Hyers-Ulam Stability and Existence of Solutions for Nigmatullin's

Hindawi Advances in Mathematical Physics Volume 2017, Article ID 9692685, 6 pages https://doi.org/10.1155/2017/9692685

Research Article Hyers-Ulam Stability and Existence of Solutions for Nigmatullin’s Fractional Diffusion Equation Zhuoyan Gao1 and JinRong Wang2 1

College of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan, Shanxi 030031, China Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, China

2

Correspondence should be addressed to JinRong Wang; [email protected] Received 2 December 2016; Accepted 26 January 2017; Published 21 February 2017 Academic Editor: Ming Mei Copyright © 2017 Zhuoyan Gao and JinRong Wang. 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 discuss stability of time-fractional order heat conduction equations and prove the Hyers-Ulam and generalized Hyers-UlamRassias stability of time-fractional order heat conduction equations via fractional Green function involving Wright function. In addition, an interesting existence result for solution is given.

𝐿

1. Introduction Motivated by a famous question of Ulam concerning the stability of group homomorphisms, Hyers and Rassias introduced the concepts of Hyers-Ulam and Hyers-Ulam-Rassias stability, respectively, in the case of the Cauchy functional equation in Banach spaces, which received a great influence in the development of the generalized Hyers-Ulam-Rassias stability of all kinds of functional equations. There are many interesting results on this topic in the case of functional equations, ordinary differential equations, partial differential equations, and impulsive differential equations; see, for example, [1–11] and the recent survey [12, 13]. Recently, Hegyi and Jung [14] presented the generalized Hyers-Ulam-Rassias stability of the classical Laplace’s equation Δ𝑢 = 0 in the class of spherically symmetric functions via harmonic functions method. Meanwhile, the same topic of fractional evolution equations via functional analysis methods has attracted attention of researchers. However, to the best of our knowledge, stability of fractional partial differential equations via direct analysis methods has not been discussed yet. In this paper, we study the stability of Nigmatullin’s timefractional order diffusion equation (see [15, Chapter 6]) 𝐿

𝛼

𝐷0+,𝑡 𝑢 (𝑥, 𝑡) = 𝜆

2𝜕

2

𝑢 (𝑥, 𝑡) , 𝑥 ∈ R, 𝑡 > 0, 𝜆 > 0, 𝜕𝑥2

𝛼−1

𝐷0+,𝑡 𝑢 (𝑥, 0) = 𝑓 (𝑥) ,

lim 𝑢 (𝑥, 𝑡) = 0,

𝑡→±∞

(1) and existence of solution to nonlinear problem 𝐿

𝛼

𝐷0+,𝑡 𝑢 (𝑥, 𝑡) = 𝜆

2𝜕

2

𝑢 (𝑥, 𝑡) + 𝑞 (𝑥, 𝑡, 𝑢 (𝑥, 𝑡)) , 𝜕𝑥2 𝑥 ∈ R, 𝑡 > 0, 𝜆 > 0, (2)

𝐿

𝛼−1

𝐷0+,𝑡 𝑢 (𝑥, 0) = 𝑓 (𝑥) ,

lim 𝑢 (𝑥, 𝑡) = 0,

𝑡→±∞

where 𝑓 is a continuous function on R and 𝑞 will be 𝛼 assumed to satisfy certain conditions and the symbol 𝐿 𝐷0+,𝑡 denotes the Riemann-Liouville time-fractional derivatives of the order 𝛼 ∈ (0, 1] (see [15, Chapter 6, p.349, (6.1.12)]) 𝐿

𝛼

𝐷0+,𝑡 𝑢 (𝑥, 𝑡) = (

𝑡 1 𝑢 (𝑥, 𝑠) 𝜕 [𝛼]+1 𝑑𝑠, ) ∫ 𝜕𝑡 Γ (1 − {𝛼}) 0 (𝑡 − 𝑠){𝛼}

(3)

𝑥 ∈ R, 𝑡 > 0, and [𝛼] and {𝛼} denote the integral and fractional parts of 𝛼 and Γ(⋅) is the Euler-Gamma function.

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Advances in Mathematical Physics

2. Preliminaries

and its solution is given by

The two parameter Mittag-Leffler functions 𝐸𝛼,𝛽 (𝑧) are 𝑖 defined 𝐸𝛼,𝛽 (𝑧) = ∑∞ 𝑖=0 (𝑧 /Γ(𝑖𝛼 + 𝛽)), 𝑧 ∈ R, and 𝛼, 𝛽 are positive real numbers. Next, 𝐸𝛼 (𝑧) = 𝐸𝛼,1 (𝑧). The solvability of (1) has been reported in [15, Chapter 6.2.1]. Here we collect the following result.

