Internat. J. Math. & Math. Sci.
Vol. 6 No. 2 (1983) 341-362
341
ON SOLVING THE PLATEAU PROBLEM
IN PARAMETRIC FORM
BARUCH CAHLON
Department of Mathematical Sciences
Oakland University
Rochester, Michigan 48063
ALAN D. SOLOMON
Mathematics and Statistics Research Division
Computer Science Division
Union Carbide Corporation-Nuclear Division
Oak Ridge, Tennessee 37830
LOUIS J. NACHMAN
Department of Mathematical Sciences
Oakland University
Rochester, Michigan 48063
(Received August 15, 1980 and in revised form February 6, 1981)
ABSTRACT.
This paper presents a numerical method for finding the solution of
Plateau’s problem in parametric form.
Using the properties of minimal surfaces we
succeded in transfering the problem of finding the minimal surface to a problem of
minimizing a functional over a class of scalar functions.
A numerical method of
minimizing a functional using the first variation is presented and convergence is
proven.
A numerical example is given.
KEY WORDS AND PHRASES. Minimal surface, algorithm, parametric form, Dirichlet’s
integral, lrmonic function.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES. 65K10
1.
INTRODUCT ION.
In this paper, we present a method for the numerical solution of the Plateau
problem in parametric form.
Sp.ecifically,
we seek a minimal surface spanning a simple
B. CAHLON, A.D. SOLOMON and L.J. NACHMAN
342
If the curve is planar then the problem reduces
closed curve in 3-dimensional space.
to that of finding a conformal mapping onto its interior.
In
To the numerical analyst the Plateau problem presents a formidable challenge.
the non-parametric case, when the surface and bounding curve admit of a single-valued
projection onto an
plane, the problem reduces to solving the minimal surface
x,y
equation
+ z y 2 )z xx
(i
for the height
z(x,y)
2z z xy + (i
x y
+
2
x
)z
yy
(.i)
0
x,y plane, and for boundary values
of the surface above the
defining the given bounding curve.
z
Finite difference iterative schemes for(l.l)have
been examined by Concus [2] and Greenspan [3], [4 ].
In the parametric case, where the surface is not assumed to admit a single valued
x(u,v)
planar projection, a vector function representation
is used.
Here
x(u,v)
is defined on a domain
structure determines that of the surface.
x(u,v)
becomes one of finding
A.
kx= O, on
B.
x
C.
x
-+2
-+2
x
0
x x
V
U
U V
maps the boundary of
P
in the
(x(u,v),y(u,v),z(u,v))
(u,v)
plane whose
By a theorem of Weierstrass [5] the problem
such that
on
onto the bounding
curve(s) of the surface in a
monotonic fashion.
A numerical scheme for simultaneously attaining
although
is in fact a boundary condition for
B
work with
cannot be easily derived, for
A,B,C
A [I], it is not clear how one may
C
In the following we introduce a method for computing the solution of this
problem.
The method depends on our recognizing the solution as a function minimizing
the Dirichlet integral over all functions satisfying
C
Similar to the method of
safe descent of R. Courant [I], we define the Dirichlet integral as a functional
on a class
is
G
of scalar functions
"sewed" onto its bounding curve.
given in section
2.
g
d(g)
which determine the manner in which the surface
The percise definition of this functional is
In section 3 the derivation of the first variation of
performed, and as an obvious consequence we see that a stationary value for
d(g)
d(g)
is
PLATEAU PROBLEM IN PARAMETRIC FORM
343
d(g)
In section 4 we define a method for minimizing
defines a minimal surface.
based on the use of the first variation; we then prove the convergence of the method
and discuss its computational implementation, which is described in section 5.
2.
DEFINITION OF d(g).
