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MittagLeffler–Hyers–Ulam stability of differential equation using Fourier transform
Advances in Difference Equations volume 2020, Article number: 389 (2020)
Abstract
This research paper aims to present the results on the MittagLeffler–Hyers–Ulam and MittagLeffler–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 farreaching 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 MittagLeffler stability of impulsive fractional difference equations; Wu, Baleanu, and Huang [9] proved the MittagLeffler stability of linear fractional delay difference equations with impulse, and Wu et al. [10] investigated the MittagLeffler stability analysis of fractional discretetime neural networks via the fixed point technique.
In this paper, we introduce some new concepts concerning the stability of differential equation in the MittagLeffler–Hyers–Ulam sense by the Fourier transform method. The Fourier transform and MittagLeffler function are effective tools for analytic expression for the solution of linear differential equation of integer or noninteger order. The MittagLeffler function \(E_{\alpha }(z^{\alpha })\) was introduced by MittagLeffler [11] in connection with the method of divergent series. The generalization and properties of \(E_{\alpha }(z^{\alpha })\) were studied and discussed in [12–15]. 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].
At present, some remarkable results to Hyers–Ulam–MittagLeffler stability of differential equation have been reported in [18–28]. In particular, Kalvandi, Eghbali, and Rassias [18] discussed MittagLeffler–Hyers–Ulam stability for the secondorder differential equation
and also proved the stability of Lane–Emden equation of second order. Existence and uniqueness of MittagLeffler–Ulam stable solution for fractional integrodifferential equation with nonlocal initial condition have been proved in [22]. In 2020, Liu et al. studied Hyers–Ulam stability and existence of solutions for fractional differential equation with MittagLeffler kernel [20]. To the best of our knowledge, there are few results on MittagLeffler–Hyers–Ulam stability of differential equation by the Fourier transform method.
Motivated by ongoing research on the stability of differential equation, in this paper, we discuss the existence and the MittagLeffler–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–MittagLeffler 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 \(\mathbb{F}\) be either a real field \(\mathbb{R}\) or a complex field \(\mathbb{C}\).
Definition 2.1
([16])
If a function \(\mathcal {H}: \mathbb{R} \rightarrow \mathbb{F}\) is piecewise continuous in each finite interval and is absolutely integrable in \(\mathbb{R}\), then the Fourier transform associated with \(\mathcal {H} \in L'(\mathbb{R})\) is a mapping \(\widehat{\mathcal {H}}(\xi ):\mathbb{R} \rightarrow \mathbb{F}\) given by the integral
Also the inverse Fourier transform associated with \(\widehat{\mathcal {H}}(\xi )\) is given by
for any \(x \in \mathbb{R}\), and the relation \(F^{1}F(\mathcal {H}) = \mathcal {H}\) holds true almost everywhere on \(\mathbb{R}\).
In the following, we give some properties of the Fourier transform which are closely related to solution process.
Lemma 2.2
Let\(\mathcal {H} \in L'(\mathbb{R})\), \(F (\mathcal {H} )(x)=\widehat{\mathcal {H}}(\xi )\), and\(\theta (x)\)be the Heaviside step function defined by\(\theta (x)=1\)for\(x \ge 0\)and\(\theta (x)=0\)for\(x<0\). Then

1.
\(F ({\mathcal {H}} (x\pm a ) ) = e^{\pm ia}\widehat{\mathcal {H}} (\xi )\);

2.
\(F (e^{zx} \theta (x) ) (\xi ) = \frac{1}{i\xi +z}\)provided that\(\operatorname{Re}(z)>0\);

3.
\(F ((ix)^{n}({\mathcal {H}}(x) ) (\xi ) = F^{n} (\xi )\);

4.
\(F( (\mathcal {H} )^{n}(x)) (\xi ) = (i\xi )^{n}\widehat{\mathcal {H}} (\xi )\).
The convolution of two functions \({\mathcal {H}}_{1}(x)\) and \({\mathcal {H}}_{2}(x)\) is defined as
We have the following theorem.
