# Fractional Fourier Transform and Ulam Stability of Fractional Differential Equation with Fractional Caputo-Type Derivative

Authors: Arunachalam Selvam; Sriramulu Sabarinathan; Samad Noeiaghdam; Vediyappan Govindan
Journal: Journal of Function Spaces
 (2022)

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## 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.


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## Body

## 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 dny/dxnd for the derivative of order n∈N≔0,1,2,⋯, when n=1/2?. 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 ϑsp′s=ps. 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 mth-order linear differential equation with initial and boundary conditions.Zhang and Li [14] studied the Ulam stabilities of m-dimensional fractional differential systems with order 1<α<2 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, Unyonget 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:(1)CD0+σps+ηps=qs,where qs is a m−times continuously differentiable function and CD0+σ is the fractional Caputo-type derivative of order σ∈m−1,m,m∈N+.


## 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 angleσ of function ps∈L′R is given by
(2)Fσpsω=p^σω=∫RKσs,ωpsds,ω∈R,where the kernel
(3)Kσs,ω=Cσeip2+ω2cotσ/2−ipωcosecσ,ifσ≠mπ,12πe−ipω,ifσ=π2,Cσ=1−icotσ2π.
As such, the inversion formula of fractional Fourier transform is given by(4)ps=12π∫RKσs,ωp^σωdω,s∈R,where the kernel
(5)Kσs,ω¯=C′σe−ip2+ω2cotσ/2+ipωcosecσ,ifσ≠mπ,12πeipω,ifσ=π2,C′σ=2π1+icotσ.Definition 2.
The Mittag-Leffler function is given in the following manner:(6)Eσs=∑m=0∞smΓσm+1,σ>0Oneparameter,Eσ,μs=∑m=0∞smΓσm+μ,σ>0,μ>0Twoparameters.where σ and μ are nonnegative constant.Definition 3 (see [28]).
The fractional integral operator of orders>0 of a function p∈L1R+ is written as
(7)I0+σps=1Γσ∫0ss−uσ−1pudu,s>0,where Γ. is the gamma function and Re>0.Definition 4 (see [28]).
The Riemann-Liouville fractional order derivative ofs>0,m−1<σ<m,m∈N, is written as
(8)RLD0+σps=1Γm−σddsm∫0ss−um−σ−1pudu,where the function ps is a continuous derivatives upto order m−1.Definition 5 (see [28]).
The fractional Caputo-type derivative of orders>0,m−1<σ<m,m∈N, is written as
(9)CD0+σps=1Γm−σ∫0ss−um−σ−1pnudu,where the function ps is a continuous derivatives up to order m−1. Then, let s>0,σ∈R,m−1<σ<m,m∈N. The relation between Caputo and Riemann-Liouville fractional derivative is given by
(10)CD0+σps=D0+σps−∑k=0m−1s−ak−σΓk−σ+1pk0.Definition 6.
Equation (1) has Ulam-Hyers-Mittag-Leffler stability, if there exist a continuously differentiable function ps satisfying the inequality
(11)CD0+σps+ηps−qs≤εEσs,∀s>0,for every ε>0, there exists a solution pσs satisfying Equation (1) such that
(12)ps−pσs≤HεEσs,where H is a nonnegative and stability constant.Definition 7.
The consideredϕ:0,∞⟶0,∞ is a function. Equation (1) has Ulam-Hyers-Rassias-Mittag-Leffler stability, if there exist a continuously differentiable function ps satisfying the inequality
(13)CD0+σps+ηps−qs≤εϕsEσs,∀s>0,for every ε>0, there exists a solution pσs satisfying Equation (1) such that
(14)ps−pσs≤HϕsεEσs,where H 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 functionps satisfies the inequality (11) for every ε>0, there exists a solution pσs satisfying Equation (1) such that
(15)ps−pσs≤HεEσs.Proof.
Let us choose a functionys follow as
(16)ys=CD0+σps+ηps−qs.
Now,(17)ys=Dσps−∑k=0m−1sk−σΓk−σ+1pk0+ηps−qs,∀s>0.
TakingFσ (the fractional Fourier transform oprator) onto both sides of Equation (17), we have
(18)Fσys=FσDσps−∑k=0m−1sk−σΓk−σ+1pk0+ηps−qs=iωn/σσFσps−eiωn/σa∑k=0m−1aksk−σΓk−σ+1iωn/σk−σ+1+ηFσps−G^aω,where pk0=ak, for k=0,1,⋯,m−1 and
(19)Fσps=Fσysiωn/σ+η+eiωn/σaiωn/σ+η∑k=0m−1akiωn/σ+η+G^aωiωn/σ+η.
Setting(20)pσs=∑k=np−1akpk0+∫0ss−νν−1Eσηs−νν−1qsdν.
By using fractional Fourier transform to (20), we have
(21)Fσpσs=eiωn/σaiωn/σ+η∑k=0m−1akiωn/σ+η+G^aωiωn/σ+η.
Hence,(22)CD0+σps+ηps=iωn/σσFσps−eiωn/σa∑k=0m−1aksk−σΓk−σ+1iωn/σk−σ+1+ηFσps−G^aω=qs.SinceFσ is one-to-one operator, CD0+σps+ηps=qs. Now, its follows form (19) and (21) that
(23)Fσps−Fσpσs=Fσysiωn/σ+η.Using the convolution property, we obtain(24)Fσps−pσs=Fσys∗1iωn/σ+η=ys∗yσs,where yσs=1/iωn/σ+η. In view of (13), we have
(25)ys≤εEσs,∀s>0.Now, applying the modules on both sides of Equation (24), we get
(26)ps−pσs=∫0ss−xσ−1Eσηs−νσ∗ysdν≤ys∫0ss−νσ−1Eσηs−νσdν≤εEσs∫0ss−νσ−1Eσηs−xσdν≤HεEσs.where H=∫0ss−xσ−1Eσηs−xσdν. Thus Equation (1) has Ulam-Hyers-Mittag-Leffler stability.Corollary 9.
The consideredϕ:0,∞⟶0,∞ is a function. If a function ps satisfies the inequality (13), for every ε>0, there exists a solution pσs satisfying Equation (1) such that
(27)ps−pas≤HϕsεEσs,∀s>0.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: whereL=xylan; M=xylose; N=productsofdecomposition; r1=releaserateofsugar; r2=decompositionrateofsugar. The model is presented in Figure 1.Figure 1
The presented model.Material balance for components:"L" and "M" for the first-order kinetic equation, we get
(28)−dNLsds=r1NLs,in which the initial concentration at s=0 is presented by NL=NL0. Also, we have the same direction for material M:
(29)−dNMsds=r1NLs−r2NMs,in which the initial concentration at s=0 is presented by NM=NM0. Equation (29) can be integrated and, using the provided boundary condition, yields
(30)NLs=NL0exp−r1s.Substituting (30) for (29) yields
(31)dNMsds+r2NMs=r1NL0exp−r1s.Now, if we take the fractional Caputo derivative in (31) instead of the classical ones, we have
(32)CDσNMs+r2NMs=r1NL0exp−r1s.Figure2 shows the solution of Equation (32) for various r1 and r2.Figure 2
Solution of Equation (32) for different values r1=0.012&r2=0.005,r1=0.014&r2=0.005, and r1=0.025&r2=0.005 with σ=1/2.


## 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.



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*Source: 3777566-2022-09-06.xml*