Laplace's equation is a second-order partial differential equation that describes the distribution of a scalar quantity in a given space, such as temperature or electric potential.
Pierre-Simon Laplace was a French mathematician and astronomer who first introduced the equation in the late 18th century. It is named after him because he extensively studied and used it in his work on celestial mechanics.
Laplace's equation has various applications in physics, engineering, and mathematics. It is used to model and solve problems related to heat transfer, fluid dynamics, electrostatics, and gravitational potential, among others.
The solutions to Laplace's equation are called harmonic functions, which are infinitely differentiable and have continuous second derivatives. These functions satisfy the equation at every point in the given space and can be represented by a series of trigonometric functions.
Some common methods for solving Laplace's equation include separation of variables, the method of images, and the use of Green's functions. These methods can be applied to both simple and complex geometries to find solutions to Laplace's equation.