Explore the fundamentals of perturbation methods for nonlinear partial differential equations in this comprehensive lecture by Vishal Vasan. Delve into the adaptation of linear equation techniques to nonlinear problems, covering topics such as multiple scales and asymptotic analysis. Gain insights into the interplay between linear waves and nonlinearity through examples from cold atoms, water waves, and atmospheric science. Learn about the Fredholm Alternative Theorem and its applications, and examine various perturbation method examples, including non-linear oscillator problems. Suitable for those with basic linear algebra knowledge, this lecture provides a unified approach to understanding seemingly disparate ad hoc methods in nonlinear equation solving.