Hindawi Publishing Corporation The Scientific World Journal Volume 2013, Article ID 978754, 3 pages http://dx.doi.org/10.1155/2013/978754
Research Article On the Stability of One-Dimensional Wave Equation Soon-Mo Jung Mathematics Section, College of Science and Technology, Hongik University, Sejong 339-701, Republic of Korea Correspondence should be addressed to Soon-Mo Jung; [email protected] Received 5 August 2013; Accepted 16 September 2013 Academic Editors: K. Ammari, I. Canak, and M. M. Cavalcanti Copyright Β© 2013 Soon-Mo Jung. 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. We prove the generalized Hyers-Ulam stability of the one-dimensional wave equation, π’π‘π‘ = π2 π’π₯π₯ , in a class of twice continuously differentiable functions.
1. Introduction In 1940, Ulam [1] gave a wide ranging talk before the mathematics club of the University of Wisconsin in which he discussed a number of important unsolved problems. Among those was the question concerning the stability of group homomorphisms: Let πΊ1 be a group and let πΊ2 be a metric group with the metric π(β , β ). Given π > 0, does there exist a πΏ > 0 such that if a function β : πΊ1 β πΊ2 satisfies the inequality π(β(π₯π¦), β(π₯)β(π¦)) < πΏ, for all π₯, π¦ β πΊ1 , then there exists a homomorphism π» : πΊ1 β πΊ2 with π(β(π₯), π»(π₯)) < π, for all π₯ β πΊ1 ? The case of approximately additive functions was solved by Hyers [2] under the assumption that πΊ1 and πΊ2 are Banach spaces. Indeed, he proved that each solution of the inequality βπ(π₯ + π¦) β π(π₯) β π(π¦)β β€ π, for all π₯ and π¦, can be approximated by an exact solution, say an additive function. In this case, the Cauchy additive functional equation, π(π₯ + π¦) = π(π₯) + π(π¦), is said to have the Hyers-Ulam stability. Rassias [3] attempted to weaken the condition for the bound of the norm of the Cauchy difference as follows: σ΅© σ΅©π σ΅© σ΅©σ΅© π (1) σ΅©σ΅©π (π₯ + π¦) β π (π₯) β π (π¦)σ΅©σ΅©σ΅© β€ π (βπ₯β + σ΅©σ΅©σ΅©π¦σ΅©σ΅©σ΅© ) and proved Hyersβ theorem. That is, Rassias proved the generalized Hyers-Ulam stability (or Hyers-Ulam-Rassias stability) of the Cauchy additive functional equation. Since then, the stability of several functional equations has been extensively investigated [4β9].
The terminologies, the generalized Hyers-Ulam stability, and the Hyers-Ulam stability can also be applied to the case of other functional equations, differential equations, and various integral equations. Given a real number π > 0, the partial differential equation π’π‘π‘ (π₯, π‘) β π2 π’π₯π₯ (π₯, π‘) = 0
(2)
is called the (one-dimensional) wave equation, where π’π‘π‘ (π₯, π‘) and π’π₯π₯ (π₯, π‘) denote the second time derivative and the second space derivative of π’(π₯, π‘), respectively. Let π : R Γ R β [0, β) be a function. If, for each twice continuously differentiable function π’ : R Γ R β C satisfying σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨π’π‘π‘ (π₯, π‘) β π2 π’π₯π₯ (π₯, π‘)σ΅¨σ΅¨σ΅¨ β€ π (π₯, π‘) σ΅¨ σ΅¨
(π₯, π‘ β R) ,
(3)
there exist a solution π’0 : R Γ R β C of the (one-dimensional) wave equation (2) and a function Ξ¦ : RΓR β [0, β) such that σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨π’ (π₯, π‘) β π’0 (π₯, π‘)σ΅¨σ΅¨σ΅¨ β€ Ξ¦ (π₯, π‘)
(π₯, π‘ β R) ,
(4)
where Ξ¦(π₯, π‘) is independent of π’(π₯, π‘) and π’0 (π₯, π‘), then we say that the wave equation (2) has the generalized HyersUlam stability (or the Hyers-Ulam-Rassias stability). In this paper, using an idea from [10], we prove the generalized Hyers-Ulam stability of the (one-dimensional) wave equation (2).