Let
C
h()
C: x
We assume that
for arc length o.
h’(2), h"(O)= h"(2)
h’(O)
(x,y,z)
be a simple closed curve in
h
Let
(u, v)
given by
2
(2.1)
< 2
0 <
is twice continuously differentiable with
t(o)
b(o)
C
binormal, normal, curvature and torsion of
circle in the
space of length
n(o)
()
respectively.
be the tangent,
P
Let
be the unit
plane
(2.2)
with boundary
F: u
A vector function of
v
u,v
(2.3)
i
on
is denoted by a lower case letter such as
while the same function referred to polar coordinates on
( r,)
For any sufficiently smooth functions
P
,
u
r cos
,
is denoted by the corresponding upper case letter:
r sin
x(u,v)
over
+ v2
U F
and closure
x(u,v)
2
(2.4)
0 <_ 8 <_ 2
0 <_ r <_ i
D
the Dirichlet integral
x,y,
of
x
is
D [x]
2
+
x
(XuYu
+
XvY v)du
(^u
(2.5)
)du dv
while the Dirichlet inner product is
D[x,y]
It is known [i] that a vector function
conditions
A,B,C
hold.
x
X
(2.6)
dv
exists on
for which
Horeover by the twice continuous differentiability of
and the extension of Kellogg’s theorem to minimal surfaces [6], [7],
(1,8)
h
has a
HIder continuous first derivative with respect to
IX 8(1,8+5)
with
,v
X
8(1,8)
<_
Sv
the HDlder constant and rdlder index, respectively.
THEOREM i.
There exists a function
x(u,v)
(2.7)
Specifically
satisfying the following conditionm
B. CAHLON, A.D. SOLOMON and L.J. NACHMAN
344
i.I.
x
1.2.
&x
1.3.
2
2
--x
x
1.4.
x x
u V
1.5.
X@
(1,@)
x
maps
is continuous on
0
u
within
t)
in
V
0
in
l)
"s tt6"lder continuous and obeys (2.7) for some values of
F
C
onto
1.7.
D[x]
1.8.
For the function
in a monotonic one-to-one fashion;
<
the Dirichlet integral (2.5) attains its
x(u,v)
least value among all functions satisfying 1.1-1.4,1.6,1.7.
LEM[A i.
Condition 1.5 implies 1.7 for any harmonic function
x
Furthermore
(2.8)
D[x] < M
M
with
a constant dependent only on
PROOF:
X(I,)
By 1.5,
expansion
v
-
admits of a uniformly convergent Fourier series
a0 +
x,e.__.
cos j8
+ 8.
sin j@)
j=l
furthermore by a simple calculation
2
0
j
(l,8))cos jede
(Xe(l,8 +)
j
=
2
1
(x(,
+)
X@(l,@))sin
j
0
whence
(2.9)
It is known [i] that the Dirichlet integral exists if and only if the series
j(j2 + j
j=l
(2.10)
converges, and if so, its value is given by the series.
(2.9) holds then (2.10) does converge, and
D[x]
<
8M2V+l
2
i
[
j=l
J
l+2v
=M
However it is clear that if
PLATEAU PROBLEM IN PARAMETRIC FORM
345
the lemma is proved.
Let
x
X
Let
be the collection of functions satisfying I.i, 1.2, 1.5, 1.6.
By 1.6 there exists a monotonic function
be any function of
0 < 0 < 2
g(0)
for which
X(1,O)
h(g(O))
(2.11)
and
g(0) =0, g(2)
Moreover, by
1.5
g(O)
t
(2.12)
has a continuous derivative
the unit tangent vector.
g’ ()
satisfying
-{(g(O))g’(O)
0(i,8)
with
2
Hence
Ie(1,0)1
Ig’ (e)
g’(e)
(2.13)
and
Ig’ (o +
Ie (,,o)ll
I1e (z,o + )1
g’ (O)
5)
<_
IXe(.,e + ) XeCZ,e)
< a5
v
we conclude:
THEOREM 2.
Any function
of
x
differentiable scalar function
g(%)
defines a monotonic r61der continuously
satisfying (2.11).
function as the boundary correspondence function for
Let
G
be the set of all functions
first derivatives, obeying (2.12).
Any
g(0)
g ( G
on
O
x
[0,2] with HDlder
continuous
defines harmonic function
satisfying (2.11). Moreover by the assumptions on h
derivative with respect to
We will refer to this
,(i,0)
x
X
has a rOlder continuous
whence by lemma i,
D[xl <
Since any function
g 6 G
defines in this way a unique
x 6
and conversely, we
can equate the problem of minimizing the Dirichlet functional over
with that of
minimizing the scalar functional
d(g)
over
g
G
D[]
(2.14)
346
3.
B. CAHLON, A.D. SOLOMON and L.J. NACHMAN
THE FIRST VARIATION OF d(g).
d(g)
We now calculate the first variation of
G
function of
for which
d(g)
assumes a stationary value.
