Abstract
We discuss stability of time-fractional order heat conduction equations and prove the Hyers-Ulam and generalized Hyers-Ulam-Rassias 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.
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 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 time-fractional order diffusion equation (see [15, Chapter 6])and existence of solution to nonlinear problemwhere is a continuous function on and will be assumed to satisfy certain conditions and the symbol denotes the Riemann-Liouville time-fractional derivatives of the order (see [15, Chapter 6, p.349, (6.1.12)])and and denote the integral and fractional parts of and is the Euler-Gamma function.
2. Preliminaries
The two parameter Mittag-Leffler functions are defined , , and , are positive real numbers. Next, .
The solvability of (1) has been reported in [15, Chapter 6.2.1]. Here we collect the following result.
Lemma 1 (see [15, Corollary 6.1] or [16, (4.19)]). Equation (1) is solvable, and its solution has the formwhere fractional Green function involving Wright function is given by provided that the integral in the right-hand side of (15) is convergent, where denotes the Hankel path of integration in the complex -plane.
Note that [17, Lemma 2] for any and , Then, we have the following estimation.
Lemma 2. For any , , , andwhere
Obviously, we have the following remarks.
Remark 3. Obviously, (7) can be fulfilled. For example, , . Moreover, we can obtain
Remark 4. If , then (1) becomes a classical heat conduction equationand its solution is given bywhereIf , then (4) satisfies the following inequality: where we use the fact that
Note that [18, Lemma 3], for all , , where . 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 .
Lemma 5. Let Then
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).
Lemma 6. Let be jointly continuous on . The solution of Cauchy problem for inhomogeneous partial differential equations of the typeis
Proof. Using the superposition principle, the following Cauchy problem (16) can be decomposed into two Cauchy problems:By Lemma 1, the solution of (18) is By virtue of homogeneous theorem the solution of (19) can be written aswhere is the solution ofBy virtue of homogeneous theorem, Lemma 1, and (21) we obtain which is the desired result.
Let and be a nonnegative and increasing function.
Consider (1) and the following two inequalities:
Now we are ready to introduce the Hyers-Ulam and generalized Hyers-Ulam-Rassias stability concepts for (1).
Definition 7. Equation (1) is Hyers-Ulam stable if there exists a number such that for each solution of inequality (23) there exists a solution of (1) with
Remark 8. A function is a solution of inequality (23) if and only if there is such that(i), , ;(ii).
Remark 9. If is a solution of inequality (23), then is a solution of the following integral inequality:
Let be a nonnegative function.
Definition 10. Equation (1) is generalized Hyers-Ulam-Rassias stable if there exists a number such that for each solution of inequality (24) there exists a solution of (1) with
Remark 11. A function is a solution of inequality (24) if and only if there is such that(i), , ;(ii).
Remark 12. If is a solution of inequality (24), then is a solution of the following integral inequality:
4. Hyers-Ulam Stability
In this section, we present the stability results.
Theorem 13. Assume that there exists such thatThen (1) is Hyers-Ulam stable on finite time interval with respect to and .
Proof. Let be a solution of inequality (23) and the solution of Cauchy problem (1), and its expression is Keeping in mind (26), we have By (30) we obtain which implies that (1) is Hyers-Ulam stable with respect to and . The proof is completed.
Example 14. Set , , , and . Then the following heat conduction equation is Ulam-Hyers stable with respect to and where we use (13) in Remark 4.
Theorem 15. Assume that there exists such thatThen (1) is generalized Hyers-Ulam-Rassias stable with respect to and .
Proof. Let be a solution of inequality (24) and the solution of Cauchy problem (1), and its expression is Keeping in mind (29), we have Hence, we have which implies that (1) is Hyers-Ulam-Rassias stable with respect to and . The proof is completed.
Remark 16. Condition (36) can be changed to which is reasonable due to the fact that (see [15, (6.3.7)])where the symbols and denote the Fourier transform and Laplace transform, respectively.
Note that, for , , one has (see [15, (1.3.55)]) Then we can choose and .
Example 17. Set and . Then the following heat conduction equationis Hyers-Ulam-Rassias stable with respect to and , where we use (13) in Remark 4 again.
5. Existence of Solution
Let be the Banach space of all bounded continuous function on with fixed equipped with the norm . Consider the Banach space of all bounded continuous function defined on equipped with the norm . Next, we introduce Banach space equipped with the norm .
We introduce the following assumptions: [H1]: is jointly continuous on with . [H2]: For any , there exists a constant such thatfor all and all with and .
Define where .
Theorem 18. Assume that [H1] and [H2] hold. Then (2) has unique solution in and .
Proof. Define given byNote that Multiplying to the above inequality and taking the supremum, we have Observing the definition of and triangular inequality, one has Next, we show that . By [H2] with , Multiplying to the above inequality and taking the supremum, we have Now we check that is contraction. Observe that Multiplying to the above inequality and taking the supremum, we have By contraction mapping principle, has a unique fixed point in which is the solution of (2).
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 (7640), and Graduate Course of Guizhou University (ZDKC003).