How To Find Finite Differences From Equation

Detached analog of a derivative

A finite divergence is a mathematical expression of the class 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 function in finite departure methods for the numerical solution of differential equations, especially boundary value issues.

The difference operator, commonly denoted ${\displaystyle \Delta }$ is the operator that maps a function f to the role ${\displaystyle \Delta [f]}$ defined by

${\displaystyle \Delta [f](x)=f(x+ane)-f(x).}$

A difference equation is a functional equation that involves the finite difference operator in the same way equally a differential equation involves derivatives. In that location are many similarities between difference equations and differential equations, peculiarly in the solving methods. Certain recurrence relations can exist written as difference equations past replacing iteration notation with finite differences.

In numerical assay, finite differences are widely used for approximating derivatives, and the term "finite difference" is often used as an abridgement of "finite difference approximation of derivatives".^{[1] [2] [iii]} Finite difference approximations are finite departure quotients in the terminology employed above.

Finite differences were introduced by Brook Taylor in 1715 and have also been studied as abstruse self-standing mathematical objects in works past George Boole (1860), L. Thousand. 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]

Basic types [edit]

The three types of the finite differences. The central difference well-nigh x gives the best approximation of the derivative of the function at x.

Three basic types are commonly considered: forward, backward, and central finite differences.^{[1] [ii] [3]}

A forrad difference, denoted ${\displaystyle \Delta _{h}[f],}$ of a part f is a function defined as

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

Depending on the application, the spacing h may exist variable or abiding. When omitted, h is taken to be i; that is,

${\displaystyle \Delta [f](x)=\Delta _{1}[f](ten)=f(10+1)-f(ten).}$

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

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

Finally, the key difference is given by

${\displaystyle \delta _{h}[f](x)=f(x+{\tfrac {h}{ii}})-f(x-{\tfrac {h}{2}})=\Delta _{h/2}[f](x)+\nabla _{h/2}[f](ten).}$

Relation with derivatives [edit]

Finite difference is often used equally an approximation of the derivative, typically in numerical differentiation.

The derivative of a function f at a point x is defined by the limit.

${\displaystyle f'(10)=\lim _{h\to 0}{\frac {f(10+h)-f(x)}{h}}.}$

If h has a stock-still (non-cypher) value instead of budgeted zero, then the correct-hand side of the above equation would exist written

${\displaystyle {\frac {f(10+h)-f(x)}{h}}={\frac {\Delta _{h}[f](10)}{h}}.}$

Hence, the forward difference divided by h approximates the derivative when h is pocket-size. The error in this approximation can exist derived from Taylor's theorem. Assuming that f is twice differentiable, we take

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

The same formula holds for the backward divergence:

${\displaystyle {\frac {\nabla _{h}[f](ten)}{h}}-f'(ten)=O(h)\to 0\quad {\text{equally }}h\to 0.}$

However, the central (also chosen centered) difference yields a more than accurate approximation. If f is 3 times differentiable,

${\displaystyle {\frac {\delta _{h}[f](x)}{h}}-f'(ten)=O\left(h^{2}\correct).}$

The main problem^{[ citation needed ]} with the central departure method, however, is that oscillating functions can yield zero derivative. If f (nh) = 1 for n odd, and f (nh) = 2 for northward even, then f ′(nh) = 0 if information technology is calculated with the fundamental departure scheme. This is particularly troublesome if the domain of f is discrete. Meet also Symmetric derivative

Authors for whom finite differences hateful finite difference approximations define the forward/backward/primal differences as the quotients given in this department (instead of employing the definitions given in the previous section).^{[1] [2] [3]}

Higher-order differences [edit]

In an coordinating fashion, i can obtain finite departure approximations to higher order derivatives and differential operators. For example, by using the in a higher place central divergence formula for f ′(x + h / 2 ) and f ′(10 − h / 2 ) and applying a central difference formula for the derivative of f ′ at x, we obtain the central difference approximation of the 2d derivative of f:

