Ockendon J., Howison S., Lacey A., Movchan A. Applied Partial Differential Equations

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228 ELLIPTIC EQUATIONS

5.17.

(i) Suppose that, in the notation of (5.58),

V2u=O inr>a,

with

g(9)

on r = a.

Using Green's theorem, show that, as r -+ oo,

u = alogrI g(O)dO+O(1).

(ii) Again using the notation of (5.58), show that, on r = a with a = 0,

er (RR')

= I1 (ItI2 +

a2 - 2aIEI cos8),

where now IfI > a, and hence that

RRl 8r

(RR')

(iii) Deduce that

27r

log

is a Green's function for the exterior Neumann problem.

5.18. Consider the interior Neumann problem

V2u=0 inr<a,

with

=g(8) onr=a,

where fo g(9) d9 = 0. Show that, when n = 0 in the notation of (5.56),x

GM = 2s log

+ constant.

Does the value of the constant matter?

5.19. From the same geometrical consideration that led to (5.58), show that the

Green's function for the Dirichlet problem for Laplace's equation in a sphere

of radius a is

4 (IX

RIIx

E'Il '

where 4' is the point inverse to 4 in the sphere.

EXERCISES 229

5.20. Suppose 8 and 9' are two-dimensional polar coordinates centred at A and B,

respectively. Show that 8-8' is a harmonic function in any doubly-connected

region surrounding, but not including, A or B. What Dirichlet data does it

satisfy on any circle through A and B? Hence find the bounded solution of

V2u = 0 for r < 1 in plane polar coordinates, with

u(1,9)-

1, 0<8<7r,

-1, -7r<9<0.

s 5.21. Suppose a solution of V2Z4 = 0 in y > 0 is such that u(x, 0) = gD(x) and

Ou/Oy (x, 0) = gN(x). Show that the Fourier transforms of 9D and ON satisfy

9N(k) = -IkI9D(k),

and hence that

9N = -ih(x) * 9D(x), 9D(x) = ih(x) * 9N(x),

where * denotes convolution and

h(k) =

1, k > 0,

1-1, k < O.

Show that

2-r

/ e-1 jkj h(k)e -ikxdk =

(ix1

+ ix1 e)

fore > 0

and take the limit as e -3 0 to deduce that h(x) _ -i/irx. Show that if

9D = 1/(1 + x2) then ON = (x2 - 1)/(x2 + 1)2, and verify (5.66) or (5.68).

Confirm the results by considering the function 1/(z + i).

5.22. Show that, if f is analytic in Izi < 1, so that

P z) = 21

f (t) dt

-1=1

then, for IzI < 1,

f (eie)eie d9

f (eie)eie dB

f (z) = 2a

eie - z

and

0 =

1 -.fete

Combine these results to give R f (z) = u(x, y), where

u(x, y)

- r2

u(cos 0, sin 0) dO

2fr

1 + r2 - 2r cos (8 -tan-1(y/x))

230

ELLIPTIC EQUATIONS

5.23. Suppose that u(x, y) satisfies

V2u=0 my>0,

with u(x, 0) = uo(x), where uo(x) -+ 0 sufficiently rapidly as x -+ ±00. Show

from the Green's function representation of the solution that

,r/2

u(x, y)

uo(x + y tan 0) dB.

-,r/2

Verify by direct differentiation that V2u = 0.

5.24. Suppose that 0(r, z) satisfies

820

180 82

8r2 +

; + . = 0 '

0ro, z) _ 0o(z)

Take Hankel transforms to show that, for z > 0,

0(r, z) =

k f (k)e ktJo(kr) dk,

where k f (k) is the inverse Laplace transform of 0(0, z). Now change the

order of integration in

Y+iao

z) =

1 f

0(0,

z)ek<-kz Jo(kr) dk d(

tai

Y-ioo

and use the result of Exercise 4.11 to show that

0(r, z) =

f 0(0,z+itcoso)d8.

