Finite difference approximation
A finite difference is a mathematical expression of the form f(x + b) − f(x + a). If a finite difference is divided by b − a, one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the numerical solution of differential equations, especially boundary value problems.
Certain recurrence relations can be written as difference equations by replacing iteration notation with finite differences.
Today, the term "finite difference" is often taken as synonymous with finite difference approximations of derivatives, especially in the context of numerical methods. Finite difference approximations are finite difference quotients in the terminology employed above.
Finite differences have also been the topic of study as abstract self-standing mathematical objects, e.g. in works by George Boole (1860), L. M. Milne-Thomson (1933), and (1939), tracing its origins back to one of Jost Bürgi's algorithms (ca. 1592) and others including Isaac Newton. In this viewpoint, the formal calculus of finite differences is an alternative to the calculus of infinitesimals.
Three forms are commonly considered: forward, backward, and central differences.
A forward difference is an expression of the form
Depending on the application, the spacing h may be variable or constant. When omitted, h is taken to be 1: .
=\Delta _{1}[f](x)](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef7a519253d38b31cb440ddc5aa961a15974f581)
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