Stability Analysis Of Delayed Fractional Integro-Differential Equations With Applications Of RLC Circuits

Abstract

This article presents the stability analysis of delay integro-differentialequations with fractional order derivative via some approximation techniques forthe derived nonlinear terms of characteristic exponents. Based on these techniques,the existence of some analytical solutions at the neighborhood of their equilibriumpoints is proved. Stability charts are constructed and so both of the critical timedelay and critical frequency formulae are obtained. The impact of this work into thegeneral RLC circuit applications exposing the delay and fractional order derivativesis discussed.

On the qualitative analyses solutions of new mathematical models of integro‐differential equations with infinite delay

“…This is a desirable fact for these kinds of works. 4) As for the claim such as the results obtained here are more effective and convenient for tests and applications, it can be found numerous functions as those are included in IDEs (2) (6)…”

“…Here, instead of g(t, 𝜂, x(t − 𝜂)) in IDE (2) , we take g(t, 𝜂, x(t − 𝜏)) in IDE (6). This modification is needed in the proof of the instability.…”

“…where I(t) is the electric current, R is the resistance, L is the inductance, C is the capacitance and the impressed voltage is E(t) (see Bohner et al [5], El-Borhamy and Ahmed [6], Hatamzadeh-Varmazyar et al [7]).…”

“…For example, as a specific example in physics, an RLC, standard closed electric circuit, can be mathematically modeled by the below linear IDE: $$$$ L\frac{d}{dt}I(t)+ RI(t)+\frac{1}{C}\int_0^tI(s) ds=E(t), $$$$ where $$$ I(t) $$$ is the electric current, $$$ R $$$ is the resistance, $$$ L $$$ is the inductance, $$$ C $$$ is the capacitance and the impressed voltage is $$$ E(t) $$$ (see Bohner et al [5], El‐Borhamy and Ahmed [6], Hatamzadeh‐Varmazyar et al [7]).…”

Stability tests and solution estimates for non-linear differential equations

“…There are many publications on fundamental properties of solutions of FDEs, ODEs and so on. We cite here the papers [1][2][3][4][5][6], [7][8][9], [10], and the books of ( [32], [33][34][35][36][37][38][39]) fully or partially devoted to fundamental motions of trajectories of solutions of these classes of equations. In particular, UAS and boundedness of solutions at the infinity describe long time behaviors of solutions.…”