Abstract
This research paper aims to present the results on the Mittag-Leffler–Hyers–Ulam and Mittag-Leffler–Hyers–Ulam–Rassias stability of linear differential equations of first, second, and nth order by the Fourier transform method. Moreover, the stability constant of such equations is obtained. Some examples are given to illustrate the main results.
Highlights
In recent years, there has been subject so far-reaching of research in derivative and differential equation because of its performance in numerous branches of pure and applied mathematics
We say that a differential equation is stable in the Hyers–Ulam sense if, for every solution of the differential equation, there exists an approximate solution of the perturbed equation that is close to it
The class of stability was first formulated by Ulam [2] for functional equation which was solved by Hyers [3] for an additive function defined on a Banach space
Summary
There has been subject so far-reaching of research in derivative and differential equation because of its performance in numerous branches of pure and applied mathematics. We introduce some new concepts concerning the stability of differential equation in the Mittag-Leffler–Hyers–Ulam sense by the Fourier transform method. The Fourier transform and Mittag-Leffler function are effective tools for analytic expression for the solution of linear differential equation of integer or noninteger order. In 2020, Liu et al studied Hyers–Ulam stability and existence of solutions for fractional differential equation with Mittag-Leffler kernel [20]. Definition 3.1 We say that linear differential equation (3.1) is said to have Mittag-Leffler– Hyers–Ulam stability if there exists a constant K > 0 with the following: for every > 0 and a continuously differentiable function H(x) satisfying the inequality. Definition 3.3 The linear differential equation (3.10) is said to have Mittag-Leffler– Hyers–Ulam stability if there exists a constant K > 0 with the following property: for every. The method of variation of constant gives the unique solution of (3.15), which is
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