Older continuously differentiable function for
and let
be any parameter.
6
assumes a stationary value for
6(6)
< 2
0 <
Let
with
()
(0)
g*
be a
be any
(2)
0
Then
d(g* +
6(6)
THEOREM 3.
Specifically, let
0
6
is differentiable for
Moreover
0
6
2
(5’(0)
Xr(l,)t(g*())(@)d
2
(3.1)
0
with
X
defined by (i0) for
PROOF
g*
g
Clearly
g*+6
(g* +
6)
(g*)
+
(3.2)
t ()de
ge
Let
_-t_l
X,X
X +
X(l,)
satisfy
6
6
for
6
(g*()), i(i,)= (g*()+
the harmonic function on
6(@))
Then
for which
g, (e)+6,q (e)
6(i, 0)
t(o)do
7
g*(e)
Clearly
6(0)
d(g*)
6(6)
d(g* + snq)
Moreover
(5(6)
(5(0) +
By Gauss’s theorem
26D[,’,6]
+
62D["’c]
(3.3)
2
D[X,Y
r
(1,e)
6(1
e)de
0
by the continuity of
t
6(1, e)
uniformly for
6
0
whence
(g,(e)),q(e)
(1,e)
(3.4)
PLATEAU PROBLEM IN PARAMETRIC FORM
f
D[-’-e]D[]
LEMMA 2.
PROOF.
As
e
r(l’e)(g*(e))(e)de
0
is uniformly bounded for all
0
347
e
the function
0
as
s
converges uniformly on
F
according to
(3.4). Using the Frenet formulas, we see
Ye(l,e)
t(g, + ) +
(o) n(o do
g
But then if
g*’ (e)
is H61der continuous with index
continuous with r61der index
v
v
then
YO
and r61der constant independent of
is rdlder
which
e
implies, by lemma i,
D[e]
M
with
in
F
a constant independent of
and hence on
D
<
M
(3.5)
e
By the uniform convergence of
e
to
Y
and the lower semi-continuity of the Dirichlet integral
(3.5) implies
D[]
(3.6)
< M
proving lemma 2.
Rewriting
(3.3) as
5()
we see by
5(o)
2D[X,Y
+
eD[e]
(3.4), (3.5), that the limit of the left hand side exists for
(3.1) is proved.
LEMLA 3.
g*
If a stationary value for
d(g)
is attained for some
g*
-
0
G
defines a minimal surface.
PROOF.
If
D[X,]
for all
()
0
then by the fundamental theorem of the calculus of variations
and
then
348
B. CAHLON, A.D. SOLOMON and L.J. NACHMAN
(1,e)-(g(e))
0
(3.7)
Xr(l,8)X8(1,8)
0
(3.8)
r
which by (23.3) implies
However, by a standard argument [8], (3) implies that
on
0
4.
AN ALGORITHM FOR MINIMIZING dg.)
deflnes a minimal surface
and the lemma is proved.
We now derive an algorithm for solving the problem
min.d (g)
(4.1)
gG
numerically.
solve
The algorithm rests upon a Raylelgh-Ritz type of approach, in which we
(4.1)over
G
a sequence of finite dimensional subsets of
yielding a sequence
At some stage in the algorithm we will need
of functions converging to the solution.
F
to require that a "three-points condition" in which three given points of
mapped into three given points of
C
is obeyed.
are
can be mapped conformal-
Since
ly onto itself by a Mobius transformation in which the images of three given points
can be preassigned, while a function
G
g
F
can be considered as mapping
onto
itself, a three points condition can always be attained through the composition of
two elements of
In addition the Dirichlet integral is invariant under the
G
Mobius transformation.
In order to guarantee convergence of the algorithm we impose
an additional smoothness assumption on the curve
functions of
G
C
and as a consequence, on the
We also assume that a minimal surface solving
(4.1)
has no branch
points on the boundary.
