Mathematical Methods for Physics and Engineering - Matematica.NET

NUMERICAL METHODSV =80−∞40.541.8 46.7 48.4 46.741.840.5∞16.8 20.4 16.8V =0Figure 27.8 The solution to exercise 27.25.27.15 1 − x 2 /2+x 4 /8 − x 6 /48; 1.0000, 0.9950, 0.9802, 0.9560, 0.9231, 0.8825; exactsolution y =exp(−x 2 /2).27.17 (b) a 1 =23/12, a 2 = −4/3, a 3 =5/12.(c) b 1 =5/12, b 2 =2/3, b 3 = −1/12.(d) ȳ(0.4) = 0.224 582, y(0.4) = 0.225 527 after correction.27.19 (a) The error is 5h 3 u (3)n /12 + O(h 4 ).(b) α = −4, β =5,µ=4andν =227.21 For λ positive the solutions are (boringly) monotonic functions of x. Withy(0)given, there are no real solutions at all for any negative λ!27.23 (a) Setting A∆t = c∆x gives, for example, u(0, 2) = (1 − c) 2 , u(1, 2) = 2c(1 − c),u(2, 2) = c 2 . For stability, 0