Page:EB1911 - Volume 08.djvu/242

interpolation, or the determination of intermediate values of u, and (iii.) relations between sums and integrals.

17. Starting from any pair of values x₀ and u₀, we may suppose the interval h from x₀ to x₁ to be divided into q equal portions. If we suppose the corresponding values of u to be obtained, and their differences taken, the successive advancing differences of u₀ being denoted by ∂u₀, ∂²u₀ ..., we have (§ 3 (ii.))

$u_{1}{=}u_{0}+q\partial u_{0}+{\frac {q\cdot q-1}{1\cdot 2}}\partial ^{2}u_{0}+\ldots .$

When q is made indefinitely great, this (writing ƒ(x) for u) becomes Taylor’s Theorem (Infinitesimal Calculus)

$f(x+h){=}f(x)+hf'(x)+{\frac {h^{2}}{1\cdot 2}}f''(x)+\ldots ,$

which, expressed in terms of operators, is

$\mathrm {E} {=}1+h\mathrm {D} +{\frac {h^{2}}{1\cdot 2}}\mathrm {D} ^{2}+{\frac {h^{3}}{1\cdot 2\cdot 3}}\mathrm {D} ^{3}+\ldots {=}e^{h\mathrm {D} }.$

This gives the relation between Δ and D. Also we have

$u_{2}{=}u_{0}+2q\partial u_{0}+{\frac {2q\cdot 2q-1}{1\cdot 2}}\partial ^{2}u_{0}+\ldots$

$u_{3}{=}u_{0}+3q\partial u_{0}+{\frac {3q\cdot 3q-1}{1\cdot 2}}\partial ^{2}u_{0}+\ldots$

$\vdots \qquad \qquad \quad \vdots$

and, if p is any integer,

$u_{\frac {p}{q}}{=}u_{0}+p\partial u_{0}+{\frac {p\cdot p-1}{1\cdot 2}}\partial ^{2}u_{0}+\ldots .$

From these equations u_p/q could be expressed in terms of u₀, u₁, u₂, ...; this is a particular case of interpolation (q.v.).

18. Differences and Differential Coefficients.—The various formulae are most quickly obtained by symbolical methods; i.e. by dealing with the operators Δ, E, D, ... as if they were algebraical quantities. Thus the relation E = e^hD (§ 17) gives

hD = log_e (1 + Δ) = Δ − 1/2Δ² + ¹⁄₃Δ³ ...

or

h(du/dx)₀ = Δu₀ − 1/2Δ²u₀ + ¹⁄₃Δ³u₀ ....

The formulae connecting central differences with differential coefficients are based on the relations μ = cosh 1/2hD = 1/2(e^1/2hD + e^-1/2hD), δ = 2 sinh 1/2hD − e^1/2hD − e^-1/2hD, and may be grouped as follows:—

u0	= u0	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$
μδu0	= (hD + 1⁄6 h3D3 + 1⁄120 h5D5 + ...)u0
δ2u0	= (h2D2 + 1⁄12 h4D4 + 1⁄360 h6D6 + ...)u0
μδ3u0	= (h3D3 + 1⁄4 h5D5 + ...)u0
δ4u0	= (h4D4 + 1⁄6 h6D6 + ...)u0
	·   ·   ·
	·   ·   ·
	·   ·   ·
μu1/2	= (1 + 1⁄8 h2D2 + 1⁄384 h4D4 + 1⁄46080 h6D6 + ...)u1/2	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$
δu1/2	= (hD + 1⁄24 h3D3 + 1⁄1920 h5D5 + ...)u1/2
μδ2u1/2	= (h2D2 + 5⁄24 h4D4 + 91⁄5760 h6D6 + ...)u1/2
δ3u1/2	= (h3D3 + 1⁄8 h5D5 + ...)u1/2
μδ4 u1/2	= (h4D4 + 7⁄24 h6D6 + ...)u1/2
	·   ·   ·
	·   ·   ·
	·   ·   ·
u0	= u0	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$
hDu0	= (μδ − 1⁄6 μδ3 + 1⁄30 μδ5 − ...)u0
h2D2u0	= (δ2 − 1⁄12 δ4 + 1⁄90 δ6 − ...)u0
h3D3u0	= (μδ3 − 1⁄4 μδ5 + ...)u0
h4D4u0	= (δ4 − 1⁄6 δ6 + ...)u0
	·   ·   ·
	·   ·   ·
	·   ·   ·
u1/2	= (μ − 1⁄8 μδ2 + 3⁄128 μδ4 − 5⁄1024 μδ6 + ...)u1/2	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$
hDu1/2	= (δ − 1⁄24 δ3 + 3⁄640 δ5 − ...)u1/2
h2D2u1/2	= (μδ2 − 5⁄24 μδ4 + 259⁄5760 μδ6 − ...)u1/2
h3D3u1/2	= (δ3 − 1⁄8 δ5 + ...)u1/2
h4D4 u1/2	= (μδ4 − 7⁄24 μδ6 + ...)u1/2
	·   ·   ·
	·   ·   ·
	·   ·   ·

