Fractional Fourier transform and stability of fractional differential equation on Lizorkin space

In the current study, we conduct an investigation into the Hyers–Ulam stability of linear fractional differential equation using the Riemann–Liouville derivatives based on fractional Fourier transform. In addition, some new results on stability conditions with respect to delay differential equation of fractional order are obtained. We establish the Hyers–Ulam–Rassias stability results as well as examine their existence and uniqueness of solutions pertaining to nonlinear problems. We provide examples that indicate the usefulness of the results presented.


Introduction
In recent years, the area related to fractional differential and integral equations has received much attention from numerous mathematicians and specialists. The derivatives of fractional order portray physical models of multiple phenomena in different fields such as biology, physics, mechanics, dynamical systems, and so on (see [1][2][3][4][5][6][7][8] and the references therein).
The possibility of fractional calculus was presented in 1695, when the notation d ν dt ν h(x) was introduced to indicate the νth order derivative of function h(x). Specifically, Leibniz composed a letter to L'Hospital in which he posed an enquiry on the derivative of order ν = 1 2 , which led to the establishment of fractional calculus. Later on, fractional derivative was presented by Lacroix [9]. Perhaps the most utilized fractional derivatives are Riemann-Liouville (R-L) and Caputo derivatives, which assume an immodest role in fractional order differential equation (FRDE).
One of the best examination regions in FRDE, which receives vast considerations by analysts, entails the existence theory of solution. This is a rapidly moving topic of investigation. For details with respect to the present hypothesis, see [10][11][12][13][14][15]. Finding an exact solution of FRDE is exceptionally troublesome, and the type of exact solution is regularly Figure 1 Flow chart of fractional Fourier transform so complicated that it is not convenient for numerical computation. Considering this, it is important to study an approximate solution with a relatively simple form and examine how close both the approximate and exact solutions are. By and large, we state that a FRDE is said to be Hyers-Ulam (H-U) stable if, for every solution of the FRDE, there exists an approximate solution of the concerned equation that is close to it.
Ulam [16] formulated the stability of a functional equation, which was solved by Hyers [17] using an additive function defined on the Banach space. This result led Rassias [18] to study and generalize the stability concept, establishing the Hyers-Ulam-Rassias stability. An integral transform (introduced by Fourier) involves a trigonometric form of the Mittag-Leffler function to identify an analytic solution with respect to a differential equation of fractional order. The Fourier transforms, Mittag-Leffler function, and fractional trigonometric function constitute an effective tool for analytic expression pertaining to the solution of differential equation of noninteger order. Indeed, the Fourier transform has become popular in view of recent developments in differential applications. It is also seen as the easiest and most effective way among many other transforms. Luchko [19] defined the fractional Fourier transform (FRFT) of real order ν, 0 < ν ≤ 1, and discussed its important properties. The application of FRFT for undertaking certain types of differential equations of fractional order has also been conducted. Indeed, there are many studies on FRFT and its applications in the literature [20][21][22][23].
In 2017, Wang et al. [24] discussed the stability of fractional differential equation based on the right-sided Riemann-Liouville fractional derivatives with respect to a continuous function space. The fixed point theorem and weighted space method were exploited. In [25], a study on the H-U stability condition was conducted, focusing on an impulsive R-L fractional neutral functional stochastic differential equation with time delays. In [26], the stability criteria pertaining to a class of fractional differential equations was investigated, in which the Krasnoselskii fixed point method was employed. Recently, Opadhyay et al. [27] discussed the R-L fractional differential equations using the Hankel transform method. At present, some remarkable results to the stability of fractional differential equation have been reported (see [28][29][30][31] and the references therein). In [32,33], the author studied the H-U stability of linear differential equation by using Fourier transform. To the best of our knowledge, there are no results on H-U stability of fractional differential equation by fractional Fourier transform (FRFT). The flow chart of fractional fourier transform is represented in Fig. 1.
Motivated by the ongoing research in this field, we examine the H-U stability and generalized H-U stability of FRDE in this study, i.e., and the delay differential equation of fractional order where D ν ϑ represents R-L fractional derivative, ξ > 0, ϑ ∈ R, and 0 < ν ≤ 1 with the help of FRFT. In our investigation, we exploit the fractional Fourier transform and present it in an integral form. Furthermore, using the convolution concept and properties of fractional Fourier transform, the solution pertaining to the stability conditions with respect to FRDE is established. Specifically, we analyze Hyers-Ulam-Rassias stability of the nonlinear FRDE and use the fixed point theorems for examining the existence and uniqueness of the solution.
We organize this article as follows.

