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Finite difference
A finite difference is a mathematical expression of the form f (x + b) − f (x + a). Finite differences (or the associated difference quotients) are often used as approximations of derivatives, such as in numerical differentiation.
The difference operator, commonly denoted , is the operator that maps a function f to the function defined by A difference equation is a functional equation that involves the finite difference operator in the same way as a differential equation involves derivatives. There are many similarities between difference equations and differential equations. Certain recurrence relations can be written as difference equations by replacing iteration notation with finite differences.
![{\displaystyle \Delta [f]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/978a8399050eeaf584e6c0f7a57196f57d9bdb71)
=f(x+1)-f(x).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/15da4d892ccab64158a6e556b809b3f365da7057)
In numerical analysis, finite differences are widely used for approximating derivatives, and the term "finite difference" is often used as an abbreviation of "finite difference approximation of derivatives".[1][2][3]
Finite differences were introduced by Brook Taylor in 1715 and have also been studied as abstract self-standing mathematical objects in works by George Boole (1860), L. M. Milne-Thomson (1933), and Károly Jordan (1939). Finite differences trace their origins back to one of Jost Bürgi's algorithms (c. 1592) and work by others including Isaac Newton. The formal calculus of finite differences can be viewed as an alternative to the calculus of infinitesimals.[4]
- ^ Paul Wilmott; Sam Howison; Jeff Dewynne (1995). The Mathematics of Financial Derivatives: A Student Introduction. Cambridge University Press. p. 137. ISBN 978-0-521-49789-3.
- ^ Peter Olver (2013). Introduction to Partial Differential Equations. Springer Science & Business Media. p. 182. ISBN 978-3-319-02099-0.
- ^ M Hanif Chaudhry (2007). Open-Channel Flow. Springer. p. 369. ISBN 978-0-387-68648-6.
- ^ Jordán, op. cit., p. 1 and Milne-Thomson, p. xxi. Milne-Thomson, Louis Melville (2000): The Calculus of Finite Differences (Chelsea Pub Co, 2000) ISBN 978-0821821077
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