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m A x Y m x m A m to she
BiiUGmm momm
" F " ™ '
i .jfT.i i or Preffeatittv
M l a o y P x e f m r o r
,7a Liu I Bireotpr oi 1sfa© Department of Mat3a©ffi&tiGS
MMBtasw
l«aa "of' tii© Graduate Sahool
AN APPROXIMATE SOLTJTIOBf TO THE
DIRICHLJBT PROBLEM
THESIS
Presented to the Graduate Council of the
lorth Texas State University in Partial
Fulfillment of the Requirements
For the Degree of
MASTER Of SCIENCE
By
Edward William Redw±ne» £,S.
Denton, Texas
August, 1964
TABLE OP CONT3SKTS
Pag©
LIS® OF ILLUSTRATIONS. . iv
Chapter
I. IN TKODU CTION . » %
II, THE PHOCliJDUHE FOR THE APPROXIMATION 5
III. SOLUTION OP THE SET Of LINEAR EQUATIONS 34
BIBLIOGRAPHY 40
ill
LIST OF ILLUSTRATIONS
Diagraa Pag©
1* Lattice Points.. • * ....... 2
2# Lattice Point wit& Thy©© Adjacent Points......» 6
3« Lattice Point with. Two Adjacent Points*........ 7
4. Lattice Point with One Adjacent Point* 8
5* Lattice Point with Pour Adjacent Points*«...••• S
6 • Lattice Points on Polygon. 9
iv
CHAPTKH I
INTRODUCTION
In the category of mathematics called partial differ-
ential equations there is a particular type of problem
called the Dirichlet problem# Proof ie given in many
partial differential equation books that every Birichlet
problem has one and only one solution. The explicit
solution is very often not easily determined» so that a
method for approximating the solution at certain points
becomes desirable, The purpose of this paper ie to present
and investigate one such method, I'he Dirichlet problem can
be stated as follows.
Definition 1.1. Let H be a simply connected* bounded 2 *•
region in E whose boundary B is a contour. Let S * RUB#
If a gllren function g(x»y) is defined and continuous on B»
then the Dirichlet problem is that of determining a
function u « f(x,y) which is;
(a) defined and continuous on Hf
(b) harmonic on E» that is, a 2 fUty) + a
2f(».y) e 0 | 9x <9y
(c) identical with g(xty) on B.
Example. Let the function g(xty) « 3x ~ y + 2 for all
points (x,y) which satisfy x 2 + y2 » 4 be given. Then the
problem of determining a function u » f(x,y) such thatt
(a) f(x,y) is continuous for all points (x*y) for
which, + y®£ 4, aad
(b) f(x,y) is harmonic at all points (x,y) for
which x2 + y 2C 4,
is a Diriohlet problem.
Before proceeding to the method of approximation, a
description of lattice point® must be given#
Definition 1.2» Let E be a region in I 2 with bound-
ary B. Let (x,y) be a point of HUB, Let L^ be the set
of all points (x + mh, y + nh), m « 0, ~1, —2, «*• ,
n « 0, —1, ~2, , which are contained in H U B for fixed
h> 0. L^ is said to be a set of lattice points, defined
on HUB, and h is the mesh width.
(-3,0)
(0,3)
*2
•" 1
3 4 5 ©
7 8 9 10 II \
12 13 14 15 16 -NI
19 20
(o,o)
21 22 23 j
24 25 26 27 28
29 •
(3,0)
(0,-3)
diagram 1—Lattice Points
Example* Let B be the oirole whose equation is
s 9 and let E be its interior, let h « 1,
(x,y) » (0,0), then is the set of points (0,3)* (-2,2),
(-1,2), (0,2), (1,2), (2,2), (-2,1), (-1,1), (0,1), (1,1),
(2,1), (-5,0), (-2,0), (-1,0), (0,0), (1,0), (2,0), (3,0),
(-2,-1), (—1,—l), (0,—1), (1,—1)* (2,—1), (-2,-2), (-1,-2),
(0,-2), (1,-2), (2,-2), (0,-3). In Diagram 1, these
points have been assigned the numbers 1 - 29* respectively.