For the purposes of this section, we will assume that the function
theorem to minimal surfaces, we see that a solution
g*
to
(4.1) has a HIder
We now redefine the collection
monotonic twice differentiable functions
continuous derivative of second order.
of (2.1)
Again using the extension of Kellogg’s
has a HDlder continuous second derivative.
continuous second derivative.
h
g
Let
G
as the set of all
obeying (2.12) and having a
HIder
349
PLATEAU PROBLEM IN PARAMETRIC FORM
inf.
g
go
Let
gS
G
be any function of
(4.2)
d(g*)
g*
go
Hence
g*
Then
HDlder continuous second derivative.
d(g)
vanishes for
go
0,2
8
and has a
has a uniformly convergent
Fourier series expansion
a
[ (aj
j=l
go
g*
cos j8
+ bj
0
iS) +2
sin
(4.3)
moreover by calculations identical to those used in deriving (2.9).
bj <--
aj
j
(4.4)
2+y
J
while
la01
where
7
< 8
2
(4.5)
are the H’Older constant and 61der exponent, respectively.
Clearly
now the series in the relation
a
go
g*
0
+--
[ (aj
+
cos j@
+ b.
J
j=l
sin j@)
(4.6)
can be differentiated termwise, since the resulting series itself converges uniform-
ly, obtaining
g*’(8)
g(e)
+
[ (-jaj
sin j
+
jbj
j-1
cos jS)
(4.7)
If we make the (reasonable) assumption that the minimal surface defined by
g*
has no branch points at the boundary, then for some possitive constant
g*’(8) >_ oo0
Let
a
S (@)
n
0
n
+
I (aj
+ b.
cos j8
3
j=l
then by (4.8) and the uniform convergence of the series
LEMMA 4.
The sequences
respectively, on
[0,2]
for
[go
n
+
(4.8)
0 <_ 8 < 2
> 0
Sn]
[g
+
S’]n
Finally, using the methods of section 2,
(4.7),
converge uniformly to
sufficiently large,
increasing.
n
sin j@)
go
+
Sn
g*
is monotonically
g*’
B. CAHLON, A.D. SOLOMON and L.J. NACBMAN
350
d(g 0 + S n)
L4MA 5.
PROOF.
d(g*)
n-
as
This assertion is easily proved by obtaining estimates of the form (2.9)
h
which under the heightened soothness assumption for
[/j
on the argument of
depend continuously (in the
h
Clearly the Fourier coefficients of
attain one higher power of
L2-norm)
while by these estimates the series (2.10) converges
h
uniformly; hence this series, which is equal to the Dirichlet integral, depends
aontinuously upon the argument of
thus proving our assertion.
h
Before describing our algorithm we will turn to some properties of functions of
the form
^
+ Sn
For any constants
A.
n
increasing.
let
J
A
+ 0
-+
g0(e)
T (0)
LEMMA 6.
B
3
Suppose that for
n
j=l
cos j0
+ B.
3
the function
0 < 0 < 2n
(4.9)
sin j0)
T (0)
n
is monotonically
Then
4
Bjl <_ n-3
IAjl
PROOF.
(Aj
If
T
n
(4.10)
is monotonically increasing, then
n
+
go(e)
For all
k
1,2,
integrating over
......
[0,2]
j;1
I + cos k0
(-jAj
> 0
sin j0
+
jBj
cos jO) >_ 0
(4.11)
Multiplying (4.11) by this function and
we obtain
2
+
2
go’(0)co"
k0 de
+ knBk
0
0
or
B
Similarly, multiplying by
>_-4/k
k
+ cos k8
-i
B
k
<
< 0
we find
4/k
or
IBkl
<_ 4/k
(4.12)
PLATEAU PROBLEM IN PARAMETRIC FORM
(+ i + sin ke)
In the same way, multiply.ing by
351
[0,2]
and integrating over
we
obtain
IAkl
<
(4.13)
4/k
and the lenna is proved.
Let
M
n
Let
satisfied.
^
+ Tn
C
n
Let
is monotonic.
denote the subset of the
n
Euclidean space of vectors
2n+l
satisfying (4.5), (4.12), (4.13).
Then
Mn
C
n
n
1,2,...,
+
for which
dimensional
(4.14)
is closed and bounded.
+ Tn
Let
Cn
is monotonic, or lies in
C
n
let
6
s
_
2n
n
is closed and convex.
For any
with
T
An’Bn)
(A0’AI’BI
denote the set of vectors (4.14) for which
Then
n
consisting of those
M
n
denote the subset of
for which (4.10) is
T (e)
denote the ollection of functions
2n+l
Tn
THEOREM 4.
