an integer larger than 2.

Theorem 4. Let n [greater than or equal to] 3 be an integer and assume that [phi], [psi] : (0, [infinity]) [right arrow] [0, [infinity]) are functions satisfying the conditions

[mathematical expression not reproducible] (23)

and

[mathematical expression not reproducible]. (24)

Suppose f : [R.sup.n] \ {(0, ..., 0)] x (0, [infinity]) [right arrow] R is a function for which there exists a continuous function g : (0, [infinity]) [right arrow] R such that

f(x, t) = k/[(4kt).sup.n/2] g([absolute value of x]/[square root of 4kt]) (25)

for all r [member of] [R.sup.n] \ {(0, ..., 0)} and t > 0 and

[mathematical expression not reproducible]. (26)

Assume moreover that the constant [C.sub.n] in (19) is chosen such that

[mathematical expression not reproducible] (27)

for some integer m [greater than or equal to] 2. For any function u [member of] [U.sup.n.sub.n-2)/2] satisfying the inequality

[absolute value of [u.sub.t] (x, t) - k [DELTA] u (x, t) - f (x, t)] [less than or equal to] [phi]([absolute value of x]/[square root of 4kt]) [psi](t) (28)

for all x [member of] [R.sup.n] \ {(0, ..., 0)} and t > 0, there exists a solution u0 : [R.sup.n] \ {(0, ..., 0)} x (0, [infinity]) [right arrow] R of the diffusion equation (4) such that [u.sub.0] [member of] [U.sup.n.sub.n-2)/2] and

[mathematical expression not reproducible]. (29)

Proof. Since u [member of] [U.sup.n.sub.n-2)/2], it follows from Lemma 1 with [alpha] = (n - 2)/2, (25), and (28) that

[mathematical expression not reproducible] (30)

for all r [member of] [R.sup.n] \ {(0, ..., 0)} and t > 0, where v : (0, [infinity]) [right arrow] R is the twice continuously differentiable function given in the definition of [U.sup.n.sub.n-2)/2] and we set r = [absolute value of x]/[square root of 4kt]. In view of (11), (24), and the last inequality, we have

[absolute value of [L.sup.2.sub.(n-2)/2] v(r) + g(r)] = v"(r) + (2r + n - 1/r) v'(r)

+ (2n - 4) v (r) + g(r) | [less than or equal to] c[phi](r) (31)

for all r > 0. We here note that {r : x [member of] [R.sup.n] {(0, ..., 0)}, t > 0} = (0, [infinity]).

On account of Lemma 2, we further have

[absolute value of {[L.sub.2r+(n-1)/r-B(r)] [omicron] [L.sub.B(r)]) v(r) + g(r)] [less than or equal to] c[phi] (r) (32)

for all r > 0, where B(r) is defined in (19). If we define a continuously differentiable function w : (0, [infinity]) [right arrow] R by w(r) := [L.sub.B(r)]v(r) = v'(r) + B(r)v(r), then it follows from the last inequality that

[absolute value of w'(r) + (2r + n - 1/r - B (r)) w (r) + g (r)] [less than or equal to] c[phi](r) (33)

for all r > 0, where B(r) is given in (19) with a positive real constant [r.sub.0].

We can now apply Theorem 3 to our inequality (33) by considering the substitutions as we see in Table 1.

Our hypothesis that n is an integer not less than 3 implies that

[mathematical expression not reproducible]. (34)

It then follows from (19) and (27) that

[mathematical expression not reproducible]. (35)

Hence, we have

[mathematical expression not reproducible], (36)

which implies that the condition (i) of Theorem 3 is satisfied. Moreover, it follows from the last inequality that

[mathematical expression not reproducible] (37)

and, by (26), we get

[mathematical expression not reproducible] (38)

for all r > 0, which means that the condition (ii) of Theorem 3 is satisfied.

Similarly, it also follows from (23) that

[mathematical expression not reproducible], (39)

by which we conclude that the condition (iii) of Theorem 3 is satisfied.

According to Theorem 3 (or [19, Theorem 1]) and (33), there exists a unique continuously differentiable function w0 : (0, [infinity]) [right arrow] R such that

[w'.sub.0] (r) + (2r + n - 1/r B(r)) [w.sub.0](r) + g(r) = 0 (40)

for all r > 0 and

[mathematical expression not reproducible] (41)

for all r > 0. In particular, by [19, Theorem 1], [w.sub.0](r) is explicitly given by

[mathematical expression not reproducible] (42)

with some [sigma] [member of] R.

By (35), we have

-B(s) < - n - 2/s

[[integral].sup.r.sub.y] B(s) ds < ln [(r/y).sup.n-2]

for any r, y > 0 with r [less than or equal to] y. Therefore, since r [less than or equal to] y < [infinity], it follows from (41) that

[mathematical expression not reproducible] (44)

or

[mathematical expression not reproducible] (45)

for all r > 0.

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

[mathematical expression not reproducible] (46)

and hence

[mathematical expression not reproducible] (47)

for all r > 0, by which we see that the condition (i) of

Theorem 3 is satisfied.

Further, it follows from the last inequality that

[mathematical expression not reproducible] (48)

for any r, [r.sub.0] > 0. By (42) and (48), we easily get

[mathematical expression not reproducible] (49)

It now follows from (26), (48), and (49) that

which means that the condition (ii) of Theorem 3 is satisfied.