Theorem 2.3
([30])
Let\({\mathcal {H}}_{1}\), \({\mathcal {H}}_{2}\)\(\in L^{1}(\mathbb{R})\). Then

1.
\(F({\mathcal {H}}_{1}*{\mathcal {H}}_{2}) = F({\mathcal {H}}_{1})F({\mathcal {H}}_{2})\);

2.
\(F^{1}({\mathcal {H}}_{1}{\mathcal {H}}_{2}) = F^{1}({\mathcal {H}}_{1})*F^{1}({\mathcal {H}}_{2})\).
Notice that if \(\theta (x)\) is the Heaviside step function, then
Definition 2.4
([11])
The MittagLeffler function of one parameter is defined as
where \(\operatorname{Re}(\alpha ) >0\) and \(z, \alpha \in {\mathbb{C}}\).
Definition 2.5
The twoparameter MittagLeffler function is denoted by \(E_{\alpha ,\beta }(z)\) and is defined as
When \(\alpha = \beta = 1\), the above equation becomes
Theorem 2.6
For any\(x, \alpha \in \mathbb{C}\)with\(\operatorname{Re}(\alpha ) >0\), the Fourier transform of MittagLeffler function is
Proof
By MittagLeffler function of one parameter for \(x \in \mathbb{R}\), we get
Taking Fourier transform of (2.3), we have
Letting \(i\xi x = z\), \(i \xi \,dx = dz\), we get
Since \(\int _{0}^{\infty } z^{k} e^{z} \,dz = \varGamma {(k+1)}= k! \), we obtain
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).
MittagLeffler–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 firstorder differential equation
where \({\mathcal {H}}(x)\) is a continuously differentiable function and a is a constant.
Definition 3.1
We say that linear differential equation (3.1) is said to have MittagLeffler–Hyers–Ulam stability if there exists a constant \(K>0\) with the following: for every \(\epsilon > 0\) and a continuously differentiable function \({\mathcal {H}}(x)\) satisfying the inequality
there exists some \({\mathcal {H}}_{o}(x)\) satisfying differential equation (3.1) such that
where K is a MittagLeffler–Hyers–Ulam stability constant.
Remark 1
If ϵ and Kϵ are replaced by continuous functions \(\phi (x)\) and \(\varPhi (x)\) in the above definition, then we say that Eq. (3.1) has Hyers–Ulam–MittagLeffler–Rassias stability.
Theorem 3.2
Letabe a scalar in\(\mathbb{F}\). Assume that, for every\(\epsilon > 0\), there exists\(K > 0\)such that\({\mathcal {H}}(x) \in L'(\mathbb{R})\)satisfying the differential inequality
for all\(x \in {\mathbb{R}}\). Then there exists a solution\({\mathcal {H}}(x) \in L'(\mathbb{R})\)of differential equation (3.1) such that
for all\(x \in {\mathbb{R}}\).
Proof
Assume that a continuously differentiable function \({\mathcal {H}}(x)\) satisfies inequality (3.3). First, let us find the classic solution of (3.1). Apply the derivative of Fourier transform
with respect to the variable x. Here \(\widehat{\mathcal {H}}(\xi )\) is the Fourier transform of \({\mathcal {H}}(x)\). Then (3.1) reduces to
Thus the solution of transformed equation (3.4) is
where C is a constant. Introduce a function \(\eta : (\infty , \infty ) \rightarrow {\mathcal {F}}\) such that
Suppose that \(\vert \eta (x) \vert \leq \epsilon E_{\alpha }(x^{\alpha })\). 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 MittagLeffler–Hyers–Ulam stability. □
Similarly, we can explore MittagLeffler–Hyers–Ulam–Rassias stability of differential equation (3.1).
