Stability of linear differential equation of higher order using Mahgoub transforms

Abstract

In this paper, by applying Mahgoub transform, we show that the n th order linear differential equationhas Hyers-Ulam stability, where a κ 's are scalars and x is an n times continuously differentiable function of exponential order.

Integral Transforms and the Hyers–Ulam Stability of Linear Differential Equations with Constant Coefficients

“…Moreover, the techniques used in this paper can be modified to obtain similar Hyers-Ulam stability results for linear constant coefficient equations of first and second order using the Laplace, Tarig, Aboodh, Mahgoub, Sawi, Shehu, and Elzaki integral transforms, respectively. For example, by including information on the best Hyers-Ulam constant, Theorem 1 slightly improves [1] (Theorem 3.3) and [8] (Theorem 3.3), which used the Laplace transform and the Maghoub transform, respectively. Moreover, Theorem 2 improves [1] (Theorem 3.4) and [8] (Theorem 3.4) for the second-order case.…”

“…For example, by including information on the best Hyers-Ulam constant, Theorem 1 slightly improves [1] (Theorem 3.3) and [8] (Theorem 3.3), which used the Laplace transform and the Maghoub transform, respectively. Moreover, Theorem 2 improves [1] (Theorem 3.4) and [8] (Theorem 3.4) for the second-order case.…”

“…The first paper of this type is by Rezaei, Jung, and Rassias [1] using the Laplace transform; other papers using the Laplace transform to investigate Hyers-Ulam stability include [2,3]. Since then, scholars have used a variety of integral transform definitions to explore Hyers-Ulam stability, including the Kamal [4], Mahgoub [5][6][7][8], Tarig [9], Shehu [10], Sawi [11], Aboodh [12][13][14], Fourier [15][16][17][18], general [19], and Elzaki [20] integral transforms. Fourier transforms require functions and equations to be defined on the whole real line (−∞, ∞), while the other transforms listed above require functions and equations to be defined on the half-line [0, ∞).…”

Mohand transforms and its application to stability of differential equation

“…In 2020, Murali et al [44] investigated the Hyers-Ulam stability of various differential equations using Fourier transform method (see also [43]). Recently, Jung et al [28] established the various forms of Hyers-Ulam stability of the firstorder linear differential equations with constant coefficients by using Mahgoub integral transform (see also [38]). Very recently, Murali et al [39] investigated the different forms of Hyers-Ulam stability and Mittag-Leffler-Hyers-Ulam stability of second order linear differential equation of the form u + µ 2 u = q(t) by using Abooth transform method (see also [37]).…”

The Aboodh Transform Techniques to Ulam Type Stability of Linear Delay Differential Equation

“…Moreover, the solution of Ulam-Hyers stability of various differential equations through the Mahgoub transform was obtained by Jung et al [14], Aruldass et al [26], Deepa et al [9] and Murali et al [19]. In [22], the authors provided the stability problems of first-order linear differential equations using the method of Fourier transform and Rassias et al [28], second order linear differential equations using the method of Fourier transform.…”