Mathematical Methods for Physics and Engineering : A Comprehensive Guide

NUMERICAL METHODS

40.5 40.5

20.4

41.8 46.7 48.4 46.7 41.8

16.8 16.8

V=80

V=0

−∞ ∞

Figure 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; exact
solutiony=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) ̄y(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=2
27.21 Forλpositive the solutions are (boringly) monotonic functions ofx.Withy(0)
given, there are no real solutions at all foranynegativeλ!
27.23 (a) SettingA∆t=c∆xgives, for example,u(0,2) = (1−c)^2 ,u(1,2) = 2c(1−c),
u(2,2) =c^2. For stability, 0<c<1.
(b)G(n, s)=[(1−c)+cs]nfor 0≤p≤n.
(c) [n!(1−c)n−pcp]/[p!(n−p)!].
(d) Whenc= 1 and the difference equation becomesu(p, n+1)=u(p− 1 ,n).
27.25 See figure 27.8.
27.27 Ifx=αythen
d^2 ψ
dy^2

−α^4

mk


^2

y^2 ψ+α^2

2 mE


^2

ψ=0.

Solutions will be either symmetric or antisymmetric withψ(0)= 0 butψ′(0) = 0

for the former and vice versa for the latter. Integration to a largish but finite

value ofyfollowed by an interpolation procedure to estimate the values ofE

that lead toψ(∞) = 0 needs to be incorporated. Simple numerical integration

such as Simpson’s rule will suffice for the normalisation integral. The solutions

should beλ=1, 3 , 5 ,....