Corollary 1
For every continuously differential function\({\mathcal {H}}(x) \in L'(\mathbb{R})\)satisfying the differential inequality
there exists a solution\({\mathcal {H}}_{o}(x) \in L'(\mathbb{R})\)of differential equation (3.1) such that
MittagLeffler–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 MittagLeffler–Hyers–Ulam stability if there exists a constant \(K>0\) with the following property: for every \(\epsilon > 0\) and a continuously differentiable function \({\mathcal {H}}(x) \in L'(\mathbb{R})\) satisfying the inequality
where \(E_{\alpha }\) is a MittagLeffler function, there exists some \({\mathcal {H}}_{o}(x) \in L'(\mathbb{R})\) satisfying differential equation (3.10) such that
Theorem 3.4
Assume that the characteristic equation of (3.10) has two different positive roots. If, for every\(\epsilon > 0\), \({\mathcal {H}}(x) \in L'(\mathbb{R})\)satisfies the inequality
then there exist some\({\mathcal {H}}_{o}(x) \in L'(\mathbb{R})\)and\(K > 0\)satisfying (3.10) such that
that is, Eq. (3.10) has MittagLeffler–Hyers–Ulam stability.
Proof
Let \(\epsilon > 0\) and \({\mathcal {H}}(x) \in L'(\mathbb{R})\) 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
where \(\widehat{\mathcal {H}}(\xi )\) is the Fourier transform of \({\mathcal {H}}(x)\), (3.10) reduces to
Let \({\mathcal {M}}_{1}\) and \({\mathcal {M}}_{2}\) be distinct roots of the characteristic equation of (3.12)
Since a, b are constant in \(\mathbb{F}\) such that
we have \((i\xi )^{2}+(i\xi )a+b = (i\xi {\mathcal {M}}_{1} ) (i\xi  {\mathcal {M}}_{2} )\).
Thus the solution of transformed equation (3.12) is
where \(C_{1}\) and \(C_{2}\) are constant. Now, we introduce the function
Next, we will show MittagLeffler–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
Set \(\widehat{Q}(\xi ) = \frac{1}{ (i\xi  {\mathcal {M}}_{1} ) (i\xi  {\mathcal {M}}_{2} )} = \frac{1}{{\mathcal {M}}_{2}{\mathcal {M}}_{1}} ( \frac{1}{(i\xi  {\mathcal {M}}_{1})(i\xi  {\mathcal {M}}_{2})} ) \).
By the inverse Fourier transform, we get
By taking account of the property of Fourier transform, we get
where \(\theta (x)\) 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. □
Similarly, we can explore that MittagLeffler–Hyers–Ulam–Rassias stability of differential equation (3.10).
Corollary 2
Letabe a scalar in\(\mathbb{F}\)and\({\mathcal {H}}(x) \in L'(\mathbb{R})\). Assume that there exists a constant\(K > 0\)such that\({\mathcal {H}}(x) \in L'(\mathbb{R})\)satisfies the differential inequality
for all\(x \in {\mathbb{R}}\). Then there exists a solution\({\mathcal {H}}_{o}(x) \in L'(\mathbb{R})\)of differential equation (3.10) such that
for all\(x \in {\mathbb{R}}\), i.e., Eq. (3.10) has MittagLeffler–Hyers–Ulam–Rassias stability.
MittagLeffler–Hyers–Ulam stability of linear differential equation of nth order
Now, we give the proof of MittagLeffler–Hyers–Ulam stability of the linear differential equation of nth order
Definition 3.5
The linear differential equation (3.18) is said to have MittagLeffler–Hyers–Ulam stability if there exists a constant \(K>0\) with the following property: for every \(\epsilon > 0\) and a continuously differentiable function \({\mathcal {H}}(x) \in L'(\mathbb{R})\) satisfying the inequality
there exists some \({\mathcal {H}}_{o}(x)\) satisfying differential equation (1.1) such that
Theorem 3.6
Let\(a_{i} \in {\mathbb{F}}\). Assume that the characteristic equation of (3.18) hasndistinct positive roots. If, for any\(\epsilon > 0\), \({\mathcal {H}} \in L'(\mathbb{R})\)satisfies the differential inequality
for all\(x >0\), then there exists a solution\({\mathcal {H}}_{o} \in L'(\mathbb{R})\)of differential equation (3.18) such that
for all\(x \in {\mathbb{R}}\).
