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e. In mathematics, a partial differential equation ( PDE) is an equation which computes a function between various partial derivatives of a multivariable function . The function is often thought of as an "unknown" to be solved for, similar to how x is thought of as an unknown number to be solved for in an algebraic equation like x2 − 3x + 2 = 0.
An ordinary differential equation ( ODE) is an equation containing an unknown function of one real or complex variable x, its derivatives, and some given functions of x. The unknown function is generally represented by a variable (often denoted y ), which, therefore, depends on x. Thus x is often called the independent variable of the equation.
In mathematics, a first-order partial differential equation is a partial differential equation that involves only first derivatives of the unknown function of n variables. The equation takes the form. Such equations arise in the construction of characteristic surfaces for hyperbolic partial differential equations, in the calculus of variations ...
The Hamilton–Jacobi equation is a first-order, non-linear partial differential equation. for a system of particles at coordinates . The function is the system's Hamiltonian giving the system's energy. The solution of the equation is the action functional, , [4] called Hamilton's principal function in older textbooks.
Specific partial differential equations. Broer–Kaup equations. Burgers' equation. Euler equations. Fokker–Planck equation. Hamilton–Jacobi equation, Hamilton–Jacobi–Bellman equation. Heat equation.
A parabolic partial differential equation is a type of partial differential equation (PDE). Parabolic PDEs are used to describe a wide variety of time-dependent phenomena in, i.a., engineering science and financial mathematics. Examples include the heat equation, time-dependent Schrödinger equation and Black–Scholes equation .
t. e. In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation.
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry .
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