2
The Scientific World Journal for any π₯, π‘ β R. Thus, it follows from inequality (6) that
2. Generalized Hyers-Ulam Stability In the following theorem, using the dβAlembert method (method of characteristic coordinates), we prove the generalized Hyers-Ulam stability of the (one-dimensional) wave equation (2). Theorem 1. Let a function π : R Γ R β [0, β) be given such that the double integral π
π
0
0
β« β« π(
π+] πβ] ) πππ] , 2 2π
(5)
1 π€+π§ π€βπ§ σ΅¨ σ΅¨σ΅¨ , ), σ΅¨σ΅¨Vπ€π§ (π€, π§)σ΅¨σ΅¨σ΅¨ β€ 2 π ( 4π 2 2π for any π€, π§ β R. Therefore, we get β
π+] πβ] 1 π§ π€ , ) πππ] β« β« π( 4π2 0 0 2 2π
σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨π’ (π₯, π‘) β π’0 (π₯, π‘)σ΅¨σ΅¨σ΅¨ β€
1 4π2
(7)
for all π₯, π‘ β R.
V (π€, π§) := π’ (
π€+π§ π€βπ§ , ). 2 2π
(8)
If we set π€ = π₯ + ππ‘ and π§ = π₯ β ππ‘, then we have π’(π₯, π‘) = V(π€, π§) and ππ€ ππ§ + Vπ§ (π€, π§) π’π‘ (π₯, π‘) = Vπ€ (π€, π§) ππ‘ ππ‘
(12)
π+] πβ] 1 π§ π€ , ) πππ] β« β« π( 2 4π 0 0 2 2π
or equivalently
ππ€ ππ§ β πVπ§π§ (π€, π§) ππ‘ ππ‘
= π2 Vπ€π€ (π€, π§) β 2π2 Vπ€π§ (π€, π§) + π2 Vπ§π§ (π€, π§) , ππ€ ππ§ π’π₯ (π₯, π‘) = Vπ€ (π€, π§) + Vπ§ (π€, π§) ππ₯ ππ₯
(9)
1 4π2
σ΅¨σ΅¨ π§ π€ σ΅¨σ΅¨ π+] πβ] σ΅¨σ΅¨ σ΅¨ , ) πππ]σ΅¨σ΅¨σ΅¨ , σ΅¨σ΅¨β« β« π ( σ΅¨σ΅¨ 2 2π σ΅¨σ΅¨ 0 0
V (π€, π§) = π’ (
π€+π§ π€βπ§ , ), 2 2π
(13)
π§ π§ V (0, π§) = π’ ( , β ) , 2 2π
π€ π€ V (π€, 0) = π’ ( , ) , 2 2π V (0, 0) = π’ (0, 0) . (14)
for all π€, π§ β R. If we set π€ = π₯ + ππ‘ and π§ = π₯ β ππ‘ in the last inequality, then we obtain σ΅¨σ΅¨ σ΅¨ σ΅¨σ΅¨π’ (π₯, π‘) β π’0 (π₯, π‘)σ΅¨σ΅¨σ΅¨ σ΅¨σ΅¨ π+] πβ] 1 σ΅¨σ΅¨σ΅¨ π₯βππ‘ π₯+ππ‘ σ΅¨ β€ 2 σ΅¨σ΅¨σ΅¨β« π( , ) πππ]σ΅¨σ΅¨σ΅¨ , β« σ΅¨σ΅¨ 4π σ΅¨σ΅¨ 0 2 2π 0
= Vπ€ (π€, π§) + Vπ§ (π€, π§) , ππ€ ππ§ + Vπ€π§ (π€, π§) ππ₯ ππ₯ ππ€ ππ§ + Vπ§π§ (π€, π§) ππ₯ ππ₯
(16)
for all π₯, π‘ β R, where we set
= Vπ€π€ (π€, π§) + 2Vπ€π§ (π€, π§) + Vπ§π§ (π€, π§) ,
π’0 (π₯, π‘) := π’ (
for all π₯, π‘ β R. Hence, we have π’π‘π‘ (π₯, π‘) β π2 π’π₯π₯ (π₯, π‘) = β4π2 Vπ€π§ (π€, π§) ,
β€
σ΅¨σ΅¨ π€ + π§ π€ β π§ σ΅¨σ΅¨ π€ π€ π§ π§ σ΅¨σ΅¨π’ ( , ) β π’ ( , ) β π’ ( , β ) + π’ (0, 0)σ΅¨σ΅¨σ΅¨σ΅¨ σ΅¨σ΅¨ 2 2π 2 2π 2 2π σ΅¨ σ΅¨ σ΅¨ σ΅¨ π§ π€ σ΅¨σ΅¨ π+] πβ] 1 σ΅¨σ΅¨ β€ 2 σ΅¨σ΅¨σ΅¨β« β« π ( , ) πππ]σ΅¨σ΅¨σ΅¨ , σ΅¨ σ΅¨σ΅¨ 4π σ΅¨ 0 0 2 2π (15)
ππ€ ππ§ + πVπ€π§ (π€, π§) ππ‘ ππ‘
+ Vπ§π€ (π€, π§)
0
Hence, it follows from (13) and the last equalities that
= πVπ€ (π€, π§) β πVπ§ (π€, π§) ,
π’π₯π₯ (π₯, π‘) = Vπ€π€ (π€, π§)
0
for all π€, π§ β R. On account of (8), we get
Proof. Let us define a function V : R Γ R β C by
