Partial Differential Equations
Partial Differential Equations involve the study of equations that contain unknown multivariable functions and their partial derivatives. These equations arise in various fields, including physics, engineering, and economics, to model phenomena involving changes in multiple variables. They are fundamental for understanding wave propagation, heat diffusion, and quantum mechanics. The subject covers methods for solving these equations, such as separation of variables, Fourier series and transforms, and numerical methods. It also includes the study of well-posedness, existence, and uniqueness of solutions, as well as their regularity properties. In this context, it is closely related to the fields of calculus of variations, functional analysis, and numerical analysis.