Mittag-Leffler–Hyers–Ulam stability of differential equation using Fourier transform

This research paper aims to present the results on the Mittag-Leffler–Hyers–Ulam and Mittag-Leffler–Hyers–Ulam–Rassias stability of linear differential equations of first, second, and nth order by the Fourier transform method. Moreover, the stability constant of such equations is obtained. Some examples are given to illustrate the main results.


Introduction
In recent years, there has been subject so far-reaching of research in derivative and differential equation because of its performance in numerous branches of pure and applied mathematics. The standards of differential equation have been unlimited and characterize physical models of many phenomena in various fields (see [1]).
As we all know, the main difficulty to find exact solution of such equation is very crucial, and the form of the exact solution (if it exists) is often so arduous that it is not appropriate for numerical calculation. In view of this, it is imperative to discuss approximate solution and ask whether it lies near the exact solution. Mostly, we say that a differential equation is stable in the Hyers-Ulam sense if, for every solution of the differential equation, there exists an approximate solution of the perturbed equation that is close to it.
The history of Hyers-Ulam stability starts from the middle of the nineteenth century. The class of stability was first formulated by Ulam [2] for functional equation which was solved by Hyers [3] for an additive function defined on a Banach space. After this result, the stability concept was investigated and generalized by Rassias [4], which is called Hyers-Ulam-Rassias stability. Further, Alsina and Ger [5] established the Hyers-Ulam stability of differential equations by replacing functional equation. Rezaei and Jung and Rassias [6] investigated the Hyers-Ulam stability of linear differential equation by applying the Laplace transform method. In [7], Algifiary and Jung gave Hyers-Ulam stability of nth order linear differential equation with the help of the Laplace transform method.
Using the Hyers-Ulam method, Wu and Baleanu [8] proved the Mittag-Leffler stability of impulsive fractional difference equations; Wu, Baleanu, and Huang [9] proved the Mittag-Leffler stability of linear fractional delay difference equations with impulse, and Wu et al. [10] investigated the Mittag-Leffler stability analysis of fractional discrete-time neural networks via the fixed point technique.
In this paper, we introduce some new concepts concerning the stability of differential equation in the Mittag-Leffler-Hyers-Ulam sense by the Fourier transform method. The Fourier transform and Mittag-Leffler function are effective tools for analytic expression for the solution of linear differential equation of integer or noninteger order. The Mittag-Leffler function E α (z α ) was introduced by Mittag-Leffler [11] in connection with the method of divergent series. The generalization and properties of E α (z α ) were studied and discussed in [12][13][14][15]. The Fourier transform is a kind of integral transform, and it was used by Fourier in 1807. It converts differential equation into simple algebraic equation. After solving the algebraic equation, we can find the solution of the original equation by inverse Fourier transform. For more details, see [16,17].
Motivated by ongoing research on the stability of differential equation, in this paper, we discuss the existence and the Mittag-Leffler-Hyers-Ulam stability of linear homogeneous differential equation with the help of Fourier transform. The contribution of the paper is outlined as follows: In Sect. 2, some definitions, lemmas, and theorems are introduced. In Sect. 3, the Hyers-Ulam-Mittag-Leffler stability of differential equation of first, second, and nth order is presented. The conclusion and examples are given in Sects. 4 and 5, respectively.

Preliminaries
In this section, we recall some basic definitions, notations, and theorems for further work. Throughout this paper, let F be either a real field R or a complex field C. (2.1) Also the inverse Fourier transform associated with H(ξ ) is given by for any x ∈ R, and the relation F -1 F(H) = H holds true almost everywhere on R.
In the following, we give some properties of the Fourier transform which are closely related to solution process.
The convolution of two functions H 1 (x) and H 2 (x) is defined as We have the following theorem.
Notice that if θ (x) is the Heaviside step function, then

Definition 2.5
The two-parameter Mittag-Leffler function is denoted by E α,β (z) and is defined as When α = β = 1, the above equation becomes

Theorem 2.6
For any x, α ∈ C with Re(α) > 0, the Fourier transform of Mittag-Leffler function is Proof By Mittag-Leffler function of one parameter for x ∈ R, we get Taking Fourier transform of (2.3), we have This completes the proof.

Main results
In this section, we study the existence and stability for differential equation (1.1). Moreover, we derive the stability constant for Eq. (1.1).