5.25. Show that, if S = f (z) is analytic, then the Neumann data satisfied by a

function on a curve in the z plane is equal to that satisfied by the function

on the corresponding curve in the C plane multiplied by I f'(z)I.

5.26. Show that, if 0 is the velocity potential in an irrotational flow and 0 + i1o =

w(z) describes a particular flow, then

(62)

W(Z) + 1

describes a flow in which V0 is tangent to jzj = a.

Note. If f (z) = u(x, y) + iv(x, y), then

7(z) = T T = u(x, -y) - iv(x, -y).

EXERCISES 231

5.27. Show that, for a > b, (5.121) can be written as

+its-

UOa+b a-io(z+ z2-c2) +ebo(z - )

a+b a-b

Show further that, if there is a circulation r around the aerofoil, so that

0 + jib = Uxze-i° + (it/27r) logz + 0(1) as jzj --> oo, then the term

log

I z

z2 --C2

27r

must be added to this formula. By considering dw/d(, show that, in the

limit b -+ 0 and c -> a, dw/dzI,,_, is finite if r = 27W, ccoef0 sin a.

5.28. Suppose we try to solve the biharmonic equation V4u = 0 in 0 < x < 1,

0 < y < 1, with u = Ou/Ox = 0 on x = 0,1, and u and Ou/8y prescribed on

y = 0, 1. Separate the variables to show that candidate solutions are

u(x, y) _ ((Ax + B) cos kx + (Cx + D) sin kx)e±k",

as are such functions multiplied by y, where A, B, C and D are constants, and

k is a complex root of k = f sink. Are these functions mutually orthogonal

for different k?

s 5.29. Consider the crack problem of §5.9.3. Show that the boundary values of the

function

On au

W(z) _

ax - lay

satisfy

Ou Ou

(-8y

-r

on the whole real axis except at x = ±c, and deduce that 3'((W(z)-ir)2) = 0

there. Use Schwarz reflection to show that W(z) can be extended to an

analytic function with singularities only at z = ±c and z = cc. Use symmetry

and the behaviour of u at infinity to show that the least singular possibility

for W(z) is

-T Z

(W(Z) - iT)2 =

z2 - r2'

and hence retrieve (5.149).101

* 5.30. By considering the limit R -> oo. show that

( d(

CI=R>c

(2 - c2 (- x

= 27ri,

where x is real and -c < x < c, and the branch cut is taken along the

interval (-c, c) of the real line. Deform the contour to one wrapped around

2 2

101 We are indebted to Dr A. K. Head for this remark.

232 ELLIPTIC EQUATIONS

the branch cut, taking care near S = x, to deduce the result of footnote 90

on p. 206.

* 5.31. Suppose that V2u(x, y, z) = 0 in z > 0, with u(x, y, 0) = gD(x, y) and u -+ 0

at infinity. Show that, if

u(k1, k2, z)

u(x, y,

z)e,(k,x+ksv)

dx dy,

00 I_00

and if

then

Deduce that

say, and also that

_ lim

H=-

(x, y, 0) = 9N (x, y),

9D=-

9N-

09D -

ik1

k1 + k2

9N = H9N1

1 1 e-c ki+k2e i(k,x+ksq) dki dk2

41r2 8x c-+0

-00 -oo

k1 + k2

- - 1

lim

(i cos(9 - tan-1(y/x)) + E)2

21r(x2 + y2)3/2

49.2 c-+0

fo,w

5.32.

(i) Show that, if V2ut = 0 in y > O and y < 0, and ut = 9D() on y = 0,

where 9D is continuous and 9D(x) -+ 0 as IxI - oo, then, on y = 0,

L +

2PV

foo

9 () -9D(x)

( - x)2

Vou

(ii) Show that, if V2uf = 0 in y > 0 and y < 0, and But/ey = 9N(X) on

y = 0, where ON -+ 0 sufficiently fast as Ixl -+ oo, then, on y = 0,

[u] ± _

009N

(e) log Ix - f I

5.33. Show that, if r K 6 « 1, the integral

9(C) dC

(z - e)2 + r2

can be approximated by

)

f a+a

g(t) dt

(

g(t) dt

J_00

z - e

J:+6

-z

Show further that the middle term is

2g(z) cosh-1 I

I = 2g(z) log 12r J

(i+o

()),

and deduce that the original integral tends to -2g(z) log r as r - 0.

