A Novel Nonlinear Memory-Driven Differential Equation and Its Analytical Solutions

This study presents a novel nonlinear memory-driven differential equation incorporating a fading memory term through an exponential kernel. The proposed model extends classical ordinary differential equations by accounting for the influence of past states on present dynamics. The equation is transformed into an equivalent system of first-order differential equations, enabling analytical treatment and solution derivation. Two representative cases were analyzed to demonstrate the effectiveness of the model. In the homogeneous case, the system yielded eigenvalues λ_1=(-3+√5)/2≈-0.38 and λ_2=(-3-√5)/2≈-2.62, confirming asymptotic stability. The solution exhibited a dual decay structure, combining fast and slow exponential modes. In the nonhomogeneous case, with forcing term e^(-t), the analytical solution included a particular solution y_p=-e^(-t), which influenced the transient response while preserving long-term stability.\r\nNumerical evaluations further supported the analytical findings, with solution values decreasing from y(0)=1 to approximately y(4)≈0.21, demonstrating consistent decay behavior. Additionally, parametric analysis showed that smaller memory parameter values (λ=0.5) produced slower decay, whereas larger values (λ=2) led to faster convergence.\r\nThe results confirm that the proposed model effectively captures memory-dependent dynamics and provides a flexible framework for analyzing systems with both internal memory effects and external forcing. This approach has potential applications in physics, engineering, and biological systems where temporal dependencies are significant.