β πVπ§π€ (π€, π§)
π€
|V (π€, π§) β V (π€, 0) β V (0, π§) + V (0, 0)|
σ΅¨σ΅¨ π₯βππ‘ π₯+ππ‘ σ΅¨σ΅¨ π+] πβ] σ΅¨σ΅¨ σ΅¨ π( , ) πππ]σ΅¨σ΅¨σ΅¨ β« σ΅¨σ΅¨β« σ΅¨σ΅¨ 0 σ΅¨σ΅¨ 2 2π 0
π’π‘π‘ (π₯, π‘) = πVπ€π€ (π€, π§)
β€
(6)
for all π₯, π‘ β R, then there exists a solution π’0 : R Γ R β C of the wave equation (2) which satisfies
π§
β€ β« β« Vπ€π§ (π, ]) πππ]
exists for all π, π β R. If a twice continuously differentiable function π’ : R Γ R β C satisfies the inequality σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨π’π‘π‘ (π₯, π‘) β π2 π’π₯π₯ (π₯, π‘)σ΅¨σ΅¨σ΅¨ β€ π (π₯, π‘) σ΅¨ σ΅¨
(11)
(10)
π₯ π π₯ π‘ + π‘, + ) 2 2 2π 2
π₯ π‘ π₯ π + π’ ( β π‘, β + ) β π’ (0, 0) . 2 2 2π 2
(17)
The Scientific World Journal
3
By some tedious calculations, we get
Proof. Since σ΅¨σ΅¨ π π σ΅¨σ΅¨ π+] πβ] σ΅¨σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨β« β« π ( σ΅¨σ΅¨ , ) πππ] σ΅¨σ΅¨ 0 0 σ΅¨σ΅¨ 2 2π σ΅¨ σ΅¨
π π’ (π₯, π‘) ππ‘ 0 π π₯ ππ‘ π₯ π₯ ππ‘ π₯ π‘ 1 π‘ = π’π₯ ( + , + ) + π’π‘ ( + , + ) 2 2 2 2π 2 2 2 2 2π 2 π₯ ππ‘ π₯ π₯ ππ‘ π₯ π π‘ 1 π‘ β π’π₯ ( β , β + ) + π’π‘ ( β , β + ) , 2 2 2 2π 2 2 2 2 2π 2 π2 π’ (π₯, π‘) ππ‘2 0 =
π2 π₯ ππ‘ π₯ π₯ ππ‘ π₯ π‘ π π‘ π’π₯π₯ ( + , + ) + π’π₯π‘ ( + , + ) 4 2 2 2π 2 2 2 2 2π 2
1 π₯ ππ‘ π₯ π₯ ππ‘ π₯ π‘ π2 π‘ + π’π‘π‘ ( + , + ) + π’π₯π₯ ( β , β + ) 4 2 2 2π 2 4 2 2 2π 2 π₯ ππ‘ π₯ π₯ ππ‘ π₯ π π‘ 1 π‘ β π’π₯π‘ ( β , β + ) + π’π‘π‘ ( β , β + ) , 2 2 2 2π 2 4 2 2 2π 2 π π’ (π₯, π‘) ππ₯ 0
π₯ ππ‘ π₯ π₯ ππ‘ π₯ 1 π‘ 1 π‘ + π’π₯ ( β , β + ) β π’π‘ ( β , β + ) , 2 2 2 2π 2 2π 2 2 2π 2 π2 π’ (π₯, π‘) ππ₯2 0 1 π₯ ππ‘ π₯ π₯ ππ‘ π₯ π‘ 1 π‘ = π’π₯π₯ ( + , + ) + π’π₯π‘ ( + , + ) 4 2 2 2π 2 2π 2 2 2π 2 π₯ ππ‘ π₯ π₯ ππ‘ π₯ 1 π‘ 1 π‘ π’ ( + , + ) + π’π₯π₯ ( β , β + ) 4π2 π‘π‘ 2 2 2π 2 4 2 2 2π 2
β
π₯ ππ‘ π₯ π₯ ππ‘ π₯ 1 π‘ 1 π‘ π’π₯π‘ ( β , β + )+ 2 π’π‘π‘ ( β , β + ) , 2π 2 2 2π 2 4π 2 2 2π 2 (18)
for all π₯, π‘ β R. Hence, we know that π2 π2 (19) π’0 (π₯, π‘) β π2 2 π’0 (π₯, π‘) = 0, 2 ππ‘ ππ₯ for any π₯, π‘ β R; that is, π’0 (π₯, π‘) is a solution of the wave equation (2). Corollary 2. Given a constant πΌ > 0, let a function π : R Γ R β [0, β) be given as 2 2
(20)
If a twice continuously differentiable function π’ : R Γ R β C satisfies inequality (6), for all π₯, π‘ β R, then there exists a solution π’0 : R Γ R β C of the wave equation (2) which satisfies σ΅¨ π₯ β ππ‘ π₯ + ππ‘ σ΅¨σ΅¨σ΅¨σ΅¨ σ΅¨ πΌπ σ΅¨σ΅¨ σ΅¨σ΅¨ ) erf ( )σ΅¨ , σ΅¨σ΅¨π’ (π₯, π‘) β π’0 (π₯, π‘)σ΅¨σ΅¨σ΅¨ β€ 2 σ΅¨σ΅¨σ΅¨erf ( β2 β2 σ΅¨σ΅¨σ΅¨ 8π σ΅¨σ΅¨ (21) for all π₯, π‘ β R.