𝑢 (𝑥, 𝑡) = ∫

−∞

𝑢 (𝑥, 𝑡) = ∫

−∞

𝛼

𝐺 (𝑥 − 𝜉, 𝑡) 𝑓 (𝜉) 𝑑𝜉,

𝐺1 (𝑥, 𝑡) =

=

(4)

|𝑢 (𝑥, 𝑡)| ≤ 𝑀,

1 ∞ 𝛼−1 ∫ 𝑡 𝐸𝛼,𝛼 (−𝜆2 𝑦2 𝑡𝛼 ) cos (𝑦𝑥) 𝑑𝑦 𝜋 0 (5)

∫

+∞

−∞

+∞

𝐺1 (𝑥 − 𝜉, 𝑡) 𝑑𝜉 = ∫

−∞ +∞

−𝛼 𝑧𝑘 1 𝜙 (𝛼, 𝛽; 𝑧) = ∑ = ∫ 𝑠−𝛽 𝑒𝑠+𝑧𝑠 𝑑𝑠, 2𝜋𝑖 𝑘!Γ (𝛼𝑘 + 𝛽) 𝐻𝑎 𝑘=0

=∫

−∞

provided that the integral in the right-hand side of (15) is convergent, where 𝐻𝑎 denotes the Hankel path of integration in the complex 𝑠-plane. Note that [17, Lemma 2] for any 𝜆 > 0 and 𝑡 > 0, 𝐸𝛼,𝛼 (−𝑡𝛼 𝜆) ≤ 1/Γ(𝛼). Then, we have the following estimation. Lemma 2. For any 𝑡 > 0, 𝑥 ∈ R, 𝐺𝛼 (𝑥, 𝑡) ≤ 𝑡𝛼 /Γ(𝛼), and (6)

where +∞

∫

−∞

󵄨 󵄨󵄨 󵄨󵄨𝑓 (𝜉)󵄨󵄨󵄨 𝑑𝜉 fl 𝑀𝑓 .

𝜕 𝜕2 𝑢 (𝑥, 𝑡) , 𝑥 ∈ R, 𝑡 > 0, 𝑢 (𝑥, 𝑡) = 𝜆2 𝜕𝑡 𝜕𝑥2 𝑢 (𝑥, 0) = 𝑓 (𝑥) ,

∞

(9)

(13) 𝜉−𝑥 ). 2𝜆√𝑡

1/𝛼

∞

1/𝛼

Lemma 5. Let 𝑚(𝛼, 𝜆) = ∫0 𝑟1/𝛼 𝑒−𝑟 𝑑𝑟/ sin(𝜋𝛼)𝜋𝛼𝜆4 . Then 𝑚 (𝛼, 𝜆) 𝑡 cos (𝑦𝑥) 𝑑𝑦. ∫ 𝜋𝑡1+𝛼 0 𝑦4 +∞

−∞

Remark 4. If 𝛼 = 1, then (1) becomes a classical heat conduction equation

1 −𝜁2 𝑒 𝑑𝜁 = 1, √𝜋

̃ (𝛼, 𝜆) = ∫0 𝑟1/𝛼 𝑒−𝑟 𝑑𝑟/ sin(𝜋𝛼)𝜋𝛼𝑧2 . ̃ (𝛼, 𝑧)/𝑡𝛼+1 , where 𝑚 𝑚 For more asymptotic expansions on Mittag-Leffler functions, one can refer to [19, Lemmas 2.2, 2.3, 2.4]. Next, we give asymptotic property of 𝐺𝛼 (𝑥, 𝑡).

𝑢 (𝑥, 𝑡) = ∫

(8)

1 −1/2 −|𝑥−𝜉|2 /4𝜆2 𝑡 𝑡 𝑒 𝑑𝜉 2𝜆√𝜋

Note that [18, Lemma 3], for all 𝑡 > 0, |𝑡𝛼−1 E𝛼,𝛼 (−𝑧𝑡𝛼 )| ≤

(7)

Remark 3. Obviously, (7) can be fulfilled. For example, 𝑓(𝜉) = 2 𝑒−(𝜉−𝜃) , 𝜃 ∈ (𝑎, 𝑏). Moreover, we can obtain 1 +∞ 𝑖𝑦𝑥 𝑡𝛼 𝛿 (𝑥) , 𝛿 (𝑥) = ∫ 𝑒 𝑑𝑦. Γ (𝛼) 2𝜋 −∞

(12)

(𝜁 =

𝐺𝛼 (𝑥, 𝑡) ≤

Obviously, we have the following remarks.

𝐺𝛼 (𝑥, 𝑡) ≤

(11)

where we use the fact that

∞

󵄨󵄨 𝑀𝑓 𝑡𝛼 󵄨󵄨 +∞ 󵄨 󵄨󵄨 𝛼 , 󵄨󵄨∫ 𝐺 (𝑥 − 𝜉, 𝑡) 𝑓 (𝜉) 𝑑𝜉󵄨󵄨󵄨 ≤ 󵄨󵄨 𝜋Γ (𝛼) 󵄨󵄨 −∞

(10)

If |𝑓(𝑥)| ≤ 𝑀, then (4) satisfies the following inequality:

1 (𝛼/2)−1 𝛼 𝛼 |𝑥| 𝜙 (− , ; − 𝑡−𝛼/2 ) 𝑡 2𝜆 2 2 𝜆

1 +∞ 𝑖𝑦𝑥 𝛼−1 = ∫ 𝑒 𝑡 𝐸𝛼,𝛼 (−𝜆2 𝑦2 𝑡𝛼 ) 𝑑𝑦, 2𝜋 −∞

1 −1/2 −|𝑥|2 /4𝜆2 𝑡 𝑡 𝑒 , 2𝜆√𝜋

2 2 1 1 |𝑥| 𝜙 (− , ; − 𝑡−1/2 ) = 𝑒−|𝑥| /4𝜆 𝑡 . 2 2 𝜆

where fractional Green function 𝐺𝛼 (𝑥, 𝑡) involving Wright function 𝜙(𝛼, 𝛽; 𝑧) is given by 𝐺𝛼 (𝑥, 𝑡) =

