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. The primary vehicle of calculus and other higher mathematics is the function. ... In mathematics, more precisely in numerical analysis, finite differences play an important role, they are one of the simplest ways of approximating a differential operator, and are extensively used in solving differential equations. ... Numerical analysis is the study of approximate methods for the problems of continuous mathematics (as distinguished from discrete mathematics). ... Visualization of airflow into a duct modelled using the Navier-Stokes equations, a set of partial differential equations. ... Shows a region where a differential equation is valid and the associated boundary values In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional restraints, called the boundary conditions. ...

In mathematical analysis, operators involving finite differences are studied. A difference operator is an operator which maps a function f to a function whose values are the corresponding finite differences. Analysis has its beginnings in the rigorous formulation of calculus. ... In mathematics, an operator is a function that performs some sort of operation on a number, variable, or function. ... In mathematics, a difference operator maps a function f(x) to another function f(x + a) − f(x + b). ...

Forward, backward, and central differences

Only 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 held constant.

A backward difference uses the function values at x and x − h, instead of the values at x + h and x:

Finally, the central difference is given by

Relation with derivatives

The derivative of a function f at a point x is defined by the limit This article is about derivatives and differentiation in mathematical calculus. ... Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements as...

If h has a fixed (non-zero) value, instead of approaching zero, then the right-hand side is

Hence, the forward difference divided by h approximates the derivative when h is small. The error in this approximation can be derived from Taylor's theorem. Assuming that f is continuously differentiable, the error is The exponential function (continuous red line) and the corresponding Taylors polynomial about a = 0 of degree four (dashed green line) In calculus, Taylors theorem, named after the mathematician Brook Taylor, who stated it in 1712, gives the approximation of a differentiable function near a point by a polynomial...

The same formula holds for the backward difference:

However, the central difference yields a more accurate approximation. Its error is proportional to square of the spacing (if f is twice continuously differentiable):

Higher-order differences

In an analogous way one can obtain finite difference approximations to higher order derivatives and differential operators. For example, by using the above central difference formula for f'(x + h / 2) and f'(x − h / 2) and applying a central difference formula for the derivative of f' at x, we obtain the central difference approximation of the second derivative of f:

More generally, the n^th-order forward, backward, and central differences are respectively given by:

Note that the central difference will, for odd n, have h multiplied by non-integers. If this is a problem (usually it is), it may be remedied taking the average of δⁿ[f](x − h / 2) and δⁿ[f](x + h / 2).

The relationship of these higher-order differences with the respective derivatives is very straightforward:

Higher-order differences can also be used to construct better approximations. As mentioned above, the first-order difference approximates the first-order derivative up to a term of order h. However, the combination

approximates f'(x) up to a term of order h². This can be proven by expanding the above expression in Taylor series, or by using the calculus of finite differences, explained below. Series expansion redirects here. ...

If necessary, the finite difference can be centered about any point by mixing forward, backward, and central differences.

Properties

• For all positive k and n

• Leibniz rule:

In calculus, the Leibniz rule, named after Gottfried Leibniz, generalizes the product rule. ...

Finite difference methods

Main article: finite difference method

An important application of finite differences is in numerical analysis, especially in numerical ordinary differential equations and numerical partial differential equations, which aim at the numerical solution of ordinary and partial differential equations respectively. The idea is to replace the derivatives appearing in the differential equation by finite differences that approximate them. The resulting methods are called finite difference methods. In mathematics, more precisely in numerical analysis, finite differences play an important role, they are one of the simplest ways of approximating a differential operator, and are extensively used in solving differential equations. ... Numerical analysis is the study of approximate methods for the problems of continuous mathematics (as distinguished from discrete mathematics). ... Numerical ordinary differential equations is the part of numerical analysis which studies the numerical solution of ordinary differential equations (ODEs). ... Numerical partial differential equations is the branch of numerical analysis that studies the numerical solution of partial differential equations. ... In mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable. ... In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ...