Proof
Let \(\epsilon > 0\) and \({\mathcal {H}}(x) \in L'(\mathbb{R})\) 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 \(\widehat{{\mathcal {H}}}(\xi )\) is the Fourier transform of \({\mathcal {H}}(x)\), (3.18) reduces to
Let \({{\mathcal {M}}}_{1}, {\mathcal {M}}_{2},\ldots , {\mathcal {M}}_{n}\) be distinct roots of the characteristic equation
Since \(a_{i}\) are constant in \(\mathbb{F}\) such that
we have
Thus the solution of transformed equation (3.19) is
where \(C_{1}, C_{2}, \ldots , C_{n}\) are constant. Now we introduce the function
Next, we will show the MittagLeffler–Hyers–Ulam stability of (3.18). By taking the Fourier transform of (3.21), it is transformed into
The method of variation of constant gives the unique solution of (3.22), which is
Set
By the inverse Fourier transform, we get
By taking account of the property of Fourier transform, we get
where \(\theta (x)\) is a Heaviside step function. So (3.23) becomes
By applying the formula of convolution, we obtain
It follows from (3.20) and (3.24) that
and so
Hence differential equation (3.18) has MittagLeffler–Hyers–Ulam stability. □
Similarly, we can prove the MittagLeffler–Hyers–Ulam–Rassias stability of Eq. (3.18).
Corollary 3
Assume that the characteristic equation of (3.18) has ‘n’ different positive roots. If, for every\(\epsilon > 0\), \({\mathcal {H}}(x) \in L'(\mathbb{R})\)satisfies the inequality
then there exist some\({\mathcal {H}}_{o}(x) \in L'(\mathbb{R})\)and\(K > 0\)satisfying (3.18) such that
Numerical examples
Example 4.1
Consider the following differential equation:
and the inequality
where \(\mathcal {H} \in L'(\mathbb{R})\).
Comparing with (3.1) and (3.3), we have, for \(\alpha = 1\), \(a=\frac{1}{\sqrt{(1+\exp (7))}}\).
The solution of Eq. (4.1) is computed and depicted in Fig. 1.
By Theorem 3.2, problem (4.1) has a solution and is Hyers–Ulam–MittagLeffler stable with
Example 4.2
Consider the following differential equation:
and the inequality
where \(\mathcal {H} \in L'(\mathbb{R})\).
Comparing with (3.10) and (3.11), we have, for \(\alpha = 2\), \(a=0\) and \(b=4i\).
Using MATLAB, the solution of Eq. (4.2) is computed and depicted in Fig. 2.
By Theorem 3.4, problem (4.2) has a solution and is Hyers–Ulam–MittagLeffler stable with
Example 4.3
Consider the following differential equation:
and the inequality
where \(\mathcal {H} \in L'(\mathbb{R})\).
Comparing with (3.10) and (3.11), we have, for \(\alpha = 2\), \(a=\frac{1}{6}\) and \(b=\frac{1}{6}\).
Using MATLAB, the solution of Eq. (4.3) is computed and depicted in Fig. 3.
By Theorem 3.4, problem (4.3) has a solution and is Hyers–Ulam–MittagLeffler stable with
where \(K=\frac{6}{5}\).
Conclusion
This research has made an attempt to analyze the MittagLeffler–Hyers–Ulam and MittagLeffler–Hyers–Ulam–Rassias stability of linear differential equation with constant coefficients. Also we have showed that the MittagLeffler 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.
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This work was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science, and Technology (NRF2017R1D1A1B04032937).
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Mohanapriya, A., Park, C., Ganesh, A. et al. MittagLeffler–Hyers–Ulam stability of differential equation using Fourier transform. Adv Differ Equ 2020, 389 (2020). https://doi.org/10.1186/s1366202002854z
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MSC
 58F15
 58F17
 53C35
 39B82
 39B62
 39B52
Keywords
 MittagLeffler–Hyers–Ulam stability
 Fourier transform
 Differential equation