𝐺1 (𝑥 − 𝜉, 𝑡) 𝑓 (𝜉) 𝑑𝜉,

where

Lemma 1 (see [15, Corollary 6.1] or [16, (4.19)]). Equation (1) is solvable, and its solution has the form +∞

+∞

(14)

𝐺𝛼 (𝑥 − 𝜉, 𝑡) 𝑓 (𝜉) 𝑑𝜉

𝑚 (𝛼, 𝜆) +∞ 𝑡 cos (𝑦 (𝑥 − 𝜉)) 𝑑𝑦𝑓 (𝜉) 𝑑𝜉. ≤ ∫ ∫ 𝜋𝑡1+𝛼 −∞ 0 𝑦4

(15)

3. Fractional Duhamel’s Principle, Stability Concepts, and Remarks The standard Duhamel principle adopts the idea form ODEs in studying Cauchy problem for inhomogeneous partial differential equations by linking Cauchy problem for corresponding homogeneous equation. In this section, we establish a fractional Duhamel principle which helps us to study Ulam’s stability of (1) and existence of solution to (2).

Advances in Mathematical Physics

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Lemma 6. Let 𝐹(𝑥, 𝑡) be jointly continuous on R × (0, +∞). The solution of Cauchy problem for inhomogeneous partial differential equations of the type 𝐿

𝛼 𝐷0,𝑡 𝑦 (𝑥, 𝑡)

=𝜆

2𝜕

2

𝑦 (𝑥, 𝑡) + 𝐹 (𝑥, 𝑡) , 𝜕𝑥2

𝛼−1

𝐷0,𝑡 𝑦 (𝑥, 0) = 𝑓 (𝑥) ,

𝑦 (𝑥, 𝑡) = ∫

−∞

+∞

0

−∞

+∫ ∫

(17) 𝐺𝛼 (𝑥 − 𝜉, 𝑡 − 𝜏) 𝐹 (𝜉, 𝜏) 𝑑𝜉 𝑑𝜏.

Proof. Using the superposition principle, the following Cauchy problem (16) can be decomposed into two Cauchy problems: 𝐿

𝐿

𝛼

𝐷0,𝑡 𝑦1 (𝑥, 𝑡) = 𝜆

2𝜕

2

𝑦1 (𝑥, 𝑡) , 𝑥 ∈ R, 𝑡 > 0, 𝜕𝑥2

𝛼

2𝜕

2

Definition 7. Equation (1) is Hyers-Ulam stable if there exists a number 𝑐 > 0 such that for each solution 𝑦 of inequality (23) there exists a solution 𝑢 of (1) with 󵄨 󵄨󵄨 󵄨󵄨𝑦 (𝑥, 𝑡) − 𝑢 (𝑥, 𝑡)󵄨󵄨󵄨 ≤ 𝑐𝜀,

(ii) 𝐿 𝐷0+,𝑡 𝑦(𝑥, 𝑡) = 𝜆2 (𝜕2 𝑦(𝑥, 𝑡)/𝜕𝑥2 ) + 𝐹(𝑥, 𝑡). Remark 9. If 𝑦(𝑥, 𝑡) is a solution of inequality (23), then 𝑦 is a solution of the following integral inequality:

𝑦2 (𝑥, 𝑡) + 𝐹 (𝑥, 𝑡) , 𝜕𝑥2

󵄨󵄨 󵄨󵄨 +∞ 󵄨 󵄨󵄨 𝛼 󵄨󵄨𝑦 (𝑥, 𝑡) − ∫ 𝐺 (𝑥 − 𝜉, 𝑡) 𝑓 (𝜉) 𝑑𝜉󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 −∞

𝑦1 (𝑥, 𝑡) = ∫

−∞

𝛼

𝐺 (𝑥 − 𝜉, 𝑡) 𝑓 (𝜉) 𝑑𝜉.

(20)

By virtue of homogeneous theorem the solution of (19) can be written as 𝑦2 (𝑥, 𝑡) = ∫ 𝜛 (𝑥, 𝑡; 𝜏) 𝑑𝜏, 0

(21)

𝐿

𝛼−1 𝐷0,𝑡 𝜛 (𝑥, 𝜏)

−∞

2𝜕

2

𝜛 (𝑥, 𝑡) , 𝑥 ∈ R, 𝑡 > 𝜏, 𝜕𝑥2

𝐺 (𝑥 − 𝜉, 𝑡 − 𝜏) 𝑑𝜉 𝑑𝜏.

