Spectral methods are numerical techniques used to solve differential equations by transforming them into a different space, typically using functions like Fourier series or polynomial expansions. These methods leverage the properties of these functions to approximate solutions more accurately and efficiently than traditional methods, especially for problems with smooth solutions. In spectral methods, the solution is expressed as a sum of basis functions, and the coefficients of these functions are determined by the original differential equation. This approach is particularly effective for problems defined on bounded domains and is widely used in fields such as fluid dynamics and quantum mechanics.