Төгсгөлөг ялгавар

Төгсгөлөг ялгавар нь f(x + b) − f(x + a) хэлбэрийн математик илэрхийлэл юм. Хэрэв төгсгөлөг ялгавар нь b − a гэснээр хуваагдвал, ялгаварын ноогдвор гарч ирнэ. Төгсгөлөг ялгавараар уламжлалыг ойролцоолох нь ялангуяа хязгаарын нөхцөлт бодлого, тэр дундаа дифференциал тэгшитгэлийн шийдийг инженер, шинжлэх ухааны хүрээнд тоон аргаар бодоход төгсгөлөг ялгаварын арга-ын гол үүргийг гүйцэтгэдэг.

Давталтын хамаарал нь ялгаварын тэгшитгэлд төгсгөлөг ялгаварын итерацийн тэмдэглэгээг оруулж өгсөнөөр төгс илэрхийлэгддэг.

Давших, Ухрах, болон Төвийн ялгаварууд

Дараах гурван төрлийн ялгаварууд ихэвчлэн хэрэглэгдэнэ. Үүнд: урагшлах, ухрах, болон төвийн ялгавар байна.

Давших ялгаварар нь дараах илэрхийллээр илэрхийлэгдэнэ.

$\Delta _{h}[f](x)=f(x+h)-f(x).\$

Энэ ялгаварыг хэрэглэх хэрэглээнээс хамаарч, h зай нь тогтмол эсвэл өөрчлөгдөж болно. Хэрэв h нь 1 тэй тэнцэх тохиолдолд: $\Delta [f](x)=\Delta _{1}[f](x)$.

Ухрах ялгавар нь x + h болон x утгын оронд x ба x − h гэсэн утгуудыг хэрэглэж дараах байдлаар илэрхийлэгдэнэ:

$\nabla _{h}[f](x)=f(x)-f(x-h).\$

Эцэст нь, Төвийн ялгавар нь дараах байдлаар хэрэглэгдэнэ.

$\delta _{h}[f](x)=f(x+{\tfrac {1}{2}}h)-f(x-{\tfrac {1}{2}}h).\$

Уламжлалтай холбогдох нь

Тэгэхээр x цэг дээрх функцын уламжлал нь хязгаараар тодорхойлогдоно.

$f'(x)=\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}.$

Хэрэв h нь тэгрүү тэмүүлэхийн оронд ямар нэгэн утга (тэгээс ялгаатай) оногдож байвал дээрхи тэгшитгэлийн баруун гар тал нь дараах байдлаар бичигдэнэ.

${\frac {f(x+h)-f(x)}{h}}={\frac {\Delta _{h}[f](x)}{h}}.$

Эндээс, h хүрэлцээтэй бага байхад h - аар хуваагдсан давших ялгавараар функцын уламжлалыг ойролцоолж болохоор харагдаж байна. Энэ ойролцооллын алдаа нь Тейлорын теоромоос тодорхойлогдох боломжтой. f нь ялгаварлагдах боломжтой гэж үзвэл, бид

${\frac {\Delta _{h}[f](x)}{h}}-f'(x)\to 0\quad {\text{as }}(h\to 0).$

гэж бичиж болно. Ижил томъёо ухрах ялгавараар бодогдож болно:

${\frac {\nabla _{h}[f](x)}{h}}-f'(x)\to 0\quad {\text{as }}(h\to 0).$

Харин, төвийн ялгавар нь илүү нарийвчлал бүхий ойролцооллийг үзүүлнэ. Хэрэв f нь хоёр удаа ялгаварлагдах бол,

${\frac {\delta _{h}[f](x)}{h}}-f'(x)=o(h).\!$

Гэхдээ, төвийн ялгаварын хамгийн гол асуудал нь хэлбэлзэл бүхий функцын уламжлал тэг байх магадлалтайд оршино. Хэрэв f(nh)=1 сондгой n үед, мөн f(nh)=2 тэгш n үед гэвэл f ' (nh)=0 болно. Иймэрхүү тохиолдлын функцад төвийн ялгаварыг хэрэглэх үед уламжлал тэг байх болно. Ялангуяа f нь дискрет хэлбэртэй бол иймэрхүү асуудал үүснэ.

Өндөр эрэмбийн ялгаварууд

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 Загвар:Mvar at Загвар:Mvar, we obtain the central difference approximation of the second derivative of Загвар:Mvar:

2-р эрэмбийн төвийн ялгавар

$f''(x)\approx {\frac {\delta _{h}^{2}[f](x)}{h^{2}}}={\frac {f(x+h)-2f(x)+f(x-h)}{h^{2}}}.$

Similarly we can apply other differencing formulas in a recursive manner.

