Partial differential equation

In mathematics, a partial differential equation (PDE) is an equation relating the partial derivatives of an unknown function of several variables. A solution of the equation is a function satisfying this relation. The idea is to try to deduce information about an unknown function by first discovering a relationship between itself and its partial derivatives in the form of a PDE. The PDE can then be used to uncover information about the unknown function, and sometimes an explicit formula for the unknown function can be discovered.

Related Topics:
Mathematics - Equation - Partial derivative - Function

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A PDE usually has many (possible infinitely many) solutions; a particular problem often requires additional boundary conditions which constrain the solution set. Where ordinary differential equations have solutions that are families with each solution characterized by the values of some parameters, for a PDE the solutions often are parametrized by functions (informally put, this means that the set of solutions is much larger).

Related Topics:
Boundary condition - Ordinary differential equation - Function

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Partial differential equations are ubiquitous in science and especially in physics, as physical laws can usually be written in form of PDEs. They describe phenomena such as fluid flow, the growth of crystals, diffusion, gravitation, and the behavior of electromagnetic fields. They are important in fields such as aircraft simulation, computer graphics, and weather prediction. The central equations of general relativity and quantum mechanics are also partial differential equations.

Related Topics:
Physics - Fluid flow - Diffusion - Gravitation - Electromagnetic field - Aircraft - Computer graphics - General relativity - Quantum mechanics

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