Abstract
In this paper, we study the Ulam-Hyers-Mittag-Leffler stability for a linear fractional order differential equation with a fractional Caputo-type derivative using the fractional Fourier transform. Finally, we provide an enumeration of the chemical reactions of the differential equation.
1. Introduction
Fractional differential equations have more attention in the research area of mathematics, and there has been significant progress in this field. However, this idea is not new and as old as differential equations. The differential equations of fractional order have proved to be valuable tools in modeling multiple phenomena in different areas of science and engineering. Indeed, it has many uses in biology, physics, electromagnetics, mechanics, electrochemistry, etc. [1–3]. Fractional calculus was initiated from a question raised by L’Hospital to Leibnitz, which related to his generalization of meaning of notation for the derivative of order , when . In his reply, dated September 30, 1695, Leibnitz wrote to L’Hospital [4], “This is an apparent paradox from which one-day useful consequences will be drawn.” Recently, Ozaktas and Kutay [5] published on this topic, dealing with different characteristics in different ways.
A functional equation is stable if for each approximate answer there is a definite quantity about it. In 1940, the simulation and a hit theory suggested by Ulam [6] prompted the study of stability issues for numerous functional equations. He gave the University of Wisconsin Mathematical Colloquium a long form of talks, presenting a variety of unresolved questions. He raised one of the questions that were connected to the stability of the functional equation: “Give conditions for a linear function near an approximately linear function to exist.” The first result concerning the stability of functional equations was presented by Hyers [7] in 1941. The stability of the form is subsequently referred to as Hyers-Ulam stability. In 1978, the generalization associated with the Hyers theorem given by Rassias [8] makes it possible for the Cauchy difference to be unbounded. In 2004, Jung [9] studied the Hyers-Ulam stability of the differential equations . Jung [10, 11] continuously published the general setting for Hyers-Ulam stability of first-order linear differential equations. In 2006, Jung [12] concentrated on the Hyers-Ulam stability of an arrangement of differential equations with coefficients through the utilization of a matrix approach. Ponmana Selvan et al. [13] have solved the different types of Ulam stability for the approximate solution of a special type of th-order linear differential equation with initial and boundary conditions.
Zhang and Li [14] studied the Ulam stabilities of -dimensional fractional differential systems with order in 2011, and in the same year, Li and Zhang [15] proved the stability of fractional order derivative for differential equations. In 2013, Ibrahim [16] investigated the Ulam-Hyers stability for iterative Cauchy fractional differential equations and Lane-Emden equations. Kalvandi et al. [17], Liu et al. [18], and Vu et al. [19] presented and proved the different types of Hyers-Ulam stability of a linear fractional differential equations.
In 2012, Wang et al. [20] carried out pioneering work on the Hyers-Ulam stability for fractional differential equations with Caputo derivative using a fixed point approach, and in the same year, Wang and Zhou [21] proved the Hyers-Ulam stability of nonlinear impulsive problems for fractional differential equations. Wang et al. [22] investigated the Mittag-Leffler-Ulam-Hyers stability of fractional evolution equations.
In 2020, Unyong et al. [23] studied Ulam stabilities of linear fractional order differential equations in Lizorkin space using the fractional Fourier transform, and in the same year, Hammachukiattikul et al. [24] derived some Ulam-Hyers stability outcomes for fractional differential equations. In the next year, Ganesh et al. [25] derived some Mittag-Leffler-Hyers-Ulam stability, which makes sure the existence and individuation of an answer for a delay fractional differential equation by using the fractional Fourier transform. In 2022, Ganesh et al. [26] carried out pioneering in the field with the Hyers-Ulam stability for fractional order implicit differential equations with two Caputo derivatives using a fractional Fourier transform.
Motivated and inspired by the above results, in this paper, because of the help of fractional Fourier transform, we would like to investigate the Ulam-Hyers-Mittag-Leffler and Ulam-Hyers-Rassias-Mittag-Leffler stability of linear fractional order differential equations with the fractional Caputo-type derivative of the form: where is a times continuously differentiable function and is the fractional Caputo-type derivative of order .
2. Preliminaries
The following definitions, theorems, notations, and lemmas will be used to obtain the main objectives of this paper.
Definition 1 (see [27]). The one dimension fractional Fourier transform with rotational angle of function is given by where the kernel As such, the inversion formula of fractional Fourier transform is given by where the kernel
Definition 2. The Mittag-Leffler function is given in the following manner: where and are nonnegative constant.
Definition 3 (see [28]). The fractional integral operator of order of a function is written as where is the gamma function and .
Definition 4 (see [28]). The Riemann-Liouville fractional order derivative of is written as where the function is a continuous derivatives upto order .
Definition 5 (see [28]). The fractional Caputo-type derivative of order , is written as where the function is a continuous derivatives up to order . Then, let . The relation between Caputo and Riemann-Liouville fractional derivative is given by
Definition 6. Equation (1) has Ulam-Hyers-Mittag-Leffler stability, if there exist a continuously differentiable function satisfying the inequality for every , there exists a solution satisfying Equation (1) such that where is a nonnegative and stability constant.
Definition 7. The considered is a function. Equation (1) has Ulam-Hyers-Rassias-Mittag-Leffler stability, if there exist a continuously differentiable function satisfying the inequality for every , there exists a solution satisfying Equation (1) such that where is a nonnegative and stability constant.
3. Main Results
In this section, we will investigate to help of fractional Fourier transform to study the Ulam-Hyers-Mittag-Leffler stability of (1).
Theorem 8. If a function satisfies the inequality (11) for every , there exists a solution satisfying Equation (1) such that
Proof. Let us choose a function follow as Now, Taking (the fractional Fourier transform oprator) onto both sides of Equation (17), we have where , for and Setting By using fractional Fourier transform to (20), we have Hence,
Since is one-to-one operator, . Now, its follows form (19) and (21) that
Using the convolution property, we obtain where . In view of (13), we have
Now, applying the modules on both sides of Equation (24), we get where . Thus Equation (1) has Ulam-Hyers-Mittag-Leffler stability.
Corollary 9. The considered is a function. If a function satisfies the inequality (13), for every , there exists a solution satisfying Equation (1) such that i.e., Equation (1) has Ulam-Hyers-Rassias-Mittag-Leffler stability.
4. Applications
In this section, the standard kinetic equation in the chemical reaction that will be used to analyze this experimental data is revealed by the equation as follows: where ; ; ; ; . The model is presented in Figure 1.
Material balance for components: and for the first-order kinetic equation, we get in which the initial concentration at is presented by . Also, we have the same direction for material : in which the initial concentration at is presented by . Equation (29) can be integrated and, using the provided boundary condition, yields
Substituting (30) for (29) yields
Now, if we take the fractional Caputo derivative in (31) instead of the classical ones, we have
Figure 2 shows the solution of Equation (32) for various and .
5. Conclusions
In this paper, the objective is investigated by using the fractional Fourier transform to study the Ulam-Hyers-Mittag-Leffler stability of linear fractional differential equations. The required outcomes have been achieved by using the fractional Fourier transform. We could reach the suitable approximation value of xylose after a certain period of time, which is crucial for analyzing the kinetic equation in the chemical reaction process.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.