EXERCISES 233

5.34. Suppose that O(r, z) satisfies

020

100

ar r 0r +

8z2

(0, z) =

1 + Z2*

Show that

0(r, z) = R

r2+(z+i)

Where are the singularities of 0?

5.35. Assume that f (x) is integrable, and differentiable in 0 < x < oo. Integrate

by parts to show that

I(k) = f x1/2 f(x)el1

(9(0)

r x-1/2eikr dx + fo

x-1 /2

(9(x) - g(0)) etk: dxf

where g(x) = f (x) + 2x f'(x). Deduce that

1(k) = 2k3/2 VF!22

ask ->oo.

Note. fo s'1/2e"ds= (1+i)

r/2.

5.36.

(i) Generalise the argument following (5.128) to show that, if V2u = 0 in

y > 0, with u -+ 0 as y i oo and u(x,0) = g(x), where g(x) - 0 as

x -+ ±oo, then u(x, y) = N w(z), where

-z

Deduce that w(z) = O(z-1) as Izj -- oo unless f o. g(C) dC = 0.

(ii) Now suppose that V'2v = 0 in y > 0, with v -4 0 as y -4 oo and

(x,0) = f(x)

for x < 0,

v(x,0) = g(x) for x > 0.

where f and g vanish at moo, respectively. Write

W(z) = F -

and let w(z) = z1/2W(z). Show that, on the real axis,

x1/29'(x),

x > 0,

Rw(z) = h(x) _

-(-x)1/2f(x), x <0,

234

ELLIPTIC EQUATIONS

and deduce that

z-1/2

h(C) dC

W(z)

1_00

C - z

Remark. This result can easily be generalised. If v and 8v/8y are prescribed

alternately on several intervals of the real axis, then the premultiplier z'/2,

whose function is to swap real and imaginary parts (to turn Neumann data

into Dirichlet), is replaced by a suitable product with square roots at the

ends of the intervals. Further, if a mixed boundary value problem of this type

is given in a domain that can be mapped onto a half-plane (for example, a

polygon, by the Schwarz-Christoffel map; see p.342), then the solution can

also be written down; in this case the Neumann data is multiplied by the

derivative of the mapping function (see Exercise 5.25).

5.37. Suppose that u(z, y) satisfies the convection-diffusion equation

V. Du=V2u

in a domain D exterior to a semi-infinite boundary r : y = ±f (x), 0 < x <

oo, where f (0) = 0 and f (x) > 0 for x > 0. Suppose also that v = V0, where

0 is given and satisfies

V20

= 0, that v -+ (1, 0) at infinity, and finally that

u = 1 on I' and u -+ 0 at infinity.

Show that, after the Boussinesq transformation, in which 0 and its harmonic

conjugate 10 (the stream function) are used as independent variables, the

problem becomes

02u 02u _ 8u

87 + 802 80

in the (¢, t) plane with the positive real axis deleted, and

u=1 on0=0,4'>0,

u--*0

at infinity.

Show that u = erfc ,I, where (£ + irq)2 = x + iy (see Exercise 5.38 for confir-

mation).

* 5.38. Suppose that

in y > 0, with

V2u

u(x,0) = 1

for x > 0, 8u(z,0)=0 for x < 0,

and ui0as x2+y2 -+ oo except on y = 0, x > 0.

(i) Show that, if z = x + iy = ( + irq)2, then

02u+02u _2/ 8u 8u\

EXERCISES 235

with

where n = r112 sin 0/2 in polar coordinates.

(ii) Derive the same result by setting u = f tee'kt dx, where

u(x,0) = f(x)

for x < 0,

(x,0) = g(x) for x > 0;

first write that

k - ik

in the notation of §5.9.4, where

k' - ik - Ikl as Iki - oo with k real,

and then Justify writing

k2 - ik = (v)+( k - i)_. Thus show that

9+ = (

i/k)+, from which the solution in (i) can be deduced by contour

integration.