𝑡

+∞

0

−∞

(27)

(𝑡 − 𝜏)𝛼 󵄨󵄨 󵄨 󵄨𝐹 (𝜉, 𝜏)󵄨󵄨󵄨 𝑑𝜉 𝑑𝜏. 𝜋Γ (𝛼) 󵄨

Let 𝜑(𝑥, 𝑡) : R × (0, +∞) → [0, +∞) be a nonnegative function.

where 𝜛 = 𝜛(𝑥, 𝑡; 𝜏) is the solution of 𝛼

0

(26)

𝛼

󵄨󵄨 󵄨󵄨 +∞ 󵄨 󵄨󵄨 𝛼 󵄨󵄨𝑦 (𝑥, 𝑡) − ∫ 𝐺 (𝑥 − 𝜉, 𝑡) 𝑓 (𝜉) 𝑑𝜉󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 −∞ 󵄨󵄨 𝑡 +∞ 󵄨󵄨 󵄨 󵄨 ≤ 󵄨󵄨󵄨∫ ∫ 𝐺𝛼 (𝑥 − 𝜉, 𝑡 − 𝜏) 𝐹 (𝜉, 𝜏) 𝑑𝜉 𝑑𝜏󵄨󵄨󵄨 󵄨󵄨 0 −∞ 󵄨󵄨 ≤ 𝜀∫ ∫

𝑡

𝐷0,𝑡 𝜛 (𝑥, 𝑡) = 𝜆

+∞

By Remark 8 and (17), we get

By Lemma 1, the solution of (18) is

𝐿

𝑡

≤ 𝜀∫ ∫

𝛼−1

𝐷0,𝑡 𝑦2 (𝑥, 0) = 0.

+∞

(25)

Remark 8. A function 𝑦(𝑥, 𝑡) is a solution of inequality (23) if and only if there is 𝐹(𝑥, 𝑡) such that

𝑥 ∈ R, 𝑡 > 0, (19) 𝐿

𝑥 ∈ R, 𝑡 > 0.

𝛼

(18)

𝛼−1

𝐷0,𝑡 𝑦2 (𝑥, 𝑡) = 𝜆

(24)

(i) |𝐹(𝑥, 𝑡)| ≤ 𝜀, 𝑥 ∈ R, 𝑡 > 0;

𝐷0,𝑡 𝑦1 (𝑥, 0) = 𝑓 (𝑥) , 𝐿

󵄨󵄨 󵄨 𝜕2 𝑦 (𝑥, 𝑡) 󵄨󵄨󵄨 󵄨󵄨𝐿 𝛼 󵄨󵄨 𝐷0+,𝑡 𝑦 (𝑥, 𝑡) − 𝜆2 󵄨󵄨 ≤ 𝜀̃ 𝜑 (𝑡) , 2 󵄨󵄨 󵄨󵄨 𝜕𝑥 󵄨 󵄨

Now we are ready to introduce the Hyers-Ulam and generalized Hyers-Ulam-Rassias stability concepts for (1).

𝐺𝛼 (𝑥 − 𝜉, 𝑡) 𝑓 (𝜉) 𝑑𝜉

𝑡

(23)

𝑥 ∈ R, 𝑡 > 0, 𝜆 > 0, 0 < 𝛼 ≤ 1.

is +∞

󵄨 󵄨󵄨 𝜕2 𝑦 (𝑥, 𝑡) 󵄨󵄨󵄨 󵄨󵄨𝐿 𝛼 󵄨󵄨 ≤ 𝜀, 󵄨󵄨 𝐷0+,𝑡 𝑦 (𝑥, 𝑡) − 𝜆2 󵄨󵄨 𝜕𝑥2 󵄨󵄨󵄨 󵄨 𝑥 ∈ R, 𝑡 > 0, 𝜆 > 0, 0 < 𝛼 ≤ 1,

𝑥 ∈ R, 𝑡 > 0, (16) 𝐿

Consider (1) and the following two inequalities:

(22)

= 𝐹 (𝑥, 𝜏) .

By virtue of homogeneous theorem, Lemma 1, and (21) we obtain 𝑦(𝑥, 𝑡) = 𝑦1 (𝑥, 𝑡) + 𝑦2 (𝑥, 𝑡) which is the desired result. ̃ (𝑡) : R → [0, +∞) be a nonnegative and Let 𝜖 > 0 and 𝜑 increasing function.

Definition 10. Equation (1) is generalized Hyers-UlamRassias stable if there exists a number 𝑐 > 0 such that for each solution 𝑦 of inequality (24) there exists a solution 𝑢 of (1) with 󵄨 󵄨󵄨 󵄨󵄨𝑦 (𝑥, 𝑡) − 𝑢 (𝑥, 𝑡)󵄨󵄨󵄨 ≤ 𝑐𝜑 (𝑥, 𝑡) , 𝑥 ∈ R, 𝑡 > 0.

(28)

Remark 11. A function 𝑦(𝑥, 𝑡) is a solution of inequality (24) if and only if there is 𝐹(𝑥, 𝑡) such that ̃ (𝑡), 𝑥 ∈ R, 𝑡 > 0; (i) |𝐹(𝑥, 𝑡)| ≤ 𝜑 𝛼

(ii) 𝐿 𝐷0+,𝑡 𝑦(𝑥, 𝑡) = 𝜆2 (𝜕2 𝑦(𝑥, 𝑡)/𝜕𝑥2 ) + 𝐹(𝑥, 𝑡).

