Mathematical Methods for Physics and Engineering - Matematica.NET

NUMERICAL METHODSderivatives beyond a particular one will vanish and there is no error in takingthe differences to obtain the derivatives.◮The following is copied from the tabulation of a second-degree polynomial f(x) at valuesof x from 1 to 12 inclusive:2, 2, ?, 8, 14, 22, 32, 46, ?, 74, 92, 112.The entries marked ? were illegible and in addition one error was made in transcription.Complete and correct the table. Would your procedure have worked if the copying errorhad been in f(6)?Write out the entries again in row (a) below, and where possible calculate first differencesin row (b) and second differences in row (c). Denote the jth entry in row (n) by(n) j .(a) 2 2 ? 8 14 22 32 46 ? 74 92 112(b) 0 ? ? 6 8 10 14 ? ? 18 20(c) ? ? ? 2 2 4 ? ? ? 2Because the polynomial is second-degree, the second differences (c) j , which are proportionalto d 2 f/dx 2 , should be constant, and clearly the constant should be 2. That is, (c) 6 shouldequal 2 and (b) 7 should equal 12 (not 14). Since all the (c) j = 2, we can conclude that(b) 2 =2,(b) 3 =4,(b) 8 = 14, and (b) 9 = 16. Working these changes back to row (a) showsthat (a) 3 =4,(a) 8 = 44 (not 46), and (a) 9 = 58.The entries therefore should read(a) 2, 2, 4, 8, 14, 22, 32, 44, 58, 74, 92, 112,where the amended entries are shown in bold type.It is easily verified that if the error were in f(6) no two computable entries in row (c)would be equal, and it would not be clear what the correct common entry should be.Nevertheless, trial and error might arrive at a self-consistent scheme. ◭27.6 Differential equationsFor the remaining sections of this chapter our attention will be on the solutionof differential equations by numerical methods. Some of the general difficultiesof applying numerical methods to differential equations will be all too apparent.Initially we consider only the simplest kind of equation – one of first order,typically represented bydy= f(x, y), (27.60)dxwhere y is taken as the dependent variable and x the independent one. If thisequation can be solved analytically then that is the best course to adopt. Butsometimes it is not possible to do so and a numerical approach becomes theonly one available. In fact, most of the examples that we will use can be solvedeasily by an explicit integration, but, for the purposes of illustration, this is anadvantage rather than the reverse since useful comparisons can then be madebetween the numerically derived solution and the exact one.1020