Conflict of Interests
Acknowledgments This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (no. 2013R1A1A2005557).
References
+
2
for all π, π β R, in view of Theorem 1, we conclude that the statement of this corollary is true.
The author declares that there is no conflict of interests regarding the publication of this paper.
1 π₯ ππ‘ π₯ π₯ ππ‘ π₯ π‘ 1 π‘ = π’π₯ ( + , + ) + π’π‘ ( + , + ) 2 2 2 2π 2 2π 2 2 2π 2
π (π₯, π‘) = πΌπβπ₯ βπ π‘ .
σ΅¨σ΅¨ π π σ΅¨σ΅¨ 2 2 σ΅¨ σ΅¨ = σ΅¨σ΅¨σ΅¨σ΅¨β« β« πΌπβπ /2β] /2 πππ]σ΅¨σ΅¨σ΅¨σ΅¨ σ΅¨σ΅¨ 0 0 σ΅¨σ΅¨ σ΅¨σ΅¨ π 2 σ΅¨σ΅¨ σ΅¨σ΅¨ π 2 σ΅¨σ΅¨ σ΅¨ σ΅¨σ΅¨ σ΅¨ = πΌ σ΅¨σ΅¨σ΅¨σ΅¨β« πβ] /2 π]σ΅¨σ΅¨σ΅¨σ΅¨ σ΅¨σ΅¨σ΅¨β« πβπ /2 ππσ΅¨σ΅¨σ΅¨ (22) σ΅¨σ΅¨ σ΅¨σ΅¨ 0 σ΅¨σ΅¨ σ΅¨σ΅¨ 0 σ΅¨σ΅¨ σ΅¨ σ΅¨σ΅¨ π σ΅¨σ΅¨σ΅¨σ΅¨ 2 π/β2 β]2 2 π/β2 βπ2 = 2πΌ σ΅¨σ΅¨( π π]) ( π ππ)σ΅¨σ΅¨σ΅¨ β« β« σ΅¨σ΅¨ βπ 0 4 σ΅¨σ΅¨σ΅¨ βπ 0 σ΅¨ σ΅¨ σ΅¨ σ΅¨ σ΅¨ πΌπ σ΅¨σ΅¨ π π σ΅¨ = σ΅¨erf ( ) erf ( )σ΅¨σ΅¨σ΅¨ < β, β2 β2 σ΅¨σ΅¨ 2 σ΅¨σ΅¨σ΅¨
[1] S. M. Ulam, A Collection of Mathematical Problems, Interscience, New York, NY, USA, 1960. [2] D. H. Hyers, βOn the stability of the linear functional equation,β Proceedings of the National Academy of Sciences of USA, vol. 27, pp. 222β224, 1941. [3] T. M. Rassias, βOn the stability of the linear mapping in Banach spaces,β Proceedings of the American Mathematical Society, vol. 72, pp. 297β300, 1978. [4] G. L. Forti, βHyers-Ulam stability of functional equations in several variables,β Aequationes Mathematicae, vol. 50, no. 1-2, pp. 143β190, 1995. [5] P. GΛavrutΛa, βA generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings,β Journal of Mathematical Analysis and Applications, vol. 184, no. 3, pp. 431β436, 1994. [6] D. H. Hyers, G. Isac, and T. M. Rassias, Stability of Functional Equations of Several Variables, Birkhauser, Boston, Mass, USA, 1998. [7] D. H. Hyers and T. M. Rassias, βApproximate homomorphisms,β Aequationes Mathematicae, vol. 44, no. 2-3, pp. 125β153, 1992. [8] S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, vol. 48 of Springer Optimization and Its Applications, Springer, New York, NY, USA, 2011. [9] T. M. Rassias, βOn the stability of functional equations and a problem of Ulam,β Acta Applicandae Mathematicae, vol. 62, no. 1, pp. 23β130, 2000. [10] B. Hegyi and S.-M. Jung, βOn the stability of Laplaceβs equation,β Applied Mathematics Letters, vol. 26, pp. 549β552, 2013.