Common applications of the finite difference method are in computational science and engineering disciplines, such as thermal engineering, fluid mechanics, etc. This box:      Fluid mechanics is the study of how fluids move and the forces on them. ...

Calculus of finite differences

Main article: difference operator

The forward difference can be considered as a difference operator, which maps the function f to Δ_h[f]. This operator satisfies In mathematics, a difference operator maps a function f(x) to another function f(x + a) − f(x + b). ...

where T_h is the shift operator with step h, defined by T_h[f](x) = f(x + h), and I is an identity operator. In mathematics, and in particular functional analysis, the shift operators are examples of linear operators, important for their simplicity and natural occurrence. ... In mathematics, an identity function, also called identity map or identity transformation, is a function that does not have any effect: it always returns the same value that was used as its argument. ...

Finite difference of higher orders can be defined in recursive manner as or, in operators notation, Another possible (and equivalent) definition is

The difference operator Δ_h is linear and satisfies Leibniz rule. Similar statements hold for the backward and central difference. In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... At least two results in calculus are called Leibnizs rule or the Leibniz rule, in honor of Gottfried Leibniz. ...

Taylor's theorem can now be expressed by the formula The exponential function (continuous red line) and the corresponding Taylors polynomial about a = 0 of degree four (dashed green line) In calculus, Taylors theorem, named after the mathematician Brook Taylor, who stated it in 1712, gives the approximation of a differentiable function near a point by a polynomial...

where D denotes the derivative operator, mapping f to its derivative f'. Formally inverting the exponential suggests that

This formula holds in the sense that both operators give the same result when applied to a polynomial. Even for analytic functions, the series on the right is not guaranteed to converge; it may be an asymptotic series. However, it can be used to obtain more accurate approximations for the derivative. For instance, retaining the first two terms of the series yields the second-order approximation to f'(x) mentioned at the end of the section Higher-order differences. In mathematics and applications, particularly the analysis of algorithms, asymptotic analysis is a method of classifying limiting behaviour, by concentrating on some trend. ...

The analogous formulas for the backward and central difference operators are

Generalizations

A generalized finite difference is usually defined as

where is its coefficients vector. An infinite difference is a further generalization, where the finite sum above is replaced by an infinite series. Another way of generalization is making coefficients μ_k depend on point x : μ_k = μ_k(x), thus considering weighted finite difference. Also one may make step h depend on point x : h = h(x). Such generalizations are useful for constructing different modulus of continuity. In mathematics, a series is often represented as the sum of a sequence of terms. ... In mathematics, the modulus of continuity is a precise way to measure the smoothness of a function. ...

See also

• Numerical differentiation
• Five-point stencil
• Divided differences
• Modulus of continuity

Numerical differentiation is a technique of numerical analysis to produce an estimate of the derivative of a mathematical function or function subroutine using values from the function and perhaps other knowledge about the function. ... In numerical analysis, given a rectangular grid in one or two dimensions, the five-point stencil of a point in the grid is made up of the point itself together with its four neighbors. In one dimension, if the spacing between points in the grid is , then the five-point... In mathematics divided differences is a recursive division process. ... In mathematics, the modulus of continuity is a precise way to measure the smoothness of a function. ...

References

• William F. Ames, Numerical Method for Partial Differential Equations, Section 1.6. Academic Press, New York, 1977. ISBN 0-12-056760-1.
• Francis B. Hildebrand, Finite-Difference Equations and Simulations, Section 2.2, Prentice-Hall, Englewood Cliffs, New Jersey, 1968.
• Boole, George, A Treatise On The Calculus of Finite Differences, 2^nd ed., Macmillan and Company, 1872. [See also: Dover edition 1960].
• Robert D. Richtmyer and K. W. Morton, Difference Methods for Initial Value Problems, 2^nd ed., Wiley, New York, 1967.

External links

• Finite Difference Method
• Finite Difference Method for Boundary Value Problems
• Table of useful finite difference formula generated using Mathematica