When u is a rational integral function of x, each of the above series is a terminating series. In other cases the series will be an infinite one, and may be divergent; but it may be used for purposes of approximation up to a certain point, and there will be a “remainder,” the limits of whose magnitude will be determinate.

19. Sums and Integrals.—The relation between a sum and an integral is usually expressed by the Euler-Maclaurin formula. The principle of this formula is that, if u_m and u_m+1, are ordinates of a curve, distant h from one another, then for a first approximation to the area of the curve between u_m and u_m+1 we have 1/2h(u_m + u_m+1), and the difference between this and the true value of the area can be expressed as the difference of two expressions, one of which is a function of x_m, and the other is the same function of x_m+1. Denoting these by φ(x_m) and φ(x_m+1), we have

$\int _{x_{m}}^{x_{m+1}}udx{=}{\tfrac {1}{2}}h(u_{m}+u_{m+1})+\phi (x_{m+1})-\phi (x_{m}).$

Adding a series of similar expressions, we find

$\int _{f}^{x_{n}}rac{x_{m}}udx{=}h{{\tfrac {1}{2}}u_{m}+u_{m+1}+u_{m+2}+\ldots +u_{n-1}+{\tfrac {1}{2}}u_{n}}+\phi (x_{n})-\phi (x_{m}).$

The function φ(x) can be expressed in terms either of differential coefficients of u or of advancing or central differences; thus there are three formulae.

(i.) The Euler-Maclaurin formula, properly so called, (due independently to Euler and Maclaurin) is

$\int ^{x_{n}}udx{=}h\cdot \mu \sigma u_{n}-{\frac {1}{12}}h^{2}{\frac {du_{n}}{dx}}{\tfrac {1}{2}}+{\tfrac {1}{720}}h^{4}{\frac {d^{3}u_{n}}{dx^{3}}}-{\tfrac {1}{30240}}h^{6}{\frac {d^{5}u_{n}}{dx^{5}}}+\ldots {=}h\cdot \mu \sigma u_{n}-{\frac {\mathrm {B} _{1}}{2!}}h_{2}{\frac {du_{n}}{dx}}+{\frac {\mathrm {B} _{2}}{4!}}h^{4}{\frac {d^{3}u_{n}}{dx^{3}}}-{\frac {\mathrm {B} _{3}}{6!}}h^{6}{\frac {d^{5}u_{n}}{dx^{5}}}+\ldots$

where B₁, B₂, B₃ ... are Bernoulli’s numbers.

(ii.) If we express differential coefficients in terms of advancing differences, we get a theorem which is due to Laplace:—

${\frac {1}{h}}\int _{x_{0}}^{x_{n}}udx{=}\mu \sigma (u_{n}-u_{0})-{\tfrac {1}{12}}(\Delta u_{n}-\Delta u_{0})+{\tfrac {1}{24}}(\Delta ^{2}u_{n}-\Delta ^{2}u_{0})-{\tfrac {19}{720}}(\Delta ^{3}u_{n}-\Delta ^{3}u_{0})+{\tfrac {3}{160}}(\Delta ^{4}u_{n}-\Delta ^{4}u_{0})-\ldots$