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Research Article On the Stability of One-Dimensional Wave Equation Soon-Mo Jung Mathematics Section, College of Science and Technology, Hongik University, Sejong 339-701, Republic of Korea Correspondence should be addressed to Soon-Mo Jung; [email protected] Received 5 August 2013; Accepted 16 September 2013 Academic Editors: K. Ammari, I. Canak, and M. M. Cavalcanti Copyright Β© 2013 Soon-Mo Jung. 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. We prove the generalized Hyers-Ulam stability of the one-dimensional wave equation, π’π‘π‘ = π2 π’π₯π₯ , in a class of twice continuously differentiable functions.
1. Introduction In 1940, Ulam [1] gave a wide ranging talk before the mathematics club of the University of Wisconsin in which he discussed a number of important unsolved problems. Among those was the question concerning the stability of group homomorphisms: Let πΊ1 be a group and let πΊ2 be a metric group with the metric π(β , β ). Given π > 0, does there exist a πΏ > 0 such that if a function β : πΊ1 β πΊ2 satisfies the inequality π(β(π₯π¦), β(π₯)β(π¦)) < πΏ, for all π₯, π¦ β πΊ1 , then there exists a homomorphism π» : πΊ1 β πΊ2 with π(β(π₯), π»(π₯)) < π, for all π₯ β πΊ1 ? The case of approximately additive functions was solved by Hyers [2] under the assumption that πΊ1 and πΊ2 are Banach spaces. Indeed, he proved that each solution of the inequality βπ(π₯ + π¦) β π(π₯) β π(π¦)β β€ π, for all π₯ and π¦, can be approximated by an exact solution, say an additive function. In this case, the Cauchy additive functional equation, π(π₯ + π¦) = π(π₯) + π(π¦), is said to have the Hyers-Ulam stability. Rassias [3] attempted to weaken the condition for the bound of the norm of the Cauchy difference as follows: σ΅© σ΅©π σ΅© σ΅©σ΅© π (1) σ΅©σ΅©π (π₯ + π¦) β π (π₯) β π (π¦)σ΅©σ΅©σ΅© β€ π (βπ₯β + σ΅©σ΅©σ΅©π¦σ΅©σ΅©σ΅© ) and proved Hyersβ theorem. That is, Rassias proved the generalized Hyers-Ulam stability (or Hyers-Ulam-Rassias stability) of the Cauchy additive functional equation. Since then, the stability of several functional equations has been extensively investigated [4β9].
The terminologies, the generalized Hyers-Ulam stability, and the Hyers-Ulam stability can also be applied to the case of other functional equations, differential equations, and various integral equations. Given a real number π > 0, the partial differential equation π’π‘π‘ (π₯, π‘) β π2 π’π₯π₯ (π₯, π‘) = 0
(2)
is called the (one-dimensional) wave equation, where π’π‘π‘ (π₯, π‘) and π’π₯π₯ (π₯, π‘) denote the second time derivative and the second space derivative of π’(π₯, π‘), respectively. Let π : R Γ R β [0, β) be a function. If, for each twice continuously differentiable function π’ : R Γ R β C satisfying σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨π’π‘π‘ (π₯, π‘) β π2 π’π₯π₯ (π₯, π‘)σ΅¨σ΅¨σ΅¨ β€ π (π₯, π‘) σ΅¨ σ΅¨
(π₯, π‘ β R) ,
(3)
there exist a solution π’0 : R Γ R β C of the (one-dimensional) wave equation (2) and a function Ξ¦ : RΓR β [0, β) such that σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨π’ (π₯, π‘) β π’0 (π₯, π‘)σ΅¨σ΅¨σ΅¨ β€ Ξ¦ (π₯, π‘)
(π₯, π‘ β R) ,
(4)
where Ξ¦(π₯, π‘) is independent of π’(π₯, π‘) and π’0 (π₯, π‘), then we say that the wave equation (2) has the generalized HyersUlam stability (or the Hyers-Ulam-Rassias stability). In this paper, using an idea from [10], we prove the generalized Hyers-Ulam stability of the (one-dimensional) wave equation (2).
2
The Scientific World Journal for any π₯, π‘ β R. Thus, it follows from inequality (6) that
2. Generalized Hyers-Ulam Stability In the following theorem, using the dβAlembert method (method of characteristic coordinates), we prove the generalized Hyers-Ulam stability of the (one-dimensional) wave equation (2). Theorem 1. Let a function π : R Γ R β [0, β) be given such that the double integral π
π
0
0
β« β« π(
π+] πβ] ) πππ] , 2 2π
(5)
1 π€+π§ π€βπ§ σ΅¨ σ΅¨σ΅¨ , ), σ΅¨σ΅¨Vπ€π§ (π€, π§)σ΅¨σ΅¨σ΅¨ β€ 2 π ( 4π 2 2π for any π€, π§ β R. Therefore, we get β
π+] πβ] 1 π§ π€ , ) πππ] β« β« π( 4π2 0 0 2 2π
σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨π’ (π₯, π‘) β π’0 (π₯, π‘)σ΅¨σ΅¨σ΅¨ β€
1 4π2
(7)
for all π₯, π‘ β R.