It is desirable now to divide into two disjoint
subsets Bjk and will be the set of lattice points
which in some manner lie near B and will be called the
lattice boundary, fhe remaining points of will make up
iijj and will be oailed the interior lattice points. The
following definitions describe how to determine and
exactly.
Definition 1.3. Let H be a region in with boundary
B and for fixed h>0 let be a set of lattice points
defined on HUB. Two points (x-j y )* (x2,y2) of are
adjacent if and only if the straight-line segment joining
them is contained in RUB and
(a) x1 * x2, jyg - y | « h, or
(b) y ^ es y^, |x2 — X x i » h .
Example. For the set of lattice points shown in Dia-
gram 1, the pairs of points 8,9; 10,16? 17*18$ 1,4 are
pairs of adjacent points, while the pairs 16,21$ 20,22$
1*5? 4,15 are not.
Definition 1.4, let H be a region in I 2 with boundary
B« If is a set of lattice points defined on R U B then
the interior of 1^ is the set of all points of which
have four adjacent points in X^.
Definition 1.5* Let tt he a region in S2 with boundary
B and for fixed h > 0 let be a set of lattice points
defined on RUB* She boundary of L^# also called the
lattice boundary, is defined by B^u m L^f **•&,
For the set of lattice point® displayed in
Diagram 1, consists of points 4, 8, 9* 10, 13# 14, 15#
16, 17# 20# 21, 22# 26, while consists of points 1, 2#
3, 5, 6, 7, 11, 12, 18, 19# 23, 24, 25, 27, 28, 29.
If u « f(x,y) is the unknown solution of a Diriohlet
problem on a plane point set H U B and is a set of lat-
tice points defined on H U B consisting of exactly M points#
then number these points in a one-to-one fashion with the
integers 1# 2, ••• , H. Denote the co-ordinates of the
point numbered k by (x^y^.) and the unknown function
u » f(x,y) at (Xfc#^) by uk, k «= 1, 2, ... , 1.
CHAPTER II
THE P OCJUURjJ ffGii THii
APPKOXIMATIQI
A method for approximating the solution to the
Dirichlet problem which is a variation of the Mebman-
Serschgorin-Gollatz approximation procedure will now he
described in four steps, fhis will be followed by a
consideration of uniqueness* convergence, and the solution
of a set of simultaneous linear equations which will give
the actual numerical values of the approximate solution
of the Dirichlet problem.
Step 1. Choose a point (I,y)€SUB and an h> 0
and then construct a set of lattice points on RWB.
will consist of N distinct points. lumber these in a
one-to-one correspondence with the integers 1, 2» ••• » B«
Step 2. For each point of the lattice boundary Bh»
approximate the analytical solution of the Dirichlet
problem as follows. Let (xs,yg) be a point of and
suppose it has been assigned the number s, such that a is
an integer and l^s^H. If (x .y_) is also a point of 3 «$ o
then set
(1) ug » g(xB#ys).
Let the four points (xs + h,y8), (xs,yg + h), (x0 - h,yg)»
6
(x »y« - &) be numbered w» p, q., z$ not necessarily respec-S 0
tlYQljt iiacli of w, p, q.i z is among the first I integers
if the point which it represents is in 1^. If
but is not a point of B» then at least one of the four
points w, p, q.» z is either not a point of BUB or is a
point of RUB which is not adjacent to (xg,ys).
Oase X. Suppose only on© of the points w, pt q,, % is
like che one described above and* without loss of general-
ity, suppose it is (x„ + h,y ) and numbered w. Then there a H
is a point (xQ + &w»yB)f 0<dw<h« which lies on B and
should be numbered 2• » such that the straight-line segment
Joining (xa»ys) &&& Cxg + y g ) is contained in HUB.
low approximate u_ by
(2) (3^ + h)us - dwtip - d ^ - dwua » Ssug,.
Note that Ugt » g{xs + which can be calculated and
given a numerical value.
