HYERS-ULAM STABILITY OF THIRD ORDER LINEAR DIFFERENTIAL EQUATIONS USING EMAD-FALIH TRANSFORM WITH APPLICATIONS TO JERK SYSTEMS

Abstract

We investigate the Hyers-Ulam stability of third order linear differential equa-tions yʹʹʹ(t) + a₂yʹʹ(t) + a₁yʹ(t) + a₀y(t) = g(t) using the Emad-Falih integral transform.The third derivative property 𝓔𝓕{fʹʹʹ(t)} = φ⁶S(φ) - φ³f(0) - φfʹ(0) - {fʹʹ(0)}over{φ} is derivedand applied to establish four types of stability. This novel approach provides sharper stabil-ity bounds compared to conventional methods and extends the applicability of Hyers-Ulamtheory to higher-order dynamical systems. The transform-based methodology oers com-putational advantages by converting dierential equations into algebraic equations whilepreserving stability characteristics. Applications to jerk dynamics, electrical transmissionlines, beam deection under distributed loads, and chaotic systems demonstrate physicalrelevance and practical utility in engineering design. Complete proofs, worked examples,and comprehensive graphical analysis validate the theoretical framework and illustrate itseectiveness in real-world scenarios.