V (π€, π§) := π’ (
π€+π§ π€βπ§ , ). 2 2π
(8)
If we set π€ = π₯ + ππ‘ and π§ = π₯ β ππ‘, then we have π’(π₯, π‘) = V(π€, π§) and ππ€ ππ§ + Vπ§ (π€, π§) π’π‘ (π₯, π‘) = Vπ€ (π€, π§) ππ‘ ππ‘
(12)
π+] πβ] 1 π§ π€ , ) πππ] β« β« π( 2 4π 0 0 2 2π
or equivalently
ππ€ ππ§ β πVπ§π§ (π€, π§) ππ‘ ππ‘
= π2 Vπ€π€ (π€, π§) β 2π2 Vπ€π§ (π€, π§) + π2 Vπ§π§ (π€, π§) , ππ€ ππ§ π’π₯ (π₯, π‘) = Vπ€ (π€, π§) + Vπ§ (π€, π§) ππ₯ ππ₯
(9)
1 4π2
σ΅¨σ΅¨ π§ π€ σ΅¨σ΅¨ π+] πβ] σ΅¨σ΅¨ σ΅¨ , ) πππ]σ΅¨σ΅¨σ΅¨ , σ΅¨σ΅¨β« β« π ( σ΅¨σ΅¨ 2 2π σ΅¨σ΅¨ 0 0
V (π€, π§) = π’ (
π€+π§ π€βπ§ , ), 2 2π
(13)
π§ π§ V (0, π§) = π’ ( , β ) , 2 2π
π€ π€ V (π€, 0) = π’ ( , ) , 2 2π V (0, 0) = π’ (0, 0) . (14)
for all π€, π§ β R. If we set π€ = π₯ + ππ‘ and π§ = π₯ β ππ‘ in the last inequality, then we obtain σ΅¨σ΅¨ σ΅¨ σ΅¨σ΅¨π’ (π₯, π‘) β π’0 (π₯, π‘)σ΅¨σ΅¨σ΅¨ σ΅¨σ΅¨ π+] πβ] 1 σ΅¨σ΅¨σ΅¨ π₯βππ‘ π₯+ππ‘ σ΅¨ β€ 2 σ΅¨σ΅¨σ΅¨β« π( , ) πππ]σ΅¨σ΅¨σ΅¨ , β« σ΅¨σ΅¨ 4π σ΅¨σ΅¨ 0 2 2π 0
= Vπ€ (π€, π§) + Vπ§ (π€, π§) , ππ€ ππ§ + Vπ€π§ (π€, π§) ππ₯ ππ₯ ππ€ ππ§ + Vπ§π§ (π€, π§) ππ₯ ππ₯
(16)
for all π₯, π‘ β R, where we set
= Vπ€π€ (π€, π§) + 2Vπ€π§ (π€, π§) + Vπ§π§ (π€, π§) ,
π’0 (π₯, π‘) := π’ (
for all π₯, π‘ β R. Hence, we have π’π‘π‘ (π₯, π‘) β π2 π’π₯π₯ (π₯, π‘) = β4π2 Vπ€π§ (π€, π§) ,
β€
σ΅¨σ΅¨ π€ + π§ π€ β π§ σ΅¨σ΅¨ π€ π€ π§ π§ σ΅¨σ΅¨π’ ( , ) β π’ ( , ) β π’ ( , β ) + π’ (0, 0)σ΅¨σ΅¨σ΅¨σ΅¨ σ΅¨σ΅¨ 2 2π 2 2π 2 2π σ΅¨ σ΅¨ σ΅¨ σ΅¨ π§ π€ σ΅¨σ΅¨ π+] πβ] 1 σ΅¨σ΅¨ β€ 2 σ΅¨σ΅¨σ΅¨β« β« π ( , ) πππ]σ΅¨σ΅¨σ΅¨ , σ΅¨ σ΅¨σ΅¨ 4π σ΅¨ 0 0 2 2π (15)
ππ€ ππ§ + πVπ€π§ (π€, π§) ππ‘ ππ‘
+ Vπ§π€ (π€, π§)
0
Hence, it follows from (13) and the last equalities that
= πVπ€ (π€, π§) β πVπ§ (π€, π§) ,
π’π₯π₯ (π₯, π‘) = Vπ€π€ (π€, π§)
0
for all π€, π§ β R. On account of (8), we get
Proof. Let us define a function V : R Γ R β C by
β πVπ§π€ (π€, π§)
π€
|V (π€, π§) β V (π€, 0) β V (0, π§) + V (0, 0)|
σ΅¨σ΅¨ π₯βππ‘ π₯+ππ‘ σ΅¨σ΅¨ π+] πβ] σ΅¨σ΅¨ σ΅¨ π( , ) πππ]σ΅¨σ΅¨σ΅¨ β« σ΅¨σ΅¨β« σ΅¨σ΅¨ 0 σ΅¨σ΅¨ 2 2π 0
π’π‘π‘ (π₯, π‘) = πVπ€π€ (π€, π§)
β€
(6)
for all π₯, π‘ β R, then there exists a solution π’0 : R Γ R β C of the wave equation (2) which satisfies
π§
β€ β« β« Vπ€π§ (π, ]) πππ]