Fundamentals
Consider L 1 (R) as the space pertaining to the complex-valued Lebesgue integrable function on the real line R with norm The definition of a Fourier transform with respect to a function h ∈ L 1 (R) is (2.1) The form of the associated inverse Fourier transform is Note that Fourier transform is useful for conversion of a function between the time and frequency domains. It adopts the principle of rotation operation on the time-frequency distribution. Given parameter ν, we can express the fractional Fourier transform of function h(x) in a one-dimensional case as follows [34]: and n is an integer, while As such, the form of the associated inverse fractional Fourier transform is

Definition 2.2
The fractional trigonometric function is denoted by Luchko et al. [36] introduced a new fractional Fourier transform F ν of order ν, (0 < ν ≤ 1), and its definition is As such, the definition of the associated inverse fractional Fourier transform is for any x ∈ R and ν > 0. If ν = 1, then H ν (ω) and the classical Fourier transform (2.1) are the same. Suppose that the space of a function with rapid decrease is denoted as S. In other words, the following relation with respect to the space of infinity differentiable functions v(x) on R is satisfied: Based on V (R), the following relation with respect to a set of functions v ∈ S is satisfied: , and it is defined as the Fourier pre-image of the space The reason for using the Lizorkin space is its convenience in using the Fourier transform as well as the inverse Fourier transform with fractional integration and differentiation operators. The properties and associated details of the Lizorkin space have been discussed in many studies (see [37][38][39]). In our study, we use F to represent the domain of either real R or complex C. According to the definition of Lizorkin space, the orthogonality condition is satisfied by any function h ∈ (R), i.e., Note that the property of invariance pertaining to the Fourier transform and its inverse holds for the space (R). In other words, both transforms are inverse of one another, i.e.,

Definition 2.3 Given a function
The inverse fractional Fourier transform for function h ∈ (R) is for any x ∈ R and ν > 0. Some properties of FRFT that are closely related to the solution in this study are given as follows. Let h, h 1 , and h 2 be functions belonging to (R). Then

Definition 2.5 ([40])
The definition of R-L fractional integral of order ν > 0 is and

Definition 2.6 ([40])
The definition of R-L fractional derivative of order ν > 0 is

Lemma 2.1 ([39]) Based on the integration by parts formula, the R-L derivative holds for
Our current study considers the definition with respect to a fractional derivative oper- where D ν and D ν + denote the left-hand and right-hand R-L fractional derivatives of order ν, in which 0 < ν < 1.

Lemma 2.2 A function that is continuously differentiable is denoted as h ∈ (R)
, ω ∈ R/{0}, and 0 < ν ≤ 1, and it is able to satisfy Proof By taking into account of the R-L fractional integral (2.7), we get (2.13) Using the variables substitution η = xt, we obtain (2.14) Both the integrals in the above equation can be evaluated by using the integral formula in [41]: Substituting equation (2.15) and (2.16) in equation (2.14), we get By the definition of R-L fractional derivative and equation (2.17), we have Hence, the proof is completed.
Similarly, the following lemma is formulated.
(2) B is an equicontinuous set, i.e., for any > 0, there exists δ > 0 such that is a bounded set. Then has at least one fixed point on ψ.
Definition 2.7 The fractional differential equation where > 0 and K > 0 is the H-U stability constant.

Definition 2.8 The fractional differential equation with delay
is satisfied by a continuously differentiable mapping h : R → F, and there exists a solution h ν : R → F of differential equation where > 0 and K > 0 is the Hyers-Ulam stability (HUS) constant.

Stability results of linear FRDE
Based on the fractional Fourier transform, we derive the results with respect to the H-U stability of FRDE in the following theorem.

Theorem 3.1 Given (R), a real continuous function is denoted by G(x)
, and assume 0 < α ≤ 1. If a function h : R → F is able to satisfy the following: for all x ∈ R and for some > 0, then, with respect to fractional differential equation

Now, taking modulus with respect to equation (3.7) on both sides gives
Similarly, we can prove that FRDE (3.1) is H-U-Rassias stable with the help of FRFT.

