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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.
Recurrence relations can be written as difference equations by replacing iteration notation with finite 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
The derivative of a function f at a point x is defined by the limit
If h has a fixed (nonzero) value, instead of approaching zero, then the righthand 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 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):
The main problem with the central difference method, however, is that oscillating functions can yield zero derivative. If h(nh)=1 for n uneven, and h(nh)=2 for n even, then f'(nh)=0 if it is calculated with the central difference scheme. This is particularly troublesome if the domain of f is discrete.
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 nonintegers. This is often a problem because it amounts to changing the interval of discretization. The problem may be remedied taking the average of δ^{n}[f](x − h / 2) and δ^{n}[f](x + h / 2).
The relationship of these higherorder differences with the respective derivatives is very straightforward:
Higherorder differences can also be used to construct better approximations. As mentioned above, the firstorder difference approximates the firstorder derivative up to a term of order h. However, the combination
approximates f'(x) up to a term of order h^{2}. This can be proven by expanding the above expression in Taylor series, or by using the calculus of finite differences, explained below.
If necessary, the finite difference can be centered about any point by mixing forward, backward, and central differences.
Using a little linear algebra, one can fairly easily construct approximations, which sample an arbitrary number of points to the left and a (possibly different) number of points to the right of the center point, for any order of derivative. This involves solving a linear system such that the Taylor expansion of the sum of those points, around the center point, well approximates the Taylor expansion of the desired derivative.
This is useful for differentiating a function on a grid, where, as one approaches the edge of the grid, one must sample fewer and fewer points on one side.
The details are outlined in these notes.
An important application of finite differences is in numerical analysis, especially in numerical 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.
Common applications of the finite difference method are in computational science and engineering disciplines, such as thermal engineering, fluid mechanics, etc.
The forward difference can be considered as a difference operator, which maps the function f to Δ_{h}[f]. This operator satisfies
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.
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.
Applying Taylor's theorem with respect to h gives the formula
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 secondorder approximation to f'(x) mentioned at the end of the section Higherorder differences.
The analogous formulas for the backward and central difference operators are
The calculus of finite differences is related to the umbral calculus in combinatorics.
The inverse operator of the forward difference operator is the indefinite sum.
In mathematics, a difference operator maps a function, ƒ(x), to another function, ƒ(x + b) − ƒ(x + a).
The forward difference operator
occurs frequently in the calculus of finite differences, where it plays a role formally similar to that of the derivative, but used in discrete circumstances. Difference equations can often be solved with techniques very similar to those for solving differential equations. This similarity led to the development of time scale calculus. Analogously we can have the backward difference operator
When restricted to polynomial functions f, the forward difference operator is a delta operator, i.e., a shiftequivariant linear operator on polynomials that reduces degree by 1.
The nth forward difference of a function f(x) is given by
where is the binomial coefficient. Forward differences applied to a sequence are sometimes called the binomial transform of the sequence, and have a number of interesting combinatorial properties.
Forward differences may be evaluated using the Nörlund–Rice integral. The integral representation for these types of series is interesting because the integral can often be evaluated using asymptotic expansion or saddlepoint techniques; by contrast, the forward difference series can be extremely hard to evaluate numerically, because the binomial coefficients grow rapidly for large n.
The Newton series or Newton forward difference equation, named after Isaac Newton, is the relationship
which holds for any polynomial function f and for some, but not all, analytic functions. Here, the expression
is the binomial coefficient, as
is the "falling factorial" or "lower factorial" and the empty product (x)_{0} defined to be 1. Note also the formal similarity of this result to Taylor's theorem; this is one of the observations that lead to the idea of umbral calculus.
In analysis with padic numbers, Mahler's theorem states that the assumption that f is a polynomial function can be weakened all the way to the assumption that f is merely continuous.
Carlson's theorem provides necessary and sufficient conditions for a Newton series to be unique, if it exists. However, a Newton series will not, in general, exist.
The Newton series, together with the Stirling series and the Selberg series, is a special case of the general difference series, all of which are defined in terms of scaled forward differences.
Analogous to rules for finding the derivative, we have:
All of the above rules apply equally well to any difference operator, including as to Δ.
The inverse operator of the forward difference operator is the indefinite sum.
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.
Finite differences can be considered in more than one variable. They are analogous to partial derivatives in several variables.
Some partial derivative approximations are:
