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 &part;u0, &part;²u0 ..., we have (§ 3 (ii.)) $u_{1} = u_{0} + q\partial u_{0} + \frac{q\cdot q - 1}{1\cdot 2}\partial^2u_{0} + \ldots .$ When q is made indefinitely great, this (writing &fnof;(x) for u) becomes Taylor’s Theorem $f(x + h) = f(x) + hf'(x) + \frac{h^2}{1\cdot 2}f''(x) + \ldots ,$ which, expressed in terms of operators, is $\mathrm{E} = 1 + h\mathrm{D} + \frac{h^2}{1\cdot 2}\mathrm{D}^2 + \frac{h^3}{1\cdot 2\cdot 3}\mathrm{D}^3 + \ldots = e^{h\mathrm{D}}.$|undefined This gives the relation between &Delta; and D. Also we have

and, if p is any integer, $u_{\frac{p}{q}} = u_{0} + p\partial u_{0} + \frac{p\cdot p - 1}{1\cdot 2}\partial^2u_{0} + \ldots .$|undefined From these equations up/q could be expressed in terms of u0, u1, u2, ...; this is a particular case of (q.v.).

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

or

The formulae connecting central differences with differential coefficients are based on the relations &mu; = cosh hD = (e1/2hD + e-1/2hD), &delta; = 2 sinh hD &minus; e1/2hD &minus; e-1/2hD, and may be grouped as follows:—

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 um and um+1, are ordinates of a curve, distant h from one another, then for a first approximation to the area of the curve between um and um+1 we have h(um + um+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 xm, and the other is the same function of xm+1. Denoting these by &phi;(xm) and &phi;(xm+1), we have $\int^{x_{m+1}}_{x_{m}}udx = \tfrac{1}{2}h(u_{m} + u_{m+1}) + \phi (x_{m+1}) - \phi (x_{m}).$|undefined Adding a series of similar expressions, we find $\int^{x_{n}}_frac{x_{m}}udx = h{\tfrac{1}{2}u_{m} + u_{m+1} + u_{m+2} + \ldots + u_{n-1} + \tfrac{1}{2}u_{n}} + \phi (x_{n}) - \phi (x_{m}).$|undefined

The function &phi;(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 $\int^{x_{n}}udx = h\cdot \mu \sigma u_{n} -\frac{1}{12} h^2\frac{du_{n}}{dx}\tfrac{1}{2} +\tfrac{1}{720} h^{4}\frac{d^3u_{n}}{dx^{3}} - \tfrac{1}{30240} h^{6}\frac{d^{5}u_{n}}{dx^{5}} + \ldots = h\cdot \mu \sigma u_{n} - \frac{\mathrm{B}_{1}}{2!}h_{2}\frac{du_{n}}{dx} + \frac{\mathrm{B}_{2}}{4!}h^{4}\frac{d^{3}u_{n}}{dx^{3}} - \frac{\mathrm{B}_{3}}{6!}h^{6}\frac{d^{5}u_{n}}{dx^{5}} + \ldots$|undefined 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:— $\frac{1}{h}\int^{x_{n}}_{x_{0}}udx = \mu \sigma (u_{n} - u_{0}) -\tfrac{1}{12}(\Delta u_{n} - \Delta u_{0}) +\tfrac{1}{24}(\Delta^2u_{n} - \Delta^2u_{0}) -\tfrac{19}{720}(\Delta^3u_{n} - \Delta^3u_{0}) +\tfrac{3}{160}(\Delta^{4}u_{n} - \Delta^{4}u_{0}) - \ldots$|undefined For practical calculations this may more conveniently be written $\frac{1}{h}\int^{x_{n}}_{x_{0}}udx = \mu \sigma (u_{n} - u_{0}) +\tfrac{1}{12}(\Delta u_{0} - \tfrac{1}{2}\Delta^2u_{0} +\tfrac{19}{60}\Delta^3u_{0} - \ldots ) +\tfrac{1}{12}(\Delta ' u_{n} - \tfrac{1}{2}\Delta '^2u_{n} +\tfrac{19}{60}\Delta '^3u_{n} - \ldots ),$|undefined where accented differences denote that the values of u are read backwards from un; i.e. &Delta;′un denotes un-1 &minus; un, not (as in § 10) un &minus; un-1.

(iii.) Expressed in terms of central differences this becomes $\frac{1}{h}\int^{x_{n}}_{x_{0}}udx = \mu \sigma (u_{n} - u_{0}) -\tfrac{1}{12}\mu \delta u_{n} +\tfrac{11}{720} \mu \delta^3u_{n} - \ldots +\tfrac{1}{12}\mu \delta u_{0} -\tfrac{11}{720} \mu \delta^3u_{0} + \ldots = \mu (\sigma -\tfrac{1}{12}\delta +\tfrac{11}{720}\delta^3 -\tfrac{191}{60480}\delta^{5} +\tfrac{2497}{3628800}\delta^{7} - \ldots )(u_{n} - u_{0}).$|undefined

(iv.) There are variants of these formulae, due to taking hum+1/2 as the first approximation to the area of the curve between um and um+1; the formulae involve the sum u1/2 + u3/2 + ... + un-1/2 &equiv; &sigma;(un &minus; u0) (see ).

20. The formulae in the last section can be obtained by symbolical methods from the relation $\frac{1}{h}\int udx = \frac{1}{h} \mathrm{D}^{-1}u = \frac{1}{h\mathrm{D}} \cdot u.$|undefined Thus for central differences, if we write &theta; &equiv; hD, we have μ = cosh &theta;, &delta; = 2 sinh &theta;, &sigma; = &delta;-1, and the result in (iii.) corresponds to the formula sinh &theta; = &theta; cosh &theta;/(1 + sinh² θ &minus; sinh4 &theta; +  sinh6 θ − . . .).

 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＝&fnof;(x)/F(y), leading to the result &fnof;F(y)dy − &fnof;f(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. 