Corollary 3.1 Given (R), a real continuous function is G(x), and suppose 0 < ν ≤ 1, the following inequality is satisfied subject to the existence of a continuous function φ(x)
for all x ∈ R. As such, a solution h ν : R → F of fractional differential equation (3.1) exists whereby where (x) = Cβ(x) R |xτ | ν-1 dτ for any x ∈ R.

Theorem 3.2 Consider the delay FRDE
where G(x) ∈ (R), then a constant K having the following property exists, i.e., for every h ∈ (R) and given > 0 satisfying Proof Let h ∈ (R) satisfy inequality (3.9) and define for each x ∈ R. We have By taking into account the formula and the property of FRFT and from equation (3.12), we can derive Applying the formula of convolution, we obtain By equation (3.12) and simple computation, we obtain Since F ν is one-to-one, it follows that (D ν From equation (3.11) and equation (3.12), we obtain

Now, taking modulus with respect to equation (3.13) on both sides gives
where K = C R |x + ξτ | ν-1 dτ for any value of x and ξ > 0. Consequently, the fractional differential equation with delay equation Similarly, we can prove that FRDE (3.8) is generalized H-U stable with the help of FRFT.

Existence and stability results of nonlinear FRDE
The current section is concerned with establishing the existence and H-U-Rassias stability to the nonlinear FRDE Denote ψ = C(R, F) as the Banach space which contains all continuous functions from R in F and its norm is Proof The definition of an operator : ψ → ψ is for any x ∈ [-X, X]. As such, is well defined because of [D 1 ]. We obtain the following for any h 1 , h 2 ∈ ψ and given x ∈ [-X, X]: From the condition LC (2X) ν (1-ν) 2πν < 1, takes the form of a contraction mapping. As a result, we are able to establish that has a unique fixed point in accordance with the Banach contraction principle, and it is a unique solution with respect to FRDE (4.1).

Theorem 4.2 Operator is compact based on hypotheses [D 1 ]-[D 3 ].
Proof Consider the definition in equation (4.2) for the operator .
Step (1): We assume that h n whereby h n → h as n → ∞ in ψ. As such, we have the following given all x ∈ [-X, X]: We have the following with respect to [D 2 ]: Consequently, Note that h n → h as n → ∞ for every x ∈ [-X, X], and based on the Lebesgue dominated convergence theorem, Therefore, is continuous.
Step (2): Here, maps a bounded set in ψ is established. For this, we just have to prove that, for any κ * > 0, there exists ρ > 0 whereby given any In fact, for any x ∈ [-X, X], from equation (4.2), we have Consequently, we have Hence (E * ) is bounded.
Step (3): We prove that the operator is equicontinuous in ψ. Suppose x 1 , x 2 ∈ [-X, X] with 0 ≤ x 1 ≤ x 2 ≤ X, as E * is a bounded set in ψ, and assume h ∈ E * .
Note that as x 1 → x 2 , and consider the above inequality, its right-hand side approximates zero, and this indicates that is equicontinuous. Based on steps (1) to step (3), it is concluded that is completely continuous. Therefore, operator is compact as in agreement with the Arzela-Ascoli theorem.
Next, by exploiting the fixed point theorem of Schaefer, we are able to show that the solution for equation (4.1) exists. Proof Now, a set B ⊂ ψ is considered, and its definition is The following is obtained for every x ∈ [-X, X]: Therefore, B is bounded. According to Theorems 2.2 and 4.2, at least one fixed point exists with respect to operator . Therefore, the considered FODE (4.1) has at least one solution in ψ.
The following inequality is used for further analysis: Based on the following condition, we examine the generalized H-U stability.

Simulations
Example 5.1 Consider Note that h 1 (x) = x 2 satisfies From (3.5), we obtain the exact solution of equation ( Hence, our result can be applied to equation (5.1).  The exact solution of equation (5.2) is
By Theorem 3.1, we get π . Therefore, we can ascertain that (5.2) is stable in the sense of H-U.

Conclusion
In this paper, the analysis of our results indicates that fractional Fourier transform constitutes a useful method for tackling linear and nonlinear FRDE in (R). In addition, our study has established the stability and existence of solution of such equations using FRFT. The effectiveness of our results has been positively shown by using two illustrative examples. Further work will focus on the application of our results to various scientific and engineering problems.