Exact Solutions of the Damped Telegrapher’s Equation with Harmonic Potential via the Generalized First Integral Method

This paper aims to develop exact analytical solutions for the telegrapher’s equation incorporating both damping and harmonic potential by employing the generalized first integral method. This approach extends the classical first integral technique through the use of Laurent polynomials, enhancing its ability to address complex nonlinear structures. The telegrapher’s equation, a fundamental model in applied mathematics and physics, describes wave propagation influenced by both dispersive and damping effects, with applications across various engineering and physical systems. By applying suitable transformations, the original nonlinear partial differential equation is reduced to an ordinary differential form. The generalized method is then utilized to derive exact solutions under different parametric conditions. These solutions offer valuable analytical insight into how damping influences wave amplitude, speed, and qualitative behavior. In particular, the method effectively captures the modulation in wave attenuation and propagation characteristics caused by dissipative effects. The key outcome of this study is the demonstration that the generalized first integral method serves as a robust and versatile analytical tool for solving nonlinear damped wave models, where conventional methods often encounter limitations. Its strength lies in simplifying complex nonlinear systems while preserving essential physical effects, providing precise analytical descriptions of wave behavior. Additionally, three-dimensional graphical visualizations of the obtained solutions offer a detailed understanding of the system’s spatial and temporal dynamics. This work contributes to the ongoing advancement of analytical techniques for nonlinear evolution equations and establishes a foundation for extending this method to other complex dynamical systems involving damping, dissipation, and external potentials.