Mathematical Methods for Physics and Engineering : A Comprehensive Guide

21.7 Hints and answers

inVand takes the specified formφ=fonS, the boundary ofV.The

Green’s function,G(r,r′), to be used satisfies

∇^2 G−m^2 G=δ(r−r′)

and vanishes whenris onS.

(b) WhenVis all space,G(r,r′) can be written asG(t)=g(t)/t,wheret=|r−r′|

andg(t) is bounded ast→∞. Find the form ofG(t).

(c) Findφ(r) in the half-spacex>0ifρ(r)=δ(r−r 1 )andφ= 0 both onx=0

and asr→∞.

21.28 Consider the PDELu(r)=ρ(r), for which the differential operatorLis given by

L=∇·[p(r)∇]+q(r),

wherep(r)andq(r) are functions of position. By proving the generalised form of

Green’s theorem,

∫

V

(φLψ−ψLφ)dV=

∮

S

p(φ∇ψ−ψ∇φ)·nˆdS ,

show that the solution of the PDE is given by

u(r 0 )=

∫

V

G(r,r 0 )ρ(r)dV(r)+

∮

S

p(r)

[

u(r)

∂G(r,r 0 )

∂n

−G(r,r 0 )

∂u(r)

∂n

]

dS(r),

whereG(r,r 0 ) is the Green’s function satisfyingLG(r,r 0 )=δ(r−r 0 ).

21.7 Hints and answers

21.1 (a)Cexp[λ(x^2 +2y)]; (b)C(x^2 y)λ.
21.3 u(x, y, t) = sin(nπx/a) sin(mπy/b)(Asinωt+Bcosωt).
21.5 (a) 6u/r^2 ,− 6 u/r^2 ,0,=2(or−3),m=0;
(b) 2u/r^2 ,(cot^2 θ−1)u/r^2 ;−u/(r^2 sin^2 θ),=1(or−2),m=±1.
21.7 Solutions of the formrgiveas− 1 , 1 , 2 ,4. Because of the asymptotic form of
ψ,anr^4 term cannot be present. The coefficients of the three remaining terms are
determined by the two boundary conditionsu= 0 on the sphere and the form of
ψfor larger.
21.9 Express cos^2 φin terms of cos 2φ;T(ρ, φ)=A+B/2+(Bρ^2 / 2 a^2 )cos2φ.
21.11 (Acosmx+Bsinmx+Ccoshmx+Dsinhmx)cos(ωt+), withm^4 a^4 =ω^2.

21.13 En=16ρA^2 c^2 /[(2n+1)^2 π^2 L];E=2ρc^2 A^2 /L=

∫A

0 [2Tv/(

1
2 L)]dv.
21.15 Note that the boundary value function is a square wave that issymmetricinφ.
21.17 Since there is no heat flow atx=±a, use a series of period 4a,u(x,0) = 100 for
0 <x≤ 2 a,u(x,0) = 0 for− 2 a≤x<0.

u(x, t)=50+

200

π

∑∞

n=0

1

2 n+1

sin

[

(2n+1)πx

2 a

]

exp

[

−

k(2n+1)^2 π^2 t

4 a^2 s

]

.

Taking only then= 0 term givest≈2300 s.
21.19 u(x, t)=[a/(a^2 +4κt)^1 /^2 ]exp[−x^2 /(a^2 +4κt)].
21.21 Fourier-transform Poisson’s equation to show that ̃ρ(α)= 0 (α^2 +q^2 )V ̃(α).
21.23 Follow the worked example that includes result (21.95). For part of the explicit
integration, substituteρ=ztanα.

Φ(0, 0 ,z)=

z(1 +z^2 )^1 /^2 −z^2 +(1+z^2 )^1 /^2 − 1

z(1 +z^2 )^1 /^2

.