(2n+l)
d(g0
(4.9), (4.14).
defined by
The function
6(2n+i)
+
(4.15)
Tn)
Using the methods of theorem 3 we find
is lower semi-continuous, and has partial
derivatives with respect to each of its independent variables; moreover
2
r(l’e)t(g 0 + Tn)COS
2
J
3B.
]
where
2
f
2
r(l’e)(go + Tn)sin je
(4.17)
de
0
(g0 + Tn)
(l,e)
6(e2n+l)
By the lower semi-continuity,
bounded set
(4.16)
je de
0
n
attains a least value on the closed
Assume this is attained at a point
n
inf. 6(
n
2n+l
e*2n+l
6(e*2n+l
whence
6"
2n+l
(4 18)
B. CAHLON, A.D. SOLOMON and L.J. NACHMAN
352
An algorithm for the solution of (I) can now be defined in the following steps:
I.
Using a gradient search type method [9] which rests on (4.16, 4.17)
find a value
II.
This value defines a monotonic function
d
minimizes
inf.
C
III.
solving (4 18)
6*2n+l
n
C
on
d(g0
Tn)
gn*
gn*
(by (4.9)) which
(4.19)
R
onto itself, derive a monotonic
satisfying the three-points condition; clearly
n
IV.
T*n
+ T*)
n
Using a Mobius transformation of
function
+
n
+
d(g0
go
X*
defines-a function
n
(4.20)
d(g)
such that
h(g*n(0))
X*(l,0)
n
and
n D[]
V.
Clearly
n+l <
Vl.
(4.21)
n
By the monotonicity of the
the three points condition and (4.22),
gn*
there is a subsequence of the functions
..
ly to a function
THEOREM 5.
PROOF.
For
D[z]
n
{Xn}
which converges uniform-
([i]).
defined by (4.2).
for
sufficiently large,
<
(4.22)
1,2,...
n
D[]
<_
go
+
Sn
belongs to
Cn
Hence
n < d(g0 + Sn)
and by lemma 5, our claim is proved.
5.
NUMERICAL EXAMPLE
As an example of an application of the results in the previous sections we
consider the followng.
PLATEAU PROBLEH I PARAHETRC F01
Let
C
(x,y,z)
be a simple closed curve in
e)
X
C
o,
e
0 <
o)
e (I-j,
,
o)
e
h(8)
(o,
e,
5
-e,
(0,
o,
5)
2R- e)
-
-]
]
]
4
.]
4
5
5
2.]
e
([5,
e
([7 "’
(1, O, O)
0 ([0,
-]
(o, l, o)
o
2
o,
(-i,
[-, -$.]
2
e ([-j,
1
o)
(5.2)
e ([,
-f.]
(o, -]., o)
e [-j4
, -j5 1
(0, O, -1)
e
(o,
T(e)
,
2
o,
l)
T2(e)
(TI(e)
[-,,
2R]
T3(e))
Our problem is to find a minimal surface spanned by a curve
Let
k
A
I
B
I
g(0)
The monotonicity of
g
B
0
k
-
be given.
A
k
+ 0 + Y.
(1emma 6) demands
4
j=l
j< A.
The function
(Aj
given by
< 2,
[,
e E[.,
e- )
(o,
/
T(e)
,
(-, o--$,
(-
h’ (E))
e ([o,
o)
2
space of length
where
(o,
33
cos j0
4
B.<
-j
+ B.
3
g(O)
C
will be
sin j0)
354
B. CAILON, A.D. SOLOMOM and L.J. NACHMAN
while we can guar ant ee
g(e)
g(2)
0
2
by choosing
k
A
0
J
j=l
(e)= (g(e))
Let the function
(5.3)
A..
-2
and
(e)
AX
0
H2(e)
(Hi(e)
H3(e))
Now we solve the Laplace equation
D
on the domain
the circles
r
e
jSe (j
0,1
Let
(ih,j6e)
<_ M
ih
O, 1,2,
(i
(see Eq. (2.11)).
plane by the points of intersection of
+
i0, i0
N)
i
and the straight lines
M)
2
3
X
X
,j, i,j’ i,j
(X
f
i
X,j
X,j
The value of
for
0 < i < i
1
0
are obtained from Poisson’s integral
1,2,3
and
e
r
(e)
X(l,e)
with the boundary condition
Define the mesh points in the
i <_ j
(5.4)
+
i
Xf
i2h
(ih)2)H()
(5.5)
d
2(ih)cos(
j6e)
(5.5) by the compound Simpson’s rule.
