A partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives. PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a relevant computer model. A special case is ordinary differential equations (ODEs), which deal with functions of a single variable and their derivatives. PDEs can be used to describe a wide variety of phenomena such as sound, heat, electrostatics, electrodynamics, fluid dynamics, elasticity, or quantum mechanics. These seemingly distinct physical phenomena can be formalized similarly in terms of PDEs.
- 1: Introduction to Partial Differential Equations
- 2: Classiﬁcation of Partial Diﬀerential Equations
- 3: Boundary and Initial Conditions
- 4: Fourier Series
- 5: Separation of Variables on Rectangular Domains
- 6: D’Alembert’s Solution to the Wave Equation
- 7: Polar and Spherical Coordinate Systems
- 8: Separation of Variables in Polar Coordinates
- 9: Series Solutions of ODEs (Frobenius’ Method)
- 10: Bessel Functions and Two-Dimensional Problems
- 11: Separation of Variables in Three Dimensions
Thumbnail: A visualization of a solution to the two-dimensional heat equation with temperature represented by the third dimension. Imaged used wth permission (Public Domain; Oleg Alexandrov). The heat equation is a parabolic partial differential equation that describes the distribution of heat (or variation in temperature) in a given region over time.