* 5.39.

(i) After removing the incident field, the Sommerfeld problem (5.102) and

(5.104) becomes

(V2+1)u=0,

with

u = -e-'x COs °

on y = 0, x < 0,

together with a radiation condition. Letting

ILY

v=o+

u(x,0) = f(x)

for x > 0,

= g(x)

for x < 0,

8,=o-

show, as in Exercise 5.38, that

__ _ 9-(k)

f+(k) +

(k-- cosa)_

2 k - 1'

where '1k2 - 1 tends to Ikl as k -+ oc with k real. Assuming that

the radiation condition can only be satisfied if

k2 - 1 is defined as

(Vly--l)_ ( k + 1 +, show that

f+(k)( k+1)++i(

k+1- c5sa+1

k - coca

g_ (k)

_ i cos a + l

(k-cosa)_'

and hence use Liouville's theorem to show that

236 ELLIPTIC EQUATIONS

(§-(k)k-1/_

-2i cos a + 1

k - cos a

Remark. The inversion for g(k) can be manipulated to give

i(l n/4)

u(r,O)

(.i(cos(j))

+Flr(-v(2-rcosCB

2a111,

where

Fr(z) =

e-iZ2 /

eit0

dt.

This solution can be shown to satisfy the radiation condition after

(5.102).

(ii) Verify that each of the two terms in u(r, B) satisfies Helmholtz' equation

by setting u = ei''v and seeking v as a function of the variable t in Ex-

ercise 5.38, taking the x axis in that exercise to lie along the boundaries

of the directly illuminated and reflected regions, respectively.

5.40. Show that, in the parabolic coordinates of Exercise 5.38, the Helmholtz equa-

tion with k = 1 becomes

a7 +l2 + 4(£2 + q2)u = 0.

Although this is not apparently a traditional separation of variables situa-

tion, show that solutions exist in which u(t, q) = U(t)V (q), where

U"+4(C2+A)U=O, V"+4(q2-A)=0,

and A is the separation constant.

The functions_U and V are called parabolic cylinder functions. Show that,

if U(t) = ei(2U(C), then

U"+4itU'+AU=0.

Verify, using the ideas of p. 109, that U has the integral representation

fe_it2/8t_1_i)dt

for an appropriate contour r.

e 5.41. The following alternative derivation of (5.112) shows that VA can be thought

of as the adjoint of and vice versa.

EXERCISES 237

(i) Suppose that V A G = 0 and V H = 0. Use the identity

to show that, if G and H vanish sufficiently rapidly at infinity, then

00 00

f. foo

(ii) Suppose that V A H = j and V - G = 6(x - F). Show that

H= J .J . /

jAGdx.

(iii) Finally, show that

G O (

4rlx-fl

is an appropriate Green's vector to deduce (5.112). Further properties

of vector distributions are discussed in Exercise 9.3.

5.42. Show that the function A in Exercise 5.1 satisfies

84A

8z2 8z2 -

and write it as

A= 2(201(Z) + Z02(z) +X1(Z) + X2(z)),

where 01,2 and X1,2 are analytic. Use the fact that A is real to show that

(z) = 01(z) and X2(z) = X1(z) in the notation of Exercise 5.26, and hence

that

A = W(IO,(z) + X1(z)).

Show also that

oz +oy = 4?-0i(z),

oz-oy-2ir=2(20'(z)+Xi(z))

5.43.

(i) Expand the equation V (lVuIP-'Vu) = 0 and use the criteria of Chap-

ter 3 to show that, if 0 < IVul < oo, then the equation is elliptic for

p > 1, parabolic for p = 1 and hyperbolic for p < 1.

(ii) Derive a variational formulation for the boundary value problem

V 1 = 0

in D

with

u = 0

on 8D.

Show that the resulting functional is neither bounded above nor below

ifp<1.