For practical calculations this may more conveniently be written

${\frac {1}{h}}\int _{x_{0}}^{x_{n}}udx{=}\mu \sigma (u_{n}-u_{0})+{\tfrac {1}{12}}(\Delta u_{0}-{\tfrac {1}{2}}\Delta ^{2}u_{0}+{\tfrac {19}{60}}\Delta ^{3}u_{0}-\ldots )+{\tfrac {1}{12}}(\Delta 'u_{n}-{\tfrac {1}{2}}\Delta '^{2}u_{n}+{\tfrac {19}{60}}\Delta '^{3}u_{n}-\ldots ),$

where accented differences denote that the values of u are read backwards from u_n; i.e. Δ′u_n denotes u_n-1 − u_n, not (as in § 10) u_n − u_n-1.

(iii.) Expressed in terms of central differences this becomes

${\frac {1}{h}}\int _{x_{0}}^{x_{n}}udx{=}\mu \sigma (u_{n}-u_{0})-{\tfrac {1}{12}}\mu \delta u_{n}+{\tfrac {11}{720}}\mu \delta ^{3}u_{n}-\ldots +{\tfrac {1}{12}}\mu \delta u_{0}-{\tfrac {11}{720}}\mu \delta ^{3}u_{0}+\ldots {=}\mu (\sigma -{\tfrac {1}{12}}\delta +{\tfrac {11}{720}}\delta ^{3}-{\tfrac {191}{60480}}\delta ^{5}+{\tfrac {2497}{3628800}}\delta ^{7}-\ldots )(u_{n}-u_{0}).$

(iv.) There are variants of these formulae, due to taking hu_m+1/2 as the first approximation to the area of the curve between u_m and u_m+1; the formulae involve the sum u_1/2 + u_3/2 + ... + u_n-1/2 ≡ σ(u_n − u₀) (see Mensuration).

20. The formulae in the last section can be obtained by symbolical methods from the relation

${\frac {1}{h}}\int udx{=}{\frac {1}{h}}\mathrm {D} ^{-1}u{=}{\frac {1}{h\mathrm {D} }}\cdot u.$

Thus for central differences, if we write θ ≡ 1/2hD, we have μ = cosh θ, δ = 2 sinh θ, σ = δ^-1, and the result in (iii.) corresponds to the formula

sinh θ = θ cosh θ/(1 + 1/3 sinh² θ − 2/3·5 sinh⁴ θ + 2·4/3·5·7 sinh⁶ θ − . . .).

References.—There is no recent English work on the theory of finite differences as a whole. G. Boole’s Finite Differences (1st ed., 1860, 2nd ed., edited by J. F. Moulton, 1872) is a comprehensive treatise, in which symbolical methods are employed very early. A. A. Markoff’s Differenzenrechnung (German trans., 1896) contains general formulae. (Both these works ignore central differences.) Encycl. der math. Wiss. vol. i. pt. 2, pp. 919-935, may also be consulted. An elementary treatment of the subject will be found in many text-books, e.g. G. Chrystal’s Algebra (pt. 2, ch. xxxi.). A. W. Sunderland, Notes on Finite Differences (1885), is intended for actuarial students. Various central-difference formulae with references are given in Proc. Lond. Math. Soc. xxxi. pp. 449-488. For other references see Interpolation.  (W. F. Sh.) 

DIFFERENTIAL EQUATION, in mathematics, a relation between one or more functions and their differential coefficients. The subject is treated here in two parts: (1) an elementary introduction dealing with the more commonly recognized types of differential equations which can be solved by rule; and (2) the general theory.

Part I.—Elementary Introduction.

Of equations involving only one independent variable, x (known as ordinary differential equations), and one dependent variable, y, and containing only the first differential coefficient dy/dx (and therefore said to be of the first order), the simplest form is that reducible to the type

dy/dx＝ƒ(x)/F(y),

leading to the result ƒF(y)dy − ƒf(x)dx＝A, where A is an arbitrary constant; this result is said to solve the differential equation, the problem of evaluating the integrals belonging to the integral calculus.