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Remark 12. If 𝑦(𝑥, 𝑡) is a solution of inequality (24), then 𝑦 is a solution of the following integral inequality: +∞ 󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨𝑦 (𝑥, 𝑡) − ∫ 𝐺𝛼 (𝑥 − 𝜉, 𝑡) 𝑓 (𝜉) 𝑑𝜉󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 −∞ 󵄨 󵄨 𝑡

≤∫ ∫ 0

+∞

−∞

󵄨 ̃ 󵄨󵄨 𝐺𝛼 (𝑥 − 𝜉, 𝑡 − 𝜏) 󵄨󵄨󵄨𝜑 (𝜉)󵄨󵄨 𝑑𝜉 𝑑𝜏

𝑡

+∞

0

−∞

≤ 𝜀∫ ∫

(29)

Theorem 15. Assume that there exists 𝑚 > 0 such that ∫

+∞

−∞

𝑢 (𝑥, 𝑡) = ∫

(𝑥,𝑡)∈R×(0,+∞) 0

−∞

󵄨 󵄨󵄨 𝛼 󵄨󵄨𝐺 (𝑥 − 𝜉, 𝑡 − 𝜏)󵄨󵄨󵄨 𝑑𝜉 𝑑𝜏 = 𝑐

󵄨󵄨 󵄨󵄨 +∞ 󵄨󵄨 󵄨 𝛼 󵄨󵄨𝑦 (𝑥, 𝑡) − ∫ 𝐺 (𝑥 − 𝜉, 𝑡) 𝑓 (𝜉) 𝑑𝜉󵄨󵄨󵄨 󵄨󵄨 −∞ 󵄨󵄨 (30)

Then (1) is Hyers-Ulam stable on finite time interval [0, 𝑇] with respect to 𝜀 and 𝑐. Proof. Let 𝑦(𝑥, 𝑡) be a solution of inequality (23) and 𝑢(𝑥, 𝑡) the solution of Cauchy problem (1), and its expression is 𝑢 (𝑥, 𝑡) = ∫

−∞

𝛼

𝐺 (𝑥 − 𝜉, 𝑡) 𝑓 (𝜉) 𝑑𝜉.

Keeping in mind (26), we have 󵄨󵄨 󵄨󵄨 +∞ 󵄨 󵄨󵄨 𝛼 󵄨󵄨𝑦 (𝑥, 𝑡) − ∫ 𝐺 (𝑥 − 𝜉, 𝑡) 𝑓 (𝜉) 𝑑𝜉󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 −∞ 𝑡

+∞

0

−∞

≤ 𝜀∫ ∫

󵄨 󵄨󵄨 𝛼 󵄨󵄨𝐺 (𝑥 − 𝜉, 𝑡 − 𝜏)󵄨󵄨󵄨 𝑑𝜉 𝑑𝜏.

By (30) we obtain 󵄨 󵄨󵄨 󵄨󵄨𝑦 (𝑥, 𝑡) − 𝑢 (𝑥, 𝑡)󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 +∞ 󵄨 󵄨 = 󵄨󵄨󵄨𝑦 (𝑥, 𝑡) − ∫ 𝐺𝛼 (𝑥 − 𝜉, 𝑡) 𝑓 (𝜉) 𝑑𝜉󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 −∞ 𝑡

+∞

0

−∞

≤ 𝜀∫ ∫

(32)

𝑡

+∞

(𝑥,𝑡)∈(0,𝑙)×(0,𝑇] 0

−∞

∫ ∫

󵄨 󵄨󵄨 1 󵄨󵄨𝐺 (𝑥 − 𝜉, 𝑡 − 𝜏)󵄨󵄨󵄨 𝑑𝜉 𝑑𝜏 󵄨 󵄨

= 𝑇, where we use (13) in Remark 4.

0

−∞

(38)

𝛼

̃ (𝜏) 𝑑𝜉 𝑑𝜏. 𝐺 (𝑥 − 𝜉, 𝑡 − 𝜏) 𝜑

Hence, we have 𝑡 +∞ (𝑡 − 𝜏)𝛼 󵄨󵄨 󵄨 󵄨󵄨 ̃ (𝜏)󵄨󵄨󵄨󵄨 𝑑𝜉 𝑑𝜏 󵄨󵄨𝜑 󵄨󵄨𝑦 (𝑥, 𝑡) − 𝑢 (𝑥, 𝑡)󵄨󵄨󵄨 ≤ ∫ ∫ 0 −∞ 𝜋Γ (𝛼)

𝑚𝑡𝛼+1 ≤ fl 𝑐𝜑 (𝑥, 𝑡) , 𝜋 (𝛼 + 1) Γ (𝛼)

(39)

which implies that (1) is Hyers-Ulam-Rassias stable with respect to 𝜑(𝑥, 𝑡) fl 𝑡𝛼+1 and 𝑐 fl 𝑚/𝜋(𝛼 + 1)Γ(𝛼). The proof is completed.

𝑡

+∞

0

−∞

∫ ∫

󵄨󵄨 𝛼 ̃ (𝜏)󵄨󵄨󵄨󵄨 𝑑𝜉 𝑑𝜏 ≤ 𝑐𝜑 (𝑥, 𝑡) , 󵄨󵄨𝐺 (𝑥 − 𝜉, 𝑡 − 𝜏) 𝜑

(40)

for each 𝑥 ∈ R, 𝑡 > 0. (33)

which is reasonable due to the fact that (see [15, (6.3.7)]) (F𝑥 L𝑡 𝐺𝛼 ) (𝑥, 𝑡) =

Example 14. Set 𝜀 > 0, 𝛼 = 1, 𝑥 ∈ (0, 𝑙), and 𝑡 ∈ (0, 𝑇]. Then the following heat conduction equation

sup

(37)

Remark 16. Condition (36) can be changed to

which implies that (1) is Hyers-Ulam stable with respect to 𝜀 and 𝑐. The proof is completed.

