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This study is devoted to developing mathematical models associated with the Helmholtz equation as a second-order oscillator involved with nonlinear Caputo fractional difference equations. This study also focuses on determining the approximate solution of this model via the Ulam stability conception. Some properties of the mathematical model dealt with in this study are also presented. Numerical simulations are presented to justify the existence of stability results. • A new formulation of the mathematical model and analysis in Helmholtz Duffing oscillator with the damping term is presented. on the natural frequencies. • Using this Helmholtz–Duffing oscillator with the damping term, we proved the Hyers–Ulam stability to ensure existence and uniqueness. • We analyzed practical examples describing the application of quadcopter. [ABSTRACT FROM AUTHOR]

In this manuscript, we describe fractional differential equations with neutral, integral boundary conditions and mixed delay using Atangana–Baleanu derivatives, which include the generalized Mittag-Leffler kernel. We determine the existence and uniqueness of results and analysis by fixed point method. Moreover, we explained the stability of the fractional differential equation in the frame of Ulam–Hyers. Then, we investigate an example with various values and illustrate the outcomes graphically. • We consider the nonlinear fractional mixed integro-differential equations with fractional boundary conditions. • The existence result is obtained by using standard fixed point theory. • An example is given to illustrate our analytical and numerical results. By using MATLAB coding graph is drawn. [ABSTRACT FROM AUTHOR]