We compute the integral of Eq.
To obtain the value of
(i2
i < j < M
N > i > i
0
for
1,2,3
we use
the following.
Consider
Laplace’s equation in polar coordinates
82X
r2
+ i_r
+ 12
8X
0r
r
Then Laplace’s equation at the point
Xi+i,j
2X’ ,j
h
+
X"-l,
2
(i,j)
I
+_
(ih)
may then be approximated by
(Xi+l,.-. Xi-l,j)
(X j + i
2
(5.6)
1,2,3
0
ih
I
g iv ing
82X
2
2h
2X",j
(e)
+
2
+
X",j-i
=0
(5.7)
PLATEAU PROBLEM IN PARAMETRIC FORM
(l
)Xi_l, j + ( + i-fi xi+. 1, j
.
1
(i58)
(i68)
1
1,2,...,M
i
i
(5.8)
0
(i58)2 xi,j+ I
If these equations are written out in detail for
j
2)X, J
2(1 +
x i,j-i +
2
355
i
0
0
+ I,...,N
and
and by using the relation
X,j
(5.9)
Xi,j+M
then it will be found that their matrix form is
Ax
where
X
d
are column vectors whose transposed are
.10 X0+I
X
(
I’
,M
2
d
d
d
2
(5.12)
I) )xi
0-I
(5.13)
0
(5.14>
H(J68)
(5.15)
2(i0
dN_ I
3
)
I
i0 <- k <_ N
i
(i
I
(H
dN
A
Xn ),
(x.,
where
The matrix
(5.10)
d
Hj
is given by
(i
i
+)I
D
(i +
2
2--2
I
(5.16)
i
2 (N
i)
DN_ 1
)I
(i
where each
D
and
I
are
M x M matrices and
i
)I
(i+
D
N
B. CABLON, A.D. SOLOMON and L.J. NACKMAN
356
a
a
-2-
t
a
t
t
ag
2ag
(5.17)
-2-
a
t
a
(e)
a
2a&
-2-
2ag
2
(5.10) we use the same method as in [i0].
To solve Eq.
We factorized the supermatrices in the form
A
U
L2
V
I
I
U
L
(5.i8)
=LU
I
V
2
2
(5.19)
U
L
N
I
UN_ I
VN_ I
U
N
and
U
m
(i-
D
I
I
i
m)m_iD
V
1
(i +
1
Vm Dm
i
)I
(i-
I
)D-I
m-l(l
i
+ 2(mi)
(5.20)
m>l
V
m
(i +
1
2-f-.m)
This method has been described by Wilson [Ii] though not actually for elliptic
difference equations but for equation of a similar form.
To solve Eq.
(5.10) we first solve the following equation
LY
The solution of Eq.
d
(5.21) is given by
Y
(YI
YN
(5.21)
PLATEAU PROBLEM IN PARAMETRIC FORM
YI dl
357
(see Eqs. 5.13-5.14)
N
Yk -Yk-1
(5.22)
(5.23)
i >_ k >_ 2
YN dN LNYN-I
(5.24)
Now we solve the equation
Y
UX
(5.25)
by solving the following equations
U-- YN
.
+
UN-j-j
X
Thus we obtain the values of
VN-jXN-j+I
,3
(5.26)
for all
I < i < N
N+I
i
(5.27)
>_ i
We do these
i <_ j <_ M
8X
In the next step we calculate the values of
i
i >_ j
1,2,3
calculations for
(i,j)
N
YN-j
at the points
,M by standard difference equation method, then
0, i
j
8X
88
8r
we compute the integral
D
=f
.2
x
[(----)
r
2] }dr de,
ox
+ E]
(5.28)
=i
by approximating it by a generalization of Simpson rule [12].
.
In the third step we compute the value
E
max
<_M
-
.X
[ [8--) N+I,j [8---)N+I,j
(see Eq. (3.8)).
In our example we first compute the value of the integral
do this by choosing of
{Aj}
{Bj}
{Aj }
satisfies Eqs.