Second-lodge central

${\displaystyle f''(x)\approx {\frac {\delta _{h}^{2}[f](x)}{h^{2}}}={\frac {{\frac {f(x+h)-f(x)}{h}}-{\frac {f(10)-f(x-h)}{h}}}{h}}={\frac {f(10+h)-2f(x)+f(x-h)}{h^{2}}}.}$

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

Second order forrard

${\displaystyle f''(x)\approx {\frac {\Delta _{h}^{2}[f](x)}{h^{two}}}={\frac {{\frac {f(x+2h)-f(10+h)}{h}}-{\frac {f(x+h)-f(ten)}{h}}}{h}}={\frac {f(ten+2h)-2f(10+h)+f(x)}{h^{two}}}.}$

Second order backward

${\displaystyle f''(x)\approx {\frac {\nabla _{h}^{2}[f](x)}{h^{2}}}={\frac {{\frac {f(10)-f(10-h)}{h}}-{\frac {f(x-h)-f(x-2h)}{h}}}{h}}={\frac {f(x)-2f(x-h)+f(10-2h)}{h^{2}}}.}$

More generally, the northwardth order forwards, backward, and primal differences are given by, respectively,

Forward

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

or for h = ane,

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

Backward

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

Key

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

These equations apply binomial coefficients subsequently the summation sign shown every bit ( ^{due north}
_i ) . Each row of Pascal's triangle provides the coefficient for each value of i.

Note that the central departure will, for odd n, have h multiplied past not-integers. This is often a problem considering information technology amounts to changing the interval of discretization. The problem may be remedied taking the average of δ^northward [f ](x − h / 2 ) and δ^northward [f ](x + h / 2 ).

Forrard 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-indicate techniques; by contrast, the forward difference serial can exist extremely hard to evaluate numerically, because the binomial coefficients grow rapidly for large n.

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

${\displaystyle {\frac {d^{n}f}{dx^{n}}}(x)={\frac {\Delta _{h}^{n}[f](10)}{h^{n}}}+O(h)={\frac {\nabla _{h}^{n}[f](x)}{h^{n}}}+O(h)={\frac {\delta _{h}^{n}[f](x)}{h^{n}}}+O\left(h^{2}\right).}$

Higher-social club differences can also be used to construct amend approximations. As mentioned above, the first-society divergence approximates the first-order derivative upwards to a term of guild h. However, the combination

${\displaystyle {\frac {\Delta _{h}[f](ten)-{\frac {1}{ii}}\Delta _{h}^{ii}[f](ten)}{h}}=-{\frac {f(x+2h)-4f(x+h)+3f(10)}{2h}}}$

approximates f ′(ten) up to a term of guild h ⁱⁱ . This tin can be proven by expanding the in a higher place 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.

Arbitrarily sized kernels [edit]

Using linear algebra 1 tin construct finite difference approximations which utilize an arbitrary number of points to the left and a (maybe different) number of points to the right of the evaluation point, for whatsoever guild derivative. This involves solving a linear system such that the Taylor expansion of the sum of those points around the evaluation point best approximates the Taylor expansion of the desired derivative. Such formulas can exist represented graphically on a hexagonal or diamond-shaped grid.^[5]

This is useful for differentiating a function on a filigree, where, equally one approaches the border of the grid, one must sample fewer and fewer points on one side.

The details are outlined in these notes.

The Finite Difference Coefficients Calculator constructs finite departure approximations for non-standard (and even not-integer) stencils given an arbitrary stencil and a desired derivative order.

Properties [edit]

• For all positive k and northward

${\displaystyle \Delta _{kh}^{n}(f,ten)=\sum \limits _{i_{i}=0}^{k-1}\sum \limits _{i_{2}=0}^{k-one}\cdots \sum \limits _{i_{n}=0}^{k-ane}\Delta _{h}^{n}\left(f,ten+i_{1}h+i_{two}h+\cdots +i_{n}h\correct).}$

• Leibniz dominion:

${\displaystyle \Delta _{h}^{n}(fg,x)=\sum \limits _{1000=0}^{north}{\binom {n}{k}}\Delta _{h}^{k}(f,10)\Delta _{h}^{northward-thousand}(grand,10+kh).}$

In differential equations [edit]

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. The idea is to replace the derivatives appearing in the differential equation by finite differences that approximate them. The resulting methods are chosen finite departure methods.

Mutual applications of the finite difference method are in computational science and technology disciplines, such every bit thermal engineering, fluid mechanics, etc.

Newton'south series [edit]

The Newton serial 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,^{[half dozen]} namely the discrete analog of the continuous Taylor expansion,

${\displaystyle f(x)=\sum _{k=0}^{\infty }{\frac {\Delta ^{chiliad}[f](a)}{k!}}\,(ten-a)_{m}=\sum _{k=0}^{\infty }{\binom {ten-a}{one thousand}}\,\Delta ^{k}[f](a),}$

which holds for any polynomial function f and for many (simply non all) analytic functions (It does not concur when f is exponential type ${\displaystyle \pi }$ . This is easily seen, as the sine function vanishes at integer multiples of ${\displaystyle \pi }$ ; the corresponding Newton series is identically zero, as all finite differences are zero in this case. Notwithstanding clearly, the sine part is not zero.). Here, the expression

${\displaystyle {\binom {x}{1000}}={\frac {(ten)_{one thousand}}{grand!}}}$

is the binomial coefficient, and

${\displaystyle (10)_{k}=x(x-1)(x-2)\cdots (10-thou+1)}$

is the "falling factorial" or "lower factorial", while the empty product (x)₀ is defined to exist i. In this particular case, there is an assumption of unit steps for the changes in the values of 10, h = one of the generalization beneath.

Notation the formal correspondence of this result to Taylor's theorem. Historically, this, besides as the Chu–Vandermonde identity,

${\displaystyle (10+y)_{northward}=\sum _{k=0}^{north}{\binom {n}{k}}(x)_{n-thousand}\,(y)_{thousand},}$

(following from it, and respective to the binomial theorem), are included in the observations that matured to the organization of umbral calculus.

Newton series expansions can be superior to Taylor series expansions when practical to discrete quantities like breakthrough spins (see Holstein–Primakoff_transformation), bosonic operator functions or discrete counting statistics.^[seven]

To illustrate how ane may use Newton's formula in bodily practice, consider the first few terms of doubling the Fibonacci sequence f = 2, 2, 4, ... 1 can find a polynomial that reproduces these values, by showtime computing a difference table, and so substituting the differences that correspond to x ₀ (underlined) into the formula as follows,

${\displaystyle {\begin{matrix}{\brainstorm{array}{|c||c|c|c|}\hline x&f=\Delta ^{0}&\Delta ^{1}&\Delta ^{ii}\\\hline 1&{\underline {2}}&&\\&&{\underline {0}}&\\2&2&&{\underline {two}}\\&&2&\\3&iv&&\\\hline \stop{array}}&\quad {\brainstorm{aligned}f(ten)&=\Delta ^{0}\cdot 1+\Delta ^{1}\cdot {\dfrac {(x-x_{0})_{1}}{1!}}+\Delta ^{ii}\cdot {\dfrac {(x-x_{0})_{2}}{ii!}}\quad (x_{0}=1)\\\\&=2\cdot 1+0\cdot {\dfrac {x-1}{one}}+2\cdot {\dfrac {(x-1)(x-two)}{2}}\\\\&=ii+(x-ane)(x-2)\\\cease{aligned}}\end{matrix}}}$

For the example of nonuniform steps in the values of ten, Newton computes the divided differences,

${\displaystyle \Delta _{j,0}=y_{j},\qquad \Delta _{j,thousand}={\frac {\Delta _{j+1,k-1}-\Delta _{j,g-one}}{x_{j+thou}-x_{j}}}\quad \ni \quad \left\{k>0,\;j\leq \max \left(j\right)-thousand\correct\},\qquad \Delta 0_{k}=\Delta _{0,m}}$

the series of products,

${\displaystyle {P_{0}}=1,\quad \quad P_{k+1}=P_{k}\cdot \left(\11 -x_{k}\right),}$

and the resulting polynomial is the scalar product,^[8]

${\displaystyle f(\xi )=\Delta 0\cdot P\left(\eleven \correct)}$ .