Mittag-Leffler-Hyers-Ulam stability of linear differential equation of first order
In this subsection, by means of Fourier transform and convolution principle, we establish the stability of the homogeneous first-order differential equation where H(x) is a continuously differentiable function and a is a constant.
where K is a Mittag-Leffler-Hyers-Ulam stability constant.
Proof Assume that a continuously differentiable function H(x) satisfies inequality ( where C is a constant. Introduce a function η : (-∞, ∞) → F such that Suppose that |η(x)| ≤ E α (x α ). By taking the Fourier transform of (3.6), it is transformed into The method of variation of constant gives the unique solution of (3.7), which is Applying the property of Fourier transform and the formula of convolution, we obtain It follows from (3.5) and (3.8) that and so for all x > 0. Clearly, this implies that the homogeneous linear differential equation (3.1) has Mittag-Leffler-Hyers-Ulam stability.

Mittag-Leffler-Hyers-Ulam stability of linear differential equation of second order
In this subsection, we are going to verify that the approximate solution is near the exact solution for the linear differential equation of second order with the help of the Fourier transform method.

Definition 3.3 The linear differential equation (3.10) is said to have Mittag-Leffler-Hyers-Ulam stability if there exists a constant K > 0 with the following property: for every > 0 and a continuously differentiable function H(x) ∈ L (R) satisfying the inequality
where E α is a Mittag-Leffler function, there exists some H o (x) ∈ L (R) satisfying differential equation (3.10) such that then there exist some H o (x) ∈ L (R) and K > 0 satisfying (3.10) such that

10) has Mittag-Leffler-Hyers-Ulam stability.
Proof Let > 0 and H(x) ∈ L (R) such that First, we will compute the classical solution of (3.10). Apply the Fourier transform with respect to variable x defined by (2.1) to (3.10). By Let M 1 and M 2 be distinct roots of the characteristic equation of (3.12) Since a, b are constant in F such that we have (-iξ ) 2 + (-iξ )a + b = (iξ -M 1 )(iξ -M 2 ). Thus the solution of transformed equation (3.12) is (3.13) where C 1 and C 2 are constant. Now, we introduce the function (3.14) Next, we will show Mittag-Leffler-Hyers-Ulam stability of (3.10). By taking the Fourier transform of (3.14), it is transformed into The method of variation of constant gives the unique solution of (3.15), which is ). By the inverse Fourier transform, we get .
By taking account of the property of Fourier transform, we get where θ (x) is a Heaviside step function. (3.16) becomes Applying the formula of convolution, we obtain It follows from (3.13) and (3.17) that for all x > 0, and so we get and so This completes the proof of the theorem.

Corollary 2 Let a be a scalar in F and H(x) ∈ L (R). Assume that there exists a constant K > 0 such that H(x) ∈ L (R) satisfies the differential inequality
for all x ∈ R. Then there exists a solution H o (x) ∈ L (R) of differential equation (3.10) such that for all x ∈ R, i.e., Eq. (3.10) has Mittag-Leffler-Hyers-Ulam-Rassias stability.

Mittag-Leffler-Hyers-Ulam stability of linear differential equation of nth order
Now, we give the proof of Mittag-Leffler-Hyers-Ulam stability of the linear differential equation of nth order Theorem 3.6 Let a i ∈ F. Assume that the characteristic equation of (3.18) has n distinct positive roots. If, for any > 0, H ∈ L (R) satisfies the differential inequality for all x > 0, then there exists a solution H o ∈ L (R) of differential equation (3.18) such that Proof Let > 0 and H(x) ∈ L (R) such that First, we will compute the classical solution of (3.18). By applying the Fourier transform with respect to variable x by using where H(ξ ) is the Fourier transform of H(x), (3.18) reduces to (-iξ ) n + a n-1 (-iξ ) n-1 + · · · + a 1 (-iξ ) + a 0 H(ξ ) = 0. (3.19) Let M 1 , M 2 , . . . , M n be distinct roots of the characteristic equation M n + a n-1 M n-1 + · · · + a 1 M + a 0 = 0.

Conclusion
This research has made an attempt to analyze the Mittag-Leffler-Hyers-Ulam and Mittag-Leffler-Hyers-Ulam-Rassias stability of linear differential equation with constant coefficients. Also we have showed that the Mittag-Leffler function and Fourier transform play an immodest role to prove the stability of differential equation. This new method of stability unifies different classes of differential equations, which may inspire further research in this domain.