Aboodh transform and the stability of second order linear differential equations

Abstract

In this paper, we introduce a new integral transform, namely Aboodh transform, and we apply the transform to investigate the Hyers–Ulam stability, Hyers–Ulam–Rassias stability, Mittag-Leffler–Hyers–Ulam stability, and Mittag-Leffler–Hyers–Ulam–Rassias stability of second order linear differential equations.

Ulam type stability of second-order linear differential equations with constant coefficients having damping term by using the Aboodh transform

“…If α = 0 and β > 0, then 4β > α 2 satisfies and there exists a positive constant γ such that β = γ 2 . Then equation (3.1) coincides whit the equation (3.2) in [55]. Thus if u 2 + γ 2 = (u − l) (u − m) with l + m = 0 and lm = γ 2 then the results of this paper coincides whit the results of [55].…”

“…Then equation (3.1) coincides whit the equation (3.2) in [55]. Thus if u 2 + γ 2 = (u − l) (u − m) with l + m = 0 and lm = γ 2 then the results of this paper coincides whit the results of [55]. If α = 0 and β < 0, then 4β < α 2 satisfies and in this case we can use the Theorem 3.2 for equation (1.1) and Theorem 4.2 for equation (1.2) but as far as we know, no study has been done in the literature that responds to this situation.…”

“…Very recently, Murali et al [55] have established the Hyers-Ulam stability and the Mittag-Leffler-Hyers-Ulam stability of the following second order linear differential equations:…”

Results on Ulam‐type stability of linear differential equation with integral transform

“…$$ In [30], the authors applied Shehu transform to solve the Ulam stability problem of differential equations. Murali et al [31] have recently focused on second‐order linear differential equations and obtained their stability results using Aboodh transform.…”

Stability of linear differential equation of higher order using Mahgoub transforms

“…In recent days, few authors have investigated the Ulam stability of the linear differential equations using various integral transform techniques, like, Fourier transform, Mahgoub transform and Aboodh transform in [12,21,22,28].…”