A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a
difference quotient. The approximation of
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
s by finite differences plays a central role in
finite difference methods for the
numerical solution of
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
s, especially
boundary value problem
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to t ...
s.
The
difference operator, commonly denoted
is the
operator that maps a function to the function