The Fourier transform approach to Hyers-Ulam stability of differential equation of second order

Abstract

The use of Fourier transform has increased in the light of recent events in different application. Fourier transform is also seen as the easiest and an effective way among the other transformation. In line with this, the research deals with the Hyers-Ulam stability of second order differential equation using Fourier transform. The study aims at deriving a generalized Hyers-Ulam results for second order differential equations with constant co-efficient H ″ ( v ) + a H ′ ( v ) + b H ( v ) = r ( t ) with the help of Fourier Transform.

Equivalence for Various Forms of Quadratic Functional Equations and Their Generalized Hyers–Ulam–Rassias Stability

“…In 2022, Jamil et al [14] established necessary and sufcient conditions for the existence Hyers-Ulam and generalized Hyers-Ulam stability for the fractional order hybrid sequential integrodiferential equations. In 2020, Mohanapriya et al [15] studied the Hyers-Ulam stability of the second-order differential equation using Fourier transform. Unyong et al [16] study the Hyers-Ulam stability with respect to the linear diferential condition of the fourth order.…”

On Fourier–Bessel matrix transforms and applications

“…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.…”

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

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.…”