< w h >
(vh>)rJ 8 'tfs' q. fi_LL
xs+4w'ys
)
z
Diagram 2.—Lattice Poinx with 2hree Adjacent Points
Oase 2. Suppose two of the points w, p, q, z, say w and
p, are suoh that each one is either not a point of RUB or is
a point of HUB which is not adjacent to Also,
suppose w and p are (xQ + h,y0) and (x8#yg + &)» respectively.
Then there are two points (x0 + <V»y8) (xgtys * dp)»
0 <d <h and 0 d < h, which should toe numbered 2' and V» w P
respectively# Then calculate ug by
(3) ( 2 ^ ^ * M w + M p ) u g - dwdpuq. " dwdpuz 18 hdptt2' + M w u 3 f
Note that Ugt » g(x0 + dwfy0) and u^, « g(xsfys + dp) and
both can be given numerical ralu.es,
(x .y +d ) w s s p
/ , x 8 V x0+ dw»y 0)
( XB" h' yB )*i ' ^
(*B»ya-h)lz
Diagram 3.—Lattice Point with Two Adjacent Points
Case 5* Suppose three of the points p> c[f z$ say
w, p, q, are such that eaoh one is either not a point of
RUB or is a point of R^3 which is not adjacent to (xm,ya)9 3 »
Also, suppose arbitrarily that w, pf $ are (x_ + hjyJ*
(x@iyg + h)* (xs - k*ys)* respectively* Then there are
three points (x& + dw>y8), (xs#ys + dp), (xB - d<jty0)*
0 <d wCh t 0<dp<h, 0<d^<ht which should be numbered 2%
3*» 4 % respectively. Then calculate ug by
(4) Cd^dpd^ +• hdv/dp + hd^d^ *f hdpd^)Ug *» d dpdgUg, * hd^d^Ugi
+ hdwdqu5,
Ohoose h small enough so that there does not exist a lattice
point in I*k which oanmot be connected to all other lattice
points in by path® which move from one lattice point to
an adjacent one*
8.
(x„iy +d ) -8**8 P
(xsfys-h)Iz
Diagram 4»—lattice Point with One Ad^aoent Point
Step 3* Let (xs»yg) be a point of and suppose it
is numbered s* Then calculate u_ by
(5) 4u — u — u — u — u » 0* a w p q. a
Hot© that this implies that ug is the mean value of u^,
u • u • u • P <1 s;
z
(xg,ys+h)
s v <•vh-ys>
(3cB»ys-
h)
Diagram 5*—lattice Point with Four Adjacent Points
Step 4« Steps 1» 2, and 3 together with subscript
notation produce a system of B linear equations in I un-
knowns u » u2, ... t Construct the N x N matrix A
by using the coefficients of the equation used to calculate
u . If siN, as the nonzero elements of row s of A, But m
the coefficient of u^, 1 <t£:N, in column t of A. Also*
construct the column matrix € by using the constants, such
that the constant from the equation used to calculate ug,
1 < la the element in row s of 0, I'hus, the final
step is the solution of the linear algebraic system which
gives an approximation to the analytical solution at each
point of
(-1
C-i.-I)
17
12
18 19
13
8 ( W
3
14 15 16
20 21
10 II
(s[h,ilh)
m-i'h)
Di&grasi 6#—Lattice Points on Polygon
Example. Let B be the polygon with side® produced
toy connecting the points (-1, 2j|) to (3>|> 2jjr) to (5^*-2^)
to to (2|»-1) to (-1,-1} to (-1,2^) with straight
lines and let E he the interior. Let g(x,y) * x + y on
all sides. Set (x,y) » (0,0) and h = 1. As show* in Dia-
gram 6, the points of are (3,-2), (-1,-1), (0,-1),
(1,-1), (2,-1), (3,-1), (-1,0), (0,0), (1,0), (2,0),
(3,0), (-1,1), (0,1), (1,1), (2,1), (3,1), (-1,2), (0,2),
(1,2), (2,2), (5,2) and are numbered 1 - 21, respectively.
