In this paper, we study the Ulam-Hyers-Mittag-Leffler stability for a linear fractional order differential equation with a fractional Caputo-type derivative using the fractional Fourier transform. Finally, we provide an enumeration of the chemical reactions of the differential equation.

The main aim of this study is to implement the general integral transform technique to determine Ulam-type stability and Ulam–Hyers–Mittag–Leffer stability. We are given suitable examples to validate and support the theoretical results. As an application, the general integral transform is used to find Ulam stability of differential equations arising in Thevenin equivalent electrical circuit system. The results are graphically represented, which provides a clear and thorough explanation of the suggested method.

The primary aim of this paper is to focus on the stability analysis of an advanced neural stochastic functional differential equation with finite delay driven by a fractional Brownian motion in a Hilbert space. We examine the existence and uniqueness of mild solution of $$ {\textrm{d}}\left[ {x}_{a}(s) + {\mathfrak {g}}(s, {x}_{a}(s - \omega (s)))\right] =\left[ {\mathfrak {I}}{x}_a(s) + {\mathfrak {f}}(s, {x}_a(s -\varrho (s)))\right] {\textrm{d}}s + \varsigma (s){\textrm{d}}\varpi ^{{\mathbb {H}}}(s),$$d x a ( s ) + g ( s , x a ( s - ω ( s ) ) ) = I x a ( s ) + f ( s , x a ( s - ϱ ( s ) ) ) d s + ς ( s ) d ϖ H ( s ) , $$0\le s\le {\mathcal {T}}$$0 ≤ s ≤ T , $${x}_a(s) = \zeta (s),\ -\rho \le s\le 0. $$x a ( s ) = ζ ( s ) , - ρ ≤ s ≤ 0 . The main goal of this paper is to investigate the Ulam–Hyers stability of the considered equation. We have also provided numerical examples to illustrate the obtained results. This article also discusses the Euler–Maruyama numerical method through two examples.