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

Abstract

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.

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

“…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.…”

On Fourier–Bessel matrix transforms and applications

“…They are also useful in evaluating integrals involving special functions. The reader may refer to, for example, previous studies 1–12 …”

“…The reader may refer to, for example, previous studies. [1][2][3][4][5][6][7][8][9][10][11][12] Among the many classical integral transforms is Fourier-Bessel transform (also designated as Hankel transform) that is a fundamental tool in many areas of mathematical statistics, physics, engineering, probability theory, analytic number theory, data analysis, and so on (see, for instance, previous studies [13][14][15][16][17][18][19] ). Many researchers regard ℌ{g(𝜂); q} = ∫ ∞ 0 q 𝔍 m (𝜂q) g(𝜂) d𝜂, q > 0, (1.1) the standard Fourier-Bessel transform involving the mth-order Bessel function of the first kind 𝔍 m (𝜂) as a kernel.…”

A New Approach to Hyers-Ulam Stability of r -Variable Quadratic Functional Equations

“…It is worth noting that among the stability problem of functional equations, the study of the Ulam stability of different types of quadratic functional equations is an important and interesting topic, and it has attracted many scholars [13][14][15][16][17][18]. In addition, very recently, authors studied various types of stability results and have been discussed with differential equation [19][20][21][22][23][24][25][26][27][28][29]. To the best of the author's knowledge, a new approach to Hyers-Ulam stability of r-variable quadratic functional equations has not been studied so far, which motivates the present study.…”