Aboodh transform and the stability of second order linear differential equations

Ramdoss Murali,Arumugam Ponmana Selvan,Choonkil Park,Jung Rye Lee

doi:10.1186/s13662-021-03451-4

Ramdoss Murali, Arumugam Ponmana Selvan + Show 2 more

Open Access

https://doi.org/10.1186/s13662-021-03451-4

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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.

Highlights

In [1], Ulam proposed the universal Ulam stability problem in metric groups
Definition 2.9 We say that differential equation (1.1) has the Mittag-Leffler–Hyers– Ulam–Rassias stability with respect to φ : (0, ∞) → (0, ∞) if there exists a positive constant Lφ satisfying the following condition: For every > 0 and some u(t) ∈ C2(I) satisfying the inequality u (t) + μ2u ≤ φ(t) Eα(t) for all t ∈ I, there exists a solution v ∈ C2(I) satisfying v (t) + μ2v = 0 and u(t) – v(t) ≤ Lφφ(t) Eα(t) for all t ∈ I
3 Hyers–Ulam stability for (1.1) we prove the Hyers–Ulam stability, Hyers–Ulam–Rassias stability, MittagLeffler–Hyers–Ulam stability, and Mittag-Leffler–Hyers–Ulam–Rassias stability of differential equation (1.1) by using the Aboodh transform

Summary

Introduction

In [1], Ulam proposed the universal Ulam stability problem in metric groups. In [2], Hyers gave the first affirmative answer to the question of Ulam for additive functional equations in Banach spaces. Definition 2.4 We say that differential equation (1.1) has the Hyers–Ulam stability if there exists a constant L > 0 satisfying the following condition: For every > 0 and some u(t) ∈

Results

Conclusion