Mathematical Methods for Physics and Engineering : A Comprehensive Guide

20.9 HINTS AND ANSWERS

20.25 The Klein–Gordon equation (which issatisfied by the quantum-mechanical wave-
function Φ(r) of a relativistic spinless particle of non-zero massm)is
∇^2 Φ−m^2 Φ=0.
Show that the solution for the scalar field Φ(r)inanyvolumeVbounded by
asurfaceSis unique if either Dirichlet or Neumann boundary conditions are
specified onS.

20.9 Hints and answers

20.1 (a) Yes,p^2 − 4 p−4; (b) no, (p−y)^2 ;(c)yes,(p^2 +4)/(2p^2 +p).
20.3 Each equation is effectively an ordinary differential equation, but with a function
of the non-integrated variable as the constant of integration;
(a)u=xy(2−lnx); (b)u=x−^1 (1−ey)+xey.
20.5 (a) (y^2 −x^2 )^1 /^2 ;(b)1+f(y^2 −x^2 ), wheref(0) = 0.
20.7 u=y+f(y−ln(sinx)); (a)u=ln(sinx); (b)u=y+[y−ln(sinx)]^2.
20.9 General solution isu(x, y)=f(x+y)+g(x+y/2). Show that 2p=−g′(p)/2,
and henceg(p)=k− 2 p^2 , whilstf(p)=p^2 −k, leading tou(x, y)=−x^2 +y^2 /2;
u(0,1) = 1/2.
20.11 p=x^2 +2y^2 ;u(x, y)=f(p)+x^2 y^2 / 2.

(a) u(x, y)=(x^2 +2y^2 +x^2 y^2 −2)/2;u(2,2) = 13. The liney= 1 cuts each

characteristic in zero or two distinct points, but this causes no difficulty with

the given boundary conditions.

(b) As in (a).

(c) The solution is defined over the space between the ellipsesp=2andp= 11;

(2,2) lies onp= 12, and sou(2,2) is undetermined.

(d)u(x, y)=(x^2 +2y^2 )^1 /^2 +x^2 y^2 /2;u(2,2) = 8 +

√

12.

(e) The liney= 0 cuts each characteristic in two distinct points. No differentiable

form off(p)givesf(±a)=±arespectively, and so there is no solution.

(f) The solution is only specified onp= 21, and sou(2,2) is undetermined.

(g) The solution is specified onp= 12, and sou(2,2) = 5 +^12 (4)(4) = 13.

20.13 The equation becomes∂^2 f/∂ξ∂η=−14, with solutionf(ξ, η)=f(ξ)+g(η)− 14 ξη,
which can be compared with the answer from the previous question;f 1 (z)=10z^2
andf 2 (z)=5z^2.
20.15 u(x, y)=f(x+iy)+g(x−iy)+(1/12)x^4 (y^2 −(1/15)x^2 ). In the last term,xand
ymay be interchanged. There are (infinitely) many other possibilities for the
specific PI, e.g. [ 15x^2 y^2 (x^2 +y^2 )−(x^6 +y^6 )]/360.
20.17 E=p^2 /(2m), the relationship between energy and momentum for a non-
relativistic particle;u(r,t)=Aexp[i(p·r−Et)/
], a plane wave of wave number
k=p/
and angular frequencyω=E/
travelling in the directionp/p.
20.19 (a)c=v±αwhereα^2 =T/ρA;
(b)u(x, t)=acos[k(x−vt)] cos(kαt)−(va/α)sin[k(x−vt)] sin(kαt).

20.21 (a)V 0

[

1 −(2/

√

π)

∫^12 x(CR/t) 1 / 2

exp(−ν^2 )dν

]

;

(b) consider the input as equivalent toV 0 applied att= 0 and continued and

−V 0 applied att=Tand continued;

V(x, t)=

2 V 0

√

π

∫ (^12) x[CR/(t−T)] 1 / 2
(^12) x(CR/t) 1 / 2
exp

(

−ν^2

)

dν;

(c) FortT, maximum atx=[2t/(CR)]^1 /^2 with value

V 0 Texp(−^12 )

(2π)^1 /^2 t

.