𝑐𝜆,1 =

+∞

(31)

󵄨 󵄨󵄨 𝛼 󵄨󵄨𝐺 (𝑥 − 𝜉, 𝑡 − 𝜏)󵄨󵄨󵄨 𝑑𝜉 𝑑𝜏 = 𝑐𝜀,

𝜕 𝜕2 𝑢 (𝑥, 𝑡) , 𝑥 ∈ (0, 𝑙) , 𝑡 ∈ (0, 𝑇] , 𝑢 (𝑥, 𝑡) = 𝜆2 𝜕𝑡 𝜕𝑥2 is Ulam-Hyers stable with respect to 𝜀 and

𝑡

≤∫ ∫

< +∞.

+∞

𝐺𝛼 (𝑥 − 𝜉, 𝑡) 𝑓 (𝜉) 𝑑𝜉.

Keeping in mind (29), we have

Theorem 13. Assume that there exists 𝑐𝜆,𝛼 > 0 such that ∫ ∫

+∞

−∞

In this section, we present the stability results.

sup

(36)

Then (1) is generalized Hyers-Ulam-Rassias stable with respect to 𝜑(𝑡) and 𝑐.

4. Hyers-Ulam Stability

+∞

for each 𝑥 ∈ R, 𝑡 > 0.

Proof. Let 𝑦(𝑥, 𝑡) be a solution of inequality (24) and 𝑢(𝑥, 𝑡) the solution of Cauchy problem (1), and its expression is

𝑀 (𝑡 − 𝜏)𝛼 󵄨󵄨 ̃ (𝜉)󵄨󵄨󵄨󵄨 𝑑𝜉 𝑑𝜏. 󵄨𝜑 𝜋Γ (𝛼) 󵄨

𝑡

󵄨󵄨 ̃ 󵄨󵄨 󵄨󵄨𝜑 (𝜉)󵄨󵄨 𝑑𝜉 𝑑𝜏 ≤ 𝑚,

(34)

𝑡𝛼

𝑡𝛼 ̃ ) (𝑥) , (F𝑥 𝜑 + 𝜆2 |𝑥|2 𝑥 ∈ R, 𝑡 > 0,

where the symbols F𝑥 and L𝑡 denote the Fourier transform and Laplace transform, respectively. ̃ (𝑡) = |𝑡|V , V ≠ 0, one has (see [15, (1.3.55)]) Note that, for 𝜑 ̃ ) (𝑥) = 2V+1 √𝜋 (F𝑥 𝜑

(35)

(41)

Γ ((V + 1) /2) −V−1 |𝑥| Γ (−V/2)

V𝜋 = −2 sin ( ) Γ (V + 1) |𝑥|−V−1 . 2

(42)

Then we can choose 𝜑(𝑥, 𝑡) = |𝑥|−V−1 𝑡𝛼 /(𝑡𝛼 + 𝜆2 |𝑥|2 ) and 𝑐 = 2|Γ(V + 1)|.

Advances in Mathematical Physics

5

̃ (𝑡) = 𝑡2 . Then the following heat Example 17. Set 𝛼 = 1 and 𝜑 conduction equation 𝜕2 𝑢 (𝑥, 𝑡) 𝜕 , 𝑢 (𝑥, 𝑡) = 𝜆2 𝜕𝑡 𝜕𝑥2

𝑥 ∈ R, 𝑡 > 0,

(43)

5. Existence of Solution Let B be the Banach space of all bounded continuous function 𝑢(𝑥, 𝑡) on R with fixed 𝑡 equipped with the norm ̃ = ‖𝑢(⋅, 𝑡)‖B = sup𝑥∈R |𝑢(𝑥, 𝑡)|. Consider the Banach space B 𝐶([0, 𝑡0 ], B) of all bounded continuous function defined on [0, 𝑇] equipped with the norm ‖𝑢(⋅, 𝑡)‖B̃ = ̃𝛼 = sup𝑡∈[0,𝑇] |𝑢(⋅, 𝑡)|. Next, we introduce Banach space B 1−𝛼 {𝑢 : 𝑡 𝑢(⋅, 𝑡) ∈ 𝐶([0, 𝑡0 ], B)} equipped with the norm ‖𝑢‖B̃ 𝛼 = sup𝑡∈[0,𝑇] 𝑡1−𝛼 |𝑢(⋅, 𝑡)|. We introduce the following assumptions: [H1]: 𝑞(𝑥, 𝑡, 𝑢(𝑥, 𝑡)) is jointly continuous on R × [0, 𝑇] × R with 𝑞(𝑥, 𝑡, 0) = 0. [H2]: For any 𝑀 > 0, there exists a constant 𝐿 > 0 such that