In analysis with p-adic numbers, Mahler'due south theorem states that the assumption that f is a polynomial function tin can be weakened all the mode to the assumption that f is only continuous.

Carlson's theorem provides necessary and sufficient weather for a Newton series to exist unique, if it exists. However, a Newton series does not, in general, exist.

The Newton series, together with the Stirling series and the Selberg serial, 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

${\displaystyle f(ten)=\sum _{thou=0}{\binom {\frac {x-a}{h}}{k}}\sum _{j=0}^{m}(-1)^{1000-j}{\binom {k}{j}}f(a+jh).}$

Calculus of finite differences [edit]

The forward difference can be considered equally an operator, chosen the difference operator, which maps the office f to Δ _h [f ].^{[9] [10]} This operator amounts to

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

where T _h is the shift operator with footstep h, defined by T _h [f ](x) = f (ten + h), and I is the identity operator.

The finite divergence of higher orders tin be defined in recursive fashion as Δ ⁿ
_h ≡ Δ _h (Δ ^{northward − ane}
_h ). Another equivalent definition is Δ ⁿ
_h = [T _h − I] ^north .

The divergence operator Δ _h is a linear operator, every bit such it satisfies Δ _h [αf + βg](x) = α Δ _h [f ](ten) + β Δ _h [thousand](x).

Information technology also satisfies a special Leibniz rule indicated above, Δ _h (f (x)thousand(x)) = (Δ _h f (x)) g(x+h) + f (10) (Δ _h thousand(x)). Similar statements agree for the astern and cardinal differences.

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

${\displaystyle \Delta _{h}=hD+{\frac {one}{2!}}h^{two}D^{two}+{\frac {1}{3!}}h^{3}D^{3}+\cdots =\mathrm {east} ^{hD}-I,}$

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 pocket-size h. Thus, T _h = e ^{hard disk} , and formally inverting the exponential yields

${\displaystyle hD=\log(1+\Delta _{h})=\Delta _{h}-{\tfrac {one}{two}}\Delta _{h}^{2}+{\tfrac {1}{iii}}\Delta _{h}^{three}-\cdots .}$

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 correct is non guaranteed to converge; it may be an asymptotic series. Even so, it can be used to obtain more than accurate approximations for the derivative. For instance, retaining the kickoff two terms of the serial yields the second-gild approximation to f ′(x) mentioned at the end of the department Higher-order differences.

The analogous formulas for the backward and fundamental deviation operators are

${\displaystyle hD=-\log(1-\nabla _{h})\quad {\text{and}}\quad hard disk=2\operatorname {arsinh} \left({\tfrac {1}{ii}}\delta _{h}\right).}$

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),

${\displaystyle \left[{\frac {\Delta _{h}}{h}},x\,T_{h}^{-1}\right]=[D,10]=I.}$

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 ⁻¹
_h ).

For instance, the umbral analog of a monomial x^north is a generalization of the above falling factorial (Pochhammer grand-symbol),

${\displaystyle ~(x)_{n}\equiv \left(xT_{h}^{-1}\correct)^{n}=10(x-h)(10-2h)\cdots {\bigl (}x-(n-1)h{\bigr )},}$

so that

${\displaystyle {\frac {\Delta _{h}}{h}}(ten)_{due north}=n(10)_{n-1},}$

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

For example, the umbral sine is

${\displaystyle \sin \left(x\,T_{h}^{-1}\right)=x-{\frac {(10)_{3}}{3!}}+{\frac {(x)_{5}}{5!}}-{\frac {(10)_{seven}}{7!}}+\cdots }$

As in the continuum limit, the eigenfunction of Δ _h / h as well happens to be an exponential,

${\displaystyle {\frac {\Delta _{h}}{h}}(i+\lambda h)^{\frac {ten}{h}}={\frac {\Delta _{h}}{h}}e^{\ln(1+\lambda h){\frac {10}{h}}}=\lambda eastward^{\ln(one+\lambda h){\frac {x}{h}}},}$

and hence Fourier sums of continuum functions are readily mapped to umbral Fourier sums faithfully, i.e., involving the aforementioned Fourier coefficients multiplying these umbral footing exponentials.^[11] This umbral exponential thus amounts to the exponential generating function of the Pochhammer symbols.