exists for all π, π β R. If a twice continuously differentiable function π’ : R Γ R β C satisfies the inequality σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨π’π‘π‘ (π₯, π‘) β π2 π’π₯π₯ (π₯, π‘)σ΅¨σ΅¨σ΅¨ β€ π (π₯, π‘) σ΅¨ σ΅¨
(11)
(10)
π₯ π π₯ π‘ + π‘, + ) 2 2 2π 2
π₯ π‘ π₯ π + π’ ( β π‘, β + ) β π’ (0, 0) . 2 2 2π 2
(17)
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3
By some tedious calculations, we get
Proof. Since σ΅¨σ΅¨ π π σ΅¨σ΅¨ π+] πβ] σ΅¨σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨β« β« π ( σ΅¨σ΅¨ , ) πππ] σ΅¨σ΅¨ 0 0 σ΅¨σ΅¨ 2 2π σ΅¨ σ΅¨
π π’ (π₯, π‘) ππ‘ 0 π π₯ ππ‘ π₯ π₯ ππ‘ π₯ π‘ 1 π‘ = π’π₯ ( + , + ) + π’π‘ ( + , + ) 2 2 2 2π 2 2 2 2 2π 2 π₯ ππ‘ π₯ π₯ ππ‘ π₯ π π‘ 1 π‘ β π’π₯ ( β , β + ) + π’π‘ ( β , β + ) , 2 2 2 2π 2 2 2 2 2π 2 π2 π’ (π₯, π‘) ππ‘2 0 =
π2 π₯ ππ‘ π₯ π₯ ππ‘ π₯ π‘ π π‘ π’π₯π₯ ( + , + ) + π’π₯π‘ ( + , + ) 4 2 2 2π 2 2 2 2 2π 2
1 π₯ ππ‘ π₯ π₯ ππ‘ π₯ π‘ π2 π‘ + π’π‘π‘ ( + , + ) + π’π₯π₯ ( β , β + ) 4 2 2 2π 2 4 2 2 2π 2 π₯ ππ‘ π₯ π₯ ππ‘ π₯ π π‘ 1 π‘ β π’π₯π‘ ( β , β + ) + π’π‘π‘ ( β , β + ) , 2 2 2 2π 2 4 2 2 2π 2 π π’ (π₯, π‘) ππ₯ 0
π₯ ππ‘ π₯ π₯ ππ‘ π₯ 1 π‘ 1 π‘ + π’π₯ ( β , β + ) β π’π‘ ( β , β + ) , 2 2 2 2π 2 2π 2 2 2π 2 π2 π’ (π₯, π‘) ππ₯2 0 1 π₯ ππ‘ π₯ π₯ ππ‘ π₯ π‘ 1 π‘ = π’π₯π₯ ( + , + ) + π’π₯π‘ ( + , + ) 4 2 2 2π 2 2π 2 2 2π 2 π₯ ππ‘ π₯ π₯ ππ‘ π₯ 1 π‘ 1 π‘ π’ ( + , + ) + π’π₯π₯ ( β , β + ) 4π2 π‘π‘ 2 2 2π 2 4 2 2 2π 2
β
π₯ ππ‘ π₯ π₯ ππ‘ π₯ 1 π‘ 1 π‘ π’π₯π‘ ( β , β + )+ 2 π’π‘π‘ ( β , β + ) , 2π 2 2 2π 2 4π 2 2 2π 2 (18)
for all π₯, π‘ β R. Hence, we know that π2 π2 (19) π’0 (π₯, π‘) β π2 2 π’0 (π₯, π‘) = 0, 2 ππ‘ ππ₯ for any π₯, π‘ β R; that is, π’0 (π₯, π‘) is a solution of the wave equation (2). Corollary 2. Given a constant πΌ > 0, let a function π : R Γ R β [0, β) be given as 2 2
(20)
If a twice continuously differentiable function π’ : R Γ R β C satisfies inequality (6), for all π₯, π‘ β R, then there exists a solution π’0 : R Γ R β C of the wave equation (2) which satisfies σ΅¨ π₯ β ππ‘ π₯ + ππ‘ σ΅¨σ΅¨σ΅¨σ΅¨ σ΅¨ πΌπ σ΅¨σ΅¨ σ΅¨σ΅¨ ) erf ( )σ΅¨ , σ΅¨σ΅¨π’ (π₯, π‘) β π’0 (π₯, π‘)σ΅¨σ΅¨σ΅¨ β€ 2 σ΅¨σ΅¨σ΅¨erf ( β2 β2 σ΅¨σ΅¨σ΅¨ 8π σ΅¨σ΅¨ (21) for all π₯, π‘ β R.