Analogously, it follows from (23) and (48) that

[mathematical expression not reproducible] (51)

by which we conclude that the condition (Hi) of Theorem 3 is satisfied.

Due to Theorem 3, (45), and (48), there exists a unique continuously differentiable function [v.sub.0] : (0, [infinity]) [right arrow] R such that

[v'.sub.0](r) + B(r) [v.sub.0](r) - [w.sub.0](r) = 0 (52)

for all r > 0 and

[mathematical expression not reproducible] (53)

for all r > 0. In particular, by [19, Theorem 1], [v.sub.0](r) is explicitly given by

[mathematical expression not reproducible] (54)

with some [tau] [member of] R. We remark that [v.sub.0](r) is indeed a twice continuously differentiable function.

Now, let us define the twice continuously differentiable function [u.sub.0]] : [R.sup.n] \ {(0, ..., 0)} x (0, [infinity]) [right arrow] R by

[u.sub.0](x, t) := 1/[(4kt).sup.(n-2)n/2], (55)

where we set r = [absolute value of x]/[square root of 4kt]. Then, [u.sub.0] [member of] [U.sup.n.sub.(n-2)/2] and inequality (29) follows directly from (53). By Lemma 1 with (n-2)/2 and [u.sub.0] and [v.sub.0] in place of [alpha], u, and v, respectively, we further have

[mathematical expression not reproducible] (56)

for all x [member of] [R.sup.n] \ {(0, ..., 0)}, t > 0, and for all r > 0 obeying the relation r = [absolute value of x]/[square root of 4kt].

Finally, it follows from Lemma 2, (11), (13), (19), (40), and (52) that

[mathematical expression not reproducible] (57)

for all r > 0. Hence, by (25), we have

[mathematical expression not reproducible] (58)

for all r [member of] [R.sup.n] \ {(0, ..., 0)} and t > 0.

If we set B(r) = [B.sub.p](r) = (n - 2)/r in the proof of Theorem 4, i.e., if we set [C.sub.n] = [infinity] in (19), then we obtain the following corollary by letting m [right arrow] [infinity] 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 [u.sub.0] : [R.sup.n] {(0, ..., 0)} x (0, [infinity]) [right arrow] R of the diffusion equation (4) such that [u.sub.0] [member of] [U.sup.n.sub.n-2)/2] and

[mathematical expression not reproducible]. (59)

We now introduce a concrete example for Theorem 4 and Corollary 5 in the following corollary.

Corollary 6. Let n [greater than or equal to] 3 be an integer and let c, [[mu].sub.1], [[mu].sub.2], [v.sub.1], [v.sub.2] be positive real constants and let a function u : [R.sup.n] \ {(0, ..., 0)} x (0, [infinity]) [right arrow] R belong to [U.sup.n.sub.n-2)/2]. If the function u satisfies the inequality

[mathematical expression not reproducible] (60)

for all x [member of] [R.sup.n]\{(0, ..., 0)} and t > 0, then there exists a solution [u.sub.0] : [R.sup.n] \ {(0, ..., 0)} x (0, [infinity]) [right arrow] R of the diffusion equation (4) such that [u.sub.0] [member of] [U.sup.n.sub.n-2)/2] and

[mathematical expression not reproducible] (61)

for all r [member of] [R.sup.n] \ {(0, ..., 0)} and t > 0.

Proof. First, we set

[mathematical expression not reproducible], (62)

Then, all the conditions and hypotheses (23), (24), (25), (26), and (28) are fulfilled.

Due to Corollary 5, there exists a solution [u.sub.0] : [R.sup.n] {(0, ..., 0)} x (0, [infinity]) [right arrow] R of the diffusion equation (4) such that [u.sub.0] [member of] [U.sup.n.sub.n-2)/2] and

[mathematical expression not reproducible] (63)

for all r [member of] [R.sup.n] \ {(0, ..., 0)} and t > 0.

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, [u.sub.tt](x,t) - [c.sup.2][u.sub.xx](x, t) = f(x,t) 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 n-dimensional diffusion equation with a source, [u.sub.t](x, t) k [DELTA] u(x, t) = f(x, t), 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 D = {x [member of] [R.sup.n] | a < [absolute value of x] < b} with 0 < a < b [less than or equal to] [infinity] and n [greater than or equal to] 2, as we see that the domain D 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).

https://doi.org/10.1155/2018/1216901

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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[20] D. Popa and I. Ra[section]a, "On the Hyers-Ulam stability of the linear differential equation," Journal of Mathematical Analysis and Applications, vol. 381, no. 2, pp. 530-537, 2011.

[21] Soon-Mo Jung and Seungwook Min, "Stability of the Wave Equation with a Source," Journal of Function Spaces, vol. 2018, Article ID 8274159, 4 pages, 2018.

Soon-Mo Jung (1) and Seungwook Min (2)

(1) Mathematics Section, College of Science and Technology, Hongik University, 30016 Sejong, Republic of Korea

(2) Division 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
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Title Annotation:Research Article
Author:Jung, Soon-Mo; Min, Seungwook
Publication:Journal of Function Spaces
Date:Jan 1, 2018
Words:4602
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