Asymptotic analysis of two nonlinear differential equations by the renormalization method based on the Taylor expansion

Abstract This study investigates the application of the renormalization method based on the Taylor expansion (TR) in the
asymptotic analysis of nonlinear differential equations. Taking the KdV equation describing water wave motion and the
Jerk equation describing various phenomena in engineering and physics, we applied TR method to obtain their global
asymptotic solutions. Furthermore, we conducted numerical simulations on the obtained asymptotic solutions and found
a very good fit with the numerical solutions. We also obtained the exact solution of the KdV equation using the complete
discrimination system for polynomials, and conducted numerical analysis and asymptotic solution, yielding satisfactory
results. These findings highlight the effectiveness of these methods in studying and solving nonlinear partial differential
equations.