Consider each point in order and applying the appropriate
steps gives the twenty-one equations:
1 0
+ H - i + k'ia * - i ' i ' i u 6
- z ) + - 2 ) +
( 5 - 1 + 1 ) U 6
« t t g .
**10
( 3 - i + D a , , -" X win 11*11'
4 U 1 5
4 u 1 4
4 U X 5
( 5 - | + 1 ) ^ -
( 3 * | ? + l ) u ^ g -
( 5 ' | + D t ^ g "
( 5 - f + 1 ) « 2 0 -
U g ** —1 "*1
u , = 0 - 1
tt4 - 1 - 1
a 2 — 1
K - K - K i •
® **X + 0
3 ~ " 7 ~ " 9 " " 1 3
1 - < 3 | - 1 )
u
l 4 - « g - U^Q - « l 4 « 0
~ ^ — H *» 9 " " U " * 1 5 - 0
$ " i 6 " i b i o " be " 1 - ( 5 I + 0 )
•"X + X U 1 2
~ u 1 2 ~ ^ 8 " ^ 4 *" *aX8 88 0
" U X 3 " n9 ~ ®X5 * * X 9 " 0
" U X 4 ~ ~ ^ 6 ~ * 2 0 m 0
• K i - K 5 - K i • 1 - ( 3 t + 1 )
ttX7 * ~ 1 + 2
' 1 * 1 7 * l ° 1 3 ~ ^ 1 9 m 1 ( 0 + 4 *
I t ^ X S ~ f ^ X # " § * 2 0 " i ' C 1 + 2 § )
' l h * i q *• ^ c " i f t o ' i « X • { 2 + 2 ^ ) C X? a, Xy c CX C.
11
< 2 > H + l - i + l ,i ) uzi - f K o - l - K 6
m * 2) 4- X*^(5 + 2^) &* *Y *T 4B*>
The following is the matrix form for these equations, such
that the first twenty-one columns make up A and the last column
is 0.
if 0 0 0 0 =1
M 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ° n
0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2
0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1
0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
*T| 0 0 0 =1 1Z 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I
0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1
0 0 -1 0 0 0 -1 4 ""1 0 0 0 -1 0 0 0 0 0 0 0 0 0
0 0 0 -1 0 0 0 *•1 •»1 0 0 0 "*1 0 0 0 0 0 0 0 0
0 0 0 0 •1 0 0 0 *•1 4 -1 0 0 0 -1 0 0 0 0 0 0 0
0 0 0 0 0 -1 0 0 0 si 11 4 4 0 0 0 0 s| 0 0 0 0 o 1|
0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 -1 0 0 0 -1 4 -1 0 0 0 -1 0 0 0 0
0 0 0 0 0 0 0 0 •1 0 0 0 *1 0 0 0 -1 0 0 0
0 0 0 0 0 0 0 0 0 -1 0 0 0 -1 4 -1 0 0 0 -1 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 si 11 0 0 0 0 z l 3L7 mj.
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1
0 0 0 0 0 0 0 0 0 0 0 0 =1 0 0 0 dk i ^ L W mmW
0 - 1
0 0 0 0 0 0 0 0 0 0 0 0 0 z
0 0 0 •»1 9 -%L
0 ^
0 0 0 0 0 0 0 0 0 0 0 0 0 0 »1 ~f 0 0 fx k Jit JSt — — ym mm— M
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 - * 0 0 0 rnrnm 1 4
12
It will now be shown that the method described above
is a reasonable technique of approximation in that
(a) if the points {xs>y0)# (xQ + h»y0)f (xg#yg + h),
(xg - h,y0), (x0,y8 - h)f numbered s, w, pf 2,
respectively, are elements of R, then (5) can be
derived within an order of h,
(b) the resulting linear algebraic system always has
one and only one solution, end
(c) the approximate solution converges to the analyt-
ical solution as the mesh width h converges to
aero#
X.et the five points UQ,y@), (xg + h*yg), (x0,ys • h),
(x0 - h,y@)t (xfi»y8 • h) be numbered s, wt p, qf z, respec-
tively, as in