Conflict of Interests
Acknowledgments This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (no. 2013R1A1A2005557).
References
+
2
for all π, π β R, in view of Theorem 1, we conclude that the statement of this corollary is true.
The author declares that there is no conflict of interests regarding the publication of this paper.
1 π₯ ππ‘ π₯ π₯ ππ‘ π₯ π‘ 1 π‘ = π’π₯ ( + , + ) + π’π‘ ( + , + ) 2 2 2 2π 2 2π 2 2 2π 2
π (π₯, π‘) = πΌπβπ₯ βπ π‘ .
σ΅¨σ΅¨ π π σ΅¨σ΅¨ 2 2 σ΅¨ σ΅¨ = σ΅¨σ΅¨σ΅¨σ΅¨β« β« πΌπβπ /2β] /2 πππ]σ΅¨σ΅¨σ΅¨σ΅¨ σ΅¨σ΅¨ 0 0 σ΅¨σ΅¨ σ΅¨σ΅¨ π 2 σ΅¨σ΅¨ σ΅¨σ΅¨ π 2 σ΅¨σ΅¨ σ΅¨ σ΅¨σ΅¨ σ΅¨ = πΌ σ΅¨σ΅¨σ΅¨σ΅¨β« πβ] /2 π]σ΅¨σ΅¨σ΅¨σ΅¨ σ΅¨σ΅¨σ΅¨β« πβπ /2 ππσ΅¨σ΅¨σ΅¨ (22) σ΅¨σ΅¨ σ΅¨σ΅¨ 0 σ΅¨σ΅¨ σ΅¨σ΅¨ 0 σ΅¨σ΅¨ σ΅¨ σ΅¨σ΅¨ π σ΅¨σ΅¨σ΅¨σ΅¨ 2 π/β2 β]2 2 π/β2 βπ2 = 2πΌ σ΅¨σ΅¨( π π]) ( π ππ)σ΅¨σ΅¨σ΅¨ β« β« σ΅¨σ΅¨ βπ 0 4 σ΅¨σ΅¨σ΅¨ βπ 0 σ΅¨ σ΅¨ σ΅¨ σ΅¨ σ΅¨ πΌπ σ΅¨σ΅¨ π π σ΅¨ = σ΅¨erf ( ) erf ( )σ΅¨σ΅¨σ΅¨ < β, β2 β2 σ΅¨σ΅¨ 2 σ΅¨σ΅¨σ΅¨
[1] S. M. Ulam, A Collection of Mathematical Problems, Interscience, New York, NY, USA, 1960. [2] D. H. Hyers, βOn the stability of the linear functional equation,β Proceedings of the National Academy of Sciences of USA, vol. 27, pp. 222β224, 1941. [3] T. M. Rassias, βOn the stability of the linear mapping in Banach spaces,β Proceedings of the American Mathematical Society, vol. 72, pp. 297β300, 1978. [4] G. L. Forti, βHyers-Ulam stability of functional equations in several variables,β Aequationes Mathematicae, vol. 50, no. 1-2, pp. 143β190, 1995. [5] P. GΛavrutΛa, βA generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings,β Journal of Mathematical Analysis and Applications, vol. 184, no. 3, pp. 431β436, 1994. [6] D. H. Hyers, G. Isac, and T. M. Rassias, Stability of Functional Equations of Several Variables, Birkhauser, Boston, Mass, USA, 1998. [7] D. H. Hyers and T. M. Rassias, βApproximate homomorphisms,β Aequationes Mathematicae, vol. 44, no. 2-3, pp. 125β153, 1992. [8] S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, vol. 48 of Springer Optimization and Its Applications, Springer, New York, NY, USA, 2011. [9] T. M. Rassias, βOn the stability of functional equations and a problem of Ulam,β Acta Applicandae Mathematicae, vol. 62, no. 1, pp. 23β130, 2000. [10] B. Hegyi and S.-M. Jung, βOn the stability of Laplaceβs equation,β Applied Mathematics Letters, vol. 26, pp. 549β552, 2013.
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