{Bj }
(j
I,
(4.12), (4.13).
k)
For
D
(5.29)
and of
E
We
in a random way and such that
IEI
not sufficiently big we stop
the random process and then we use gradient method [9].
To use the gradient method we calculate the value of the
mating the integrals
gadient
by approxi-
B. CAHLON, A.D. SOLOMON and L.J. NACHMAN
358
(5.30)
2
0B0- -
/
2
=I
j
____OX (l,B)T(g(B))sin
(5.31)
dB
j
0
As before we approximate the integrals (5.30), (5.31) by the compound Simpson’s rule.
We halted our process when the values of
IEI
were smaller than
In the following table we see the numerical results for
and
N
M
21
31
i
0
2
10
-3
k
I0
I0
In Table I we present a selected result that was obtained by random choices of
Aj
Bj
method.
In Table II we see selected results that were obtained by using gradient
The
intial value of
{Aj}
results obtained by random selection.
drawn from the values of
X,j
{Bj}
for the gradient method are the best
In Figure I, we show the =Linimal surface
and using the closed curve
Figure
C
given in (5.1).
PLATEAU PROBLEM IN PARAMETRIC FORM
Table I
Calculations by Random Choice for the Coefficients of
Value of
Integral D
Value of
IEl
134598. 84615
30000. 70486
12036. 91971
246.44 340
18 ii. 33271
1238. 02080
711.49210
18.52716
272.28744
78.04352
256.88041
42.43274
181.60385
43. 24106
i01.73822
59. 3020]
73. 75771
20.54170
53.35582
9.87578
49.01988
13.00610
33.56237
16.20803
24.76756
9.08349
14.99357
12.11406
9.34689
6.42093
8.00960
9.95630
7.97180
4.27323
7.00766
6.51285
6.67786
3.15938
5.42465
3.54083
(Aj,Bj)
359
360
B. CAHLON, A.D. SOMOMON and L.J. NACHMAN
Table II
Calculations by Gradient Method for the Coefficients of
Value of
Integral D
Value of
El
5.42465
3.54083
5. 33672
2.86892
5.11811
2.57031
4.96691
1.05302
4.87354
0.47714
4.77961
0.34489
4.75412
0.68156
4.703091
0.81391
4.59962
0.46404
4.58185
0.45846
4. 50000
O. 39333
4.41726
0.32073
4.38645
0.22813
4.32602
0.19501
4.30832
0.24215
4.28768
0.36908
4.24679
0.22170
4.23814
0.13596
4.13554
0.02041
4.085048
0.00174
(Aj,Bj)
PLATEAU PROBLEM IN PARAMETRIC FORM
361
REFERENCES
i.
Courant, R.,
Dirich.let’.s
principle
c0nfprmal mapping. and minimal surfaces,
Interscience, New York, 1950.
2.
Concus, P., Numerical solution of Plateau’s problem, Mathematics of Computation_,
Vol. 2_I, 340-350 (1967).
3.
Greenspan, D., On approximating extremals of functions, Part I: The method and
examples for boundary problems, ICC Bulletin, Vol._4 99-120 (1965).
4.
Greenspan, D., On approximating extremals of functions, Part II: Theory and
generalization related to boundary value problems for nonlinear differential
equations, .l.ntern.ational J..ournal.of Enineerin S.cience_ Vol. 5, 571-588 (1967).
5.
Rado, T., On the problem of Plateau, Chelsea, New York, 1951.
6.
Evrafov, M., Analytic functions, W. Saunders, Philadelphia, 1966.
7.
Heinz, E., Uber das randverhalten quasilinearer elliptischer systems mit
isothermen parametern, Math. Zeitschrift, Vol. 11__3, 99-105 (1970).
8.
Courant R. and Hilbert, D., Methods of mathematical physics, Vol. 2, Interscience,
New York, 1962.
9.
Daniel, J., The approximate minimization of functionals, Prentice-Hall,
Englewood Cliffs, 1971.
i0.
Keller, H.B., Numerical methods for two point boundary value r.0b_!_e_ms
Blaisdell Publishing Co., 1968.
Ii.
Wilson, L.C., Solution of certain large sets of equations on Pegasus using
matrix methods, Compute.r 3. _2, 130-133.
12.
Carnahan, B., Luther, H.A., and Wilkes, J.O., Applied Numerical Methods,
J. Wiley and Sons, Inc., New York 1969.