2-р эрэмбийн давших ялгавар

$f''(x)\approx {\frac {\Delta _{h}^{2}[f](x)}{h^{2}}}={\frac {f(x+2h)-2f(x+h)+f(x)}{h^{2}}}.$

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

Давших ялгавар

$\Delta _{h}^{n}[f](x)=\sum _{i=0}^{n}(-1)^{i}{\binom {n}{i}}f(x+(n-i)h),$

or for h=1,

$\Delta ^{n}[f](x)=\sum _{k=0}^{n}{\binom {n}{k}}(-1)^{n-k}f(x+k)$

Ухрах

$\nabla _{h}^{n}[f](x)=\sum _{i=0}^{n}(-1)^{i}{\binom {n}{i}}f(x-ih),$

Төвийн

$\delta _{h}^{n}[f](x)=\sum _{i=0}^{n}(-1)^{i}{\binom {n}{i}}f\left(x+\left({\frac {n}{2}}-i\right)h\right).$

These equations are using binomial coefficients after the summation sign shown as $\ {\binom {n}{i}}$. Each row of Pascal's triangle provides the coefficient for each value of i.

Note that the central difference will, for odd Загвар:Mvar, have Загвар:Mvar multiplied by non-integers. This is often a problem because it amounts to changing the interval of discretization. The problem may be remedied taking the average of $\delta ^{n}[f](x-h/2)$ and $\delta ^{n}[f](x+h/2)$.

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 saddle-point techniques; by contrast, the forward difference series can be extremely hard to evaluate numerically, because the binomial coefficients grow rapidly for large Загвар:Mvar.

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

${\frac {d^{n}f}{dx^{n}}}(x)={\frac {\Delta _{h}^{n}[f](x)}{h^{n}}}+O(h)={\frac {\nabla _{h}^{n}[f](x)}{h^{n}}}+O(h)={\frac {\delta _{h}^{n}[f](x)}{h^{n}}}+O(h^{2}).$

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 Загвар:Mvar. However, the combination

${\frac {\Delta _{h}[f](x)-{\frac {1}{2}}\Delta _{h}^{2}[f](x)}{h}}=-{\frac {f(x+2h)-4f(x+h)+3f(x)}{2h}}$

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.

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

Дурын хэмжээтэй цөмүүд

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.

Шинж чанар

• For all positive k and n

$\Delta _{kh}^{n}(f,x)=\sum \limits _{i_{1}=0}^{k-1}\sum \limits _{i_{2}=0}^{k-1}\cdots \sum \limits _{i_{n}=0}^{k-1}\Delta _{h}^{n}(f,x+i_{1}h+i_{2}h+\cdots +i_{n}h).$

• Leibniz rule:

$\Delta _{h}^{n}(fg,x)=\sum \limits _{k=0}^{n}{\binom {n}{k}}\Delta _{h}^{k}(f,x)\Delta _{h}^{n-k}(g,x+kh).$

Төгсгөлөг ялгаварын аргууд

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.

Ньютоны цуваа

The Newton series consists of the terms of the Newton forward difference equation, named after Isaac Newton; in essence, it is the Newton interpolation formula, first published in his Principia Mathematica in 1687, namely the discrete analog of the continuum Taylor expansion,

Загвар:Equation box 1 which holds for any polynomial function f and for most (but not all) analytic functions. Here, the expression

${x \choose k}={\frac {(x)_{k}}{k!}}$

is the binomial coefficient, and

$(x)_{k}=x(x-1)(x-2)\cdots (x-k+1)$

is the "falling factorial" or "lower factorial", while the empty product (x)₀ is defined to be 1. In this particular case, there is an assumption of unit steps for the changes in the values of x, h = 1 of the generalization below.

Note also the formal correspondence of this result to Taylor's theorem. Historically, this, as well as the Chu–Vandermonde identity,

$(x+y)_{n}=\sum _{k=0}^{n}{n \choose k}(x)_{n-k}~(y)_{k}~,$

(following from it, and corresponding to the binomial theorem), are included in the observations which matured to the system of the umbral calculus.

To illustrate how one may use Newton's formula in actual practice, consider the first few terms of doubling the Fibonacci sequence Загвар:Mvar = 2, 2, 4, ... One can find a polynomial that reproduces these values, by first computing a difference table, and then substituting the differences which correspond to x₀ (underlined) into the formula as follows,

${\begin{matrix}{\begin{array}{|c||c|c|c|}\hline x&f=\Delta ^{0}&\Delta ^{1}&\Delta ^{2}\\\hline 1&{\underline {2}}&&\\&&{\underline {0}}&\\2&2&&{\underline {2}}\\&&2&\\3&4&&\\\hline \end{array}}&\quad {\begin{aligned}f(x)&=\Delta ^{0}\cdot 1+\Delta ^{1}\cdot {\dfrac {(x-x_{0})_{1}}{1!}}+\Delta ^{2}\cdot {\dfrac {(x-x_{0})_{2}}{2!}}\quad (x_{0}=1)\\\\&=2\cdot 1+0\cdot {\dfrac {x-1}{1}}+2\cdot {\dfrac {(x-1)(x-2)}{2}}\\\\&=2+(x-1)(x-2)\\\end{aligned}}\end{matrix}}$

For the case of nonuniform steps in the values of x, Newton computes the divided differences,

$\Delta _{j,0}=y_{j},\quad \quad \Delta _{j,k}={\frac {\Delta _{j+1,k-1}-\Delta _{j,k-1}}{x_{j+k}-x_{j}}}\quad \ni \quad \left\{k>0,\ \ j\leq \max \left(j\right)-k\right\},\quad \quad \Delta 0_{k}=\Delta _{0,k}$

the series of products,

${P_{0}}=1,\quad \quad P_{k+1}=P_{k}\cdot \left(\xi -x_{k}\right)~,$

and the resulting polynomial is the scalar product, $f(\xi )=\Delta 0\cdot P\left(\xi \right)$ .

In analysis with p-adic 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 suitably scaled forward differences.

In a compressed and slightly more general form and equidistant nodes the formula reads

$f(x)=\sum _{k=0}{{\frac {x-a}{h}} \choose k}\sum _{j=0}^{k}(-1)^{k-j}{k \choose j}f(a+jh).$

Төгсгөлөг ялгаварын тооцоолол

The forward difference can be considered as a difference operator, which maps the function Загвар:Mvar to Δ_h[f ]. This operator amounts to

$\Delta _{h}=T_{h}-I,\,$

where T_h is the shift operator with step h, defined by T_h[f ](x) = f(x+h), and Загвар:Mvar is the identity operator.

The finite difference of higher orders can be defined in recursive manner as Δ_hⁿ ≡ Δ_h (Δ_hⁿ⁻¹). Another equivalent definition is Δ_hⁿ = [T_h −I]ⁿ.

The difference operator Δ_h is a linear operator and it satisfies a special Leibniz rule indicated above, Δ_h(f(x)g(x)) = (Δ_hf(x)) g(x+h) + f(x) (Δ_hg(x)). Similar statements hold for the backward and central differences.

Formally applying the Taylor series with respect to h, yields the formula

$\Delta _{h}=hD+{\frac {1}{2}}h^{2}D^{2}+{\frac {1}{3!}}h^{3}D^{3}+\cdots =\mathrm {e} ^{hD}-I~,$

where D denotes the continuum derivative operator, mapping f to its derivative f'. The expansion is valid when both sides act on analytic functions, for sufficiently small h. Thus, T_h=e^hD, and formally inverting the exponential yields

$hD=\log(1+\Delta _{h})=\Delta _{h}-{\tfrac {1}{2}}\Delta _{h}^{2}+{\tfrac {1}{3}}\Delta _{h}^{3}+\cdots .\,$

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.

The analogous formulas for the backward and central difference operators are

$hD=-\log(1-\nabla _{h})\quad {\text{and}}\quad hD=2\,\operatorname {arsinh} ({\tfrac {1}{2}}\delta _{h}).$

The calculus of finite differences is related to the umbral calculus of combinatorics. This remarkably systematic correspondence is due to the identity of the commutators of the umbral quantities to their continuum analogs (h→0 limits),

A large number of formal differential relations of standard calculus involving functions f(x) thus map systematically to umbral finite-difference analogs involving f(xT_h⁻¹).

For instance, the umbral analog of a monomial xⁿ is a generalization of the above falling factorial (Pochhammer k-symbol),

$~(x)_{n}\equiv (xT_{h}^{-1})^{n}=x(x-h)(x-2h)\cdots (x-(n-1)h)$ ,

so that

${\frac {\Delta _{h}}{h}}~(x)_{n}=n~(x)_{n-1}~,$

hence the above Newton interpolation formula (by matching coefficients in the expansion of an arbitrary function f(x) in such symbols), and so on.

For example, the umbral sine is

$\sin(x\,T_{h}^{-1})=x-{\frac {(x)_{3}}{3!}}+{\frac {(x)_{5}}{5!}}-{\frac {(x)_{7}}{7!}}+\cdots .$

As in the continuum limit, the eigenfunction of Δ_h /h also happens to be an exponential,

${\frac {\Delta _{h}}{h}}~(1+\lambda h)^{x/h}={\frac {\Delta _{h}}{h}}~e^{\ln(1+\lambda h)~x/h}=\lambda ~e^{\ln(1+\lambda h)~x/h}~,$

and hence Fourier sums of continuum functions are readily mapped to umbral Fourier sums faithfully, i.e., involving the same Fourier coefficients multiplying these umbral basis exponentials. This umbral exponential thus amounts to the exponential generating function of the Pochhammer symbols.

Thus, for instance, the Dirac delta function maps to its umbral correspondent, the cardinal sine function,

$\delta (x)\mapsto {\frac {\sin {\bigl [}{\frac {\pi }{2}}(1+x/h){\bigr ]}}{\pi (x+h)}}~,$

and so forth. Difference equations can often be solved with techniques very similar to those for solving differential equations.

The inverse operator of the forward difference operator, so then the umbral integral, is the indefinite sum or antidifference operator.

Төгсгөлөг ялгаварын операторын тооцоолох дүрмүүд

Analogous to rules for finding the derivative, we have:

• Constant rule: If c is a constant, then

$\Delta c=0{\,}$

• Linearity: if a and b are constants,

$\Delta (af+bg)=a\,\Delta f+b\,\Delta g$

All of the above rules apply equally well to any difference operator, including $\nabla$ as to $\Delta$.

• Product rule:

$\Delta (fg)=f\,\Delta g+g\,\Delta f+\Delta f\,\Delta g$

$\nabla (fg)=f\,\nabla g+g\,\nabla f-\nabla f\,\nabla g$

• Quotient rule:

$\nabla \left({\frac {f}{g}}\right)={\frac {1}{g}}\det {\begin{bmatrix}\nabla f&\nabla g\\f&g\end{bmatrix}}\left(\det {\begin{bmatrix}g&\nabla g\\1&1\end{bmatrix}}\right)^{-1}$

or

$\nabla \left({\frac {f}{g}}\right)={\frac {g\,\nabla f-f\,\nabla g}{g\cdot (g-\nabla g)}}$

$\Delta \left({\frac {f}{g}}\right)={\frac {g\,\Delta f-f\,\Delta g}{g\cdot (g+\Delta g)}}$

• Summation rules:

$\sum _{n=a}^{b}\Delta f(n)=f(b+1)-f(a)$

$\sum _{n=a}^{b}\nabla f(n)=f(b)-f(a-1)$

Ерөнхийлөл буюу нэгтгэл

• A generalized finite difference is usually defined as

$\Delta _{h}^{\mu }[f](x)=\sum _{k=0}^{N}\mu _{k}f(x+kh),$

where $\mu =(\mu _{0},\ldots ,\mu _{N})$ 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 $\mu _{k}$ depend on point $x$ : $\mu _{k}=\mu _{k}(x)$, 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.

• The generalized difference can be seen as the polynomial rings $R[T_{h}]$ . It leads to difference algebras.

• Difference operator generalizes to Möbius inversion over a partially ordered set.

• As a convolution operator: Via the formalism of incidence algebras, difference operators and other Möbius inversion can be represented by convolution with a function on the poset, called the Möbius function μ; for the difference operator, μ is the sequence (1, −1, 0, 0, 0, ...).

Зарим өөрчлөлтүүд дахь төгсгөлөг ялгавар

Finite differences can be considered in more than one variable. They are analogous to partial derivatives in several variables.

Some partial derivative approximations are (using central step method):

$f_{x}(x,y)\approx {\frac {f(x+h,y)-f(x-h,y)}{2h}}\$

$f_{y}(x,y)\approx {\frac {f(x,y+k)-f(x,y-k)}{2k}}\$

$f_{xx}(x,y)\approx {\frac {f(x+h,y)-2f(x,y)+f(x-h,y)}{h^{2}}}\$

$f_{yy}(x,y)\approx {\frac {f(x,y+k)-2f(x,y)+f(x,y-k)}{k^{2}}}\$

$f_{xy}(x,y)\approx {\frac {f(x+h,y+k)-f(x+h,y-k)-f(x-h,y+k)+f(x-h,y-k)}{4hk}}~.$

Alternatively, for applications in which the computation of Загвар:Mvar is the most costly step, and both first and second derivatives must be computed, a more efficient formula for the last case is

$f_{xy}(x,y)\approx {\frac {f(x+h,y+k)-f(x+h,y)-f(x,y+k)+2f(x,y)-f(x-h,y)-f(x,y-k)+f(x-h,y-k)}{2hk}}~,$

since the only values to be computed which are not already needed for the previous four equations are f(x+h, y+k) and f(x−h, y−k).

Мөн үзэх

• Carlson's theorem
• Divided differences
• Finite difference coefficients
• Finite difference method
• Five-point stencil
• Gilbreath's conjecture
• Lagrange polynomial
• Central differencing scheme
• Upwind differencing scheme for convection
• Modulus of continuity
• Newton polynomial
• Nörlund–Rice integral
• Numerical differentiation
• Sheffer sequence
• Summation by parts
• Table of Newtonian series
• Taylor series
• Time scale calculus
• Umbral calculus

Гадаад линк

• Загвар:Springer
• Table of useful finite difference formula generated using Mathematica
• Finite Calculus: A Tutorial for Solving Nasty Sums
• Discrete Second Derivative from Unevenly Spaced Points