≤ 𝐿 ‖𝑢 (⋅, 𝑡) − V (⋅, 𝑡)‖B

(44)

for all 𝑡 ∈ [0, 𝑇] and all 𝑢, V ∈ B with ‖𝑢(⋅, 𝑡)‖B ≤ 𝑀 and ‖V(⋅, 𝑡)‖B ≤ 𝑀. Define 󵄩󵄩 󵄩󵄩 +∞ ̃𝛼 : 󵄩󵄩󵄩𝑢 (⋅, ⋅) − ∫ 𝐺𝛼 (⋅ − 𝜉, ⋅) 𝑓 (𝜉) 𝑑𝜉󵄩󵄩󵄩 Υ = {𝑢 ∈ B 󵄩󵄩 ̃ 󵄩󵄩 −∞ 󵄩B𝛼 󵄩

≤∫

(45)

󵄩 󵄩 ≤ 󵄩󵄩󵄩𝑓󵄩󵄩󵄩B , 𝑡 ∈ [0, 𝑡𝑚 ]} ,

Observing the definition of Υ and triangular inequality, one has 1 󵄩 󵄩 ) 󵄩󵄩𝑓󵄩󵄩 . (49) ‖𝑢‖B̃ 𝛼 ≤ (1 + Γ (𝛼) 󵄩 󵄩B 0,

Next, we show that Q : Υ → Υ. By [H2] with 𝑞(𝑥, 𝑡, 0) = 󵄩󵄩 󵄩󵄩 +∞ 󵄩 󵄩󵄩 𝛼 󵄩󵄩(Q𝑢) (⋅, 𝑡) − ∫ 𝐺 (⋅ − 𝜉, 𝑡) 𝑓 (𝜉) 𝑑𝜉󵄩󵄩󵄩 󵄩󵄩B −∞ 󵄩󵄩 ≤∫

(𝑡 − 𝜏)𝛼−1 󵄩󵄩 󵄩 󵄩𝑞 (⋅, 𝜏, 𝑢 (⋅, 𝜏))󵄩󵄩󵄩B 𝑑𝜏 Γ (𝛼) 󵄩

𝑡 𝐿 󵄩 󵄩 ≤ ∫ (𝑡 − 𝜏)𝛼−1 𝑑𝜏 󵄩󵄩󵄩𝑢 (⋅, 𝜉)󵄩󵄩󵄩B Γ (𝛼) 0

=

Multiplying 𝑡1−𝛼 to the above inequality and taking the supremum, we have 󵄩󵄩 󵄩󵄩 +∞ 󵄩 󵄩󵄩 𝛼 󵄩󵄩Q𝑢 − ∫ 𝐺 (⋅ − 𝜉, ⋅) 𝑓 (𝜉) 𝑑𝜉󵄩󵄩󵄩 −∞ 󵄩󵄩B̃ 𝛼 󵄩󵄩 (51) 𝐿𝑡 1 ≤ ‖𝑢 (⋅, ⋅)‖B̃ ≤ ‖𝑢 (⋅, ⋅)‖B̃ < ‖𝑢 (⋅, ⋅)‖B̃ . Γ (𝛼 + 1) 2 Now we check that Q is contraction. Observe that ‖(Q𝑢) (⋅, 𝑡) − (QV) (⋅, 𝑡)‖B 𝑡

0

𝐿 (𝑡 − 𝜏)𝛼−1 ‖𝑢 (⋅, 𝜏) − V (⋅, 𝜏)‖B 𝑑𝜏 Γ (𝛼)

Q (𝑢) (𝑥, 𝑡)

𝐿𝑡𝛼 ‖𝑢 (⋅, 𝑡) − V (⋅, 𝑡)‖B . Γ (𝛼 + 1)

‖Q𝑢 − QV‖B̃ 𝛼 ≤

𝑡

+∞

0

−∞

𝐺𝛼 (𝑥 − 𝜉, 𝑡 − 𝜏) 𝑞 (𝜉, 𝜏, 𝑢 (𝜉, 𝜏)) 𝑑𝜉 𝑑𝜏.

(52)

Multiplying 𝑡1−𝛼 to the above inequality and taking the supremum, we have

̃𝛼 given by Proof. Define Q : Υ → B

+∫ ∫

(50)

𝐿𝑡𝛼 󵄩󵄩 󵄩 󵄩𝑢 (⋅, 𝜉)󵄩󵄩󵄩B . Γ (𝛼 + 1) 󵄩

≤

−∞

𝑡

0

Theorem 18. Assume that [H1] and [H2] hold. Then (2) has unique solution in Υ and ‖𝑢‖B̃ 𝛼 ≤ (1 + 1/Γ(𝛼))‖𝑓‖B .

𝐺𝛼 (𝑥 − 𝜉, 𝑡) 𝑓 (𝜉) 𝑑𝜉

(47)

Multiplying 𝑡1−𝛼 to the above inequality and taking the supremum, we have 󵄩󵄩 󵄩󵄩󵄩 +∞ 𝛼 󵄩󵄩∫ 𝐺 (⋅ − 𝜉, ⋅) 𝑓 (𝜉) 𝑑𝜉󵄩󵄩󵄩 ≤ 1 󵄩󵄩󵄩𝑓󵄩󵄩󵄩 ̃ . (48) 󵄩󵄩 −∞ 󵄩󵄩 ̃ 󵄩 󵄩B 󵄩 󵄩B𝛼 Γ (𝛼)

≤∫

+∞

𝑡𝛼 󵄩󵄩 󵄩 󵄩𝛿 (⋅ − 𝜉) 𝑓 (𝜉)󵄩󵄩󵄩B 𝑑𝜉 Γ (𝛼) 󵄩

𝑡𝛼 󵄩󵄩 󵄩 󵄩𝑓 (𝜉)󵄩󵄩󵄩B . Γ (𝛼) 󵄩

=

where 𝑡𝑚 = min{Γ(𝛼 + 1)/2𝐿, (Γ(𝛼 + 1)/2𝐿)1/𝛼 }.

=∫

+∞

−∞

is Hyers-Ulam-Rassias stable with respect to 𝜑(𝑥, 𝑡) = |𝑥|−3 𝑡2 /(𝑡2 + 𝜆2 |𝑥|2 ) and 𝑐 = 2, where we use (13) in Remark 4 again.

󵄩 󵄩󵄩 󵄩󵄩𝑞 (⋅, 𝑡, 𝑢 (⋅, 𝑡)) − 𝑞 (⋅, 𝑡, V (⋅, 𝑡))󵄩󵄩󵄩B

Note that 󵄩󵄩 󵄩󵄩 +∞ 󵄩 󵄩󵄩 𝛼 󵄩󵄩∫ 𝐺 (⋅ − 𝜉, 𝑡) 𝑓 (𝜉) 𝑑𝜉󵄩󵄩󵄩 󵄩󵄩B 󵄩󵄩 −∞

(46)

𝐿𝑡𝛼 ‖𝑢 (⋅, ⋅) − V (⋅, ⋅)‖B̃ 𝛼 Γ (𝛼 + 1)

(53) 1 ≤ ‖𝑢 (⋅, ⋅) − V (⋅, ⋅)‖B̃ 𝛼 . 2 By contraction mapping principle, Q has a unique fixed point in Υ which is the solution of (2).

6

Competing Interests The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments This work is supported by National Natural Science Foundation of China (11661016), Training Object of High Level and Innovative Talents of Guizhou Province ((2016)4006), Unite Foundation of Guizhou Province ([2015]7640), and Graduate Course of Guizhou University (ZDKC[2015]003).

References [1] 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. [2] I. A. Rus, “Ulam stabilities of ordinary differential equations in a Banach space,” Carpathian Journal of Mathematics, vol. 26, no. 1, pp. 103–107, 2010. [3] D. Popa and I. Ras¸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. [4] S.-M. Jung, “Hyers-Ulam stability of linear partial differential equations of first order,” Applied Mathematics Letters, vol. 22, no. 1, pp. 70–74, 2009. [5] D. S. Cimpean and D. Popa, “Hyers-Ulam stability of Euler’s equation,” Applied Mathematics Letters, vol. 24, no. 9, pp. 1539– 1543, 2011. [6] N. Lungu and D. Popa, “Hyers-Ulam stability of a first order partial differential equation,” Journal of Mathematical Analysis and Applications, vol. 385, no. 1, pp. 86–91, 2012. [7] J. Wang, M. Feˇckan, and Y. Zhou, “Ulam’s type stability of impulsive ordinary differential equations,” Journal of Mathematical Analysis and Applications, vol. 395, no. 1, pp. 258–264, 2012. [8] S. d. Abbas and M. Benchohra, “Ulam-Hyers stability for the Darboux problem for partial fractional differential and integro-differential equations via Picard operators,” Results in Mathematics, vol. 65, no. 1-2, pp. 67–79, 2014. [9] J. Wang and X. Li, “A uniform method to Ulam-Hyers stability for some linear fractional equations,” Mediterranean Journal of Mathematics, vol. 13, no. 2, pp. 625–635, 2016. [10] J. Huang and Y. Li, “Hyers-Ulam stability of delay differential equations of first order,” Mathematische Nachrichten, vol. 289, no. 1, pp. 60–66, 2016. [11] J. Brzdęk and N. Eghbali, “On approximate solutions of some delayed fractional differential equations,” Applied Mathematics Letters, vol. 54, pp. 31–35, 2016. [12] J. Brzdęk, W. Fechner, M. S. Moslehian, and J. Sikorska, “Recent developments of the conditional stability of the homomorphism equation,” Banach Journal of Mathematical Analysis, vol. 9, no. 3, pp. 278–326, 2015. [13] J. Brzdęk, L. C˘adariu, K. Ciepli´nski, A. Foˇsner, and Z. Le´sniak, “Survey on recent ulam stability results concerning derivations,” Journal of Function Spaces, vol. 2016, Article ID 1235103, 9 pages, 2016. [14] B. Hegyi and S.-M. Jung, “On the stability of Laplace’s equation,” Applied Mathematics Letters, vol. 26, no. 5, pp. 549–552, 2013. [15] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science, 2006.

Advances in Mathematical Physics [16] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. [17] J. Wang, M. Feˇckan, and Y. Zhou, “Presentation of solutions of impulsive fractional Langevin equations and existence results: impulsive fractional Langevin equations,” European Physical Journal: Special Topics, vol. 222, no. 8, pp. 1857–1874, 2013. [18] N. D. Cong, T. S. Doan, S. Siegmund, and H. T. Tuan, “On stable manifolds for planar fractional differential equations,” Applied Mathematics and Computation, vol. 226, pp. 157–168, 2014. [19] J. Wang, M. Feˇckan, and Y. Zhou, “Center stable manifold for planar fractional damped equations,” Applied Mathematics and Computation, vol. 296, pp. 257–269, 2017.

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