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

${\displaystyle \delta (x)\mapsto {\frac {\sin \left[{\frac {\pi }{ii}}\left(ane+{\frac {ten}{h}}\right)\correct]}{\pi (x+h)}},}$

and so forth.^[12] Deviation equations can often be solved with techniques very like to those for solving differential equations.

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

Rules for calculus of finite divergence operators [edit]

Coordinating to rules for finding the derivative, we take:

• Constant rule: If c is a constant, and then

${\displaystyle \Delta c=0}$

• Linearity: if a and b are constants,

${\displaystyle \Delta (af+bg)=a\,\Delta f+b\,\Delta m}$

All of the to a higher place rules utilize equally well to whatever difference operator, including ∇ as to Δ.

• Product rule:

${\displaystyle {\brainstorm{aligned}\Delta (fg)&=f\,\Delta g+k\,\Delta f+\Delta f\,\Delta g\\\nabla (fg)&=f\,\nabla g+1000\,\nabla f-\nabla f\,\nabla k\end{aligned}}}$

• Quotient rule:

${\displaystyle \nabla \left({\frac {f}{thou}}\right)={\frac {1}{g}}\det {\begin{bmatrix}\nabla f&\nabla g\\f&one thousand\end{bmatrix}}\left(\det {\begin{bmatrix}k&\nabla g\\i&1\end{bmatrix}}\right)^{-1}}$

or

${\displaystyle \nabla \left({\frac {f}{g}}\right)={\frac {1000\,\nabla f-f\,\nabla g}{g\cdot (yard-\nabla g)}}}$

• Summation rules:

${\displaystyle {\begin{aligned}\sum _{n=a}^{b}\Delta f(n)&=f(b+1)-f(a)\\\sum _{north=a}^{b}\nabla f(northward)&=f(b)-f(a-1)\stop{aligned}}}$

Meet references.^{[13] [fourteen] [fifteen] [16]}

Generalizations [edit]

• A generalized finite difference is unremarkably defined as

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

  

  where μ = (μ ₀, …, μ_North ) is its coefficient vector. An infinite divergence is a further generalization, where the finite sum higher up is replaced by an space series. Some other way of generalization is making coefficients μ_k depend on point ten: μ_k = μ_k (ten), thus considering weighted finite deviation. As well one may brand the step h depend on signal x: h = h(x). Such generalizations are useful for constructing unlike modulus of continuity.
• The generalized difference can exist seen as the polynomial rings R[T_h ]. It leads to difference algebras.
• Divergence operator generalizes to Möbius inversion over a partially ordered set up.
• As a convolution operator: Via the formalism of incidence algebras, deviation operators and other Möbius inversion tin can be represented by convolution with a function on the poset, called the Möbius function μ; for the divergence operator, μ is the sequence (1, −1, 0, 0, 0, …).

Multivariate finite differences [edit]

Finite differences can exist considered in more than 1 variable. They are analogous to partial derivatives in several variables.

Some partial derivative approximations are:

${\displaystyle {\begin{aligned}f_{10}(x,y)&\approx {\frac {f(ten+h,y)-f(x-h,y)}{2h}}\\f_{y}(ten,y)&\approx {\frac {f(x,y+k)-f(x,y-m)}{2k}}\\f_{xx}(x,y)&\approx {\frac {f(x+h,y)-2f(x,y)+f(ten-h,y)}{h^{ii}}}\\f_{yy}(x,y)&\approx {\frac {f(x,y+grand)-2f(x,y)+f(x,y-1000)}{thousand^{ii}}}\\f_{xy}(x,y)&\approx {\frac {f(x+h,y+chiliad)-f(x+h,y-k)-f(x-h,y+k)+f(x-h,y-k)}{4hk}}.\stop{aligned}}}$

Alternatively, for applications in which the ciphering of f is the most costly pace, and both first and second derivatives must be computed, a more than efficient formula for the last instance is

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

since the only values to compute that are non already needed for the previous four equations are f (10 + h, y + k) and f (x − h, y − k).

Encounter besides [edit]

• Discrete calculus
• Divided differences
• Finite-deviation fourth dimension-domain method (FDTD)
• Finite volume method
• FTCS scheme
• Gilbreath'southward conjecture
• Sheffer sequence
• Summation by parts
• Time scale calculus
• Upwind differencing scheme for convection

References [edit]

1. ^ ^{a b c} Paul Wilmott; Sam Howison; Jeff Dewynne (1995). The Mathematics of Financial Derivatives: A Student Introduction . Cambridge University Press. p. 137. ISBN978-0-521-49789-3.
2. ^ ^{a b c} Peter Olver (2013). Introduction to Partial Differential Equations. Springer Science & Business Media. p. 182. ISBN978-3-319-02099-0.
3. ^ ^{a b c} Yard Hanif Chaudhry (2007). Open-Aqueduct Flow. Springer. p. 369. ISBN978-0-387-68648-6.
4. ^ 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
5. ^ Fraser, Duncan C. (Jan 1, 1909). "On the Graphic Depiction of Interpolation Formulæ". Periodical of the Constitute of Actuaries. 43 (ii): 235–241. doi:10.1017/S002026810002494X. Retrieved April 17, 2017.
6. ^ Newton, Isaac, (1687). Principia, Book 3, Lemma V, Case 1
7. ^ Jürgen König and Alfred Hucht, SciPost Phys. x, 007 (2021) doi: 10.21468/SciPostPhys.10.ane.007
8. ^ Richtmeyer, D. and Morton, K.West., (1967). Deviation Methods for Initial Value Bug, 2nd ed., Wiley, New York.
9. ^ Boole, George, (1872). A Treatise On The Calculus of Finite Differences, second ed., Macmillan and Company. On line. Likewise, [Dover edition 1960]
10. ^ Hashemite kingdom of jordan, Charles, (1939/1965). "Calculus of Finite Differences", Chelsea Publishing. On-line: [i]
11. ^ Zachos, C. (2008). "Umbral Deformations on Discrete Space-Time". International Journal of Modern Physics A. 23 (thirteen): 2005–2014. arXiv:0710.2306. Bibcode:2008IJMPA..23.2005Z. doi:10.1142/S0217751X08040548. S2CID 16797959.
12. ^ Curtright, T. L.; Zachos, C. K. (2013). "Umbral Vade Mecum". Frontiers in Physics. i: xv. arXiv:1304.0429. Bibcode:2013FrP.....1...15C. doi:x.3389/fphy.2013.00015. S2CID 14106142.
13. ^ Levy, H.; Lessman, F. (1992). Finite Difference Equations. Dover. ISBN0-486-67260-three.
14. ^ Ames, West. F., (1977). Numerical Methods for Partial Differential Equations, Section 1.6. Academic Printing, New York. ISBN 0-12-056760-1.
15. ^ Hildebrand, F. B., (1968). Finite-Difference Equations and Simulations, Section 2.two, Prentice-Hall, Englewood Cliffs, New Jersey.
16. ^ Flajolet, Philippe; Sedgewick, Robert (1995). "Mellin transforms and asymptotics: Finite differences and Rice's integrals" (PDF). Theoretical Computer Science. 144 (1–ii): 101–124. doi:10.1016/0304-3975(94)00281-M. .

• Richardson, C. H. (1954): An Introduction to the Calculus of Finite Differences (Van Nostrand (1954) online copy
• Mickens, R. E. (1991): Difference Equations: Theory and Applications (Chapman and Hall/CRC) ISBN 978-0442001360

External links [edit]

• "Finite-difference calculus", Encyclopedia of Mathematics, European monetary system Press, 2001 [1994]
• Tabular array of useful finite deviation formula generated using Mathematica
• D. Gleich (2005), Finite Calculus: A Tutorial for Solving Nasty Sums
• Discrete Second Derivative from Unevenly Spaced Points

Source: https://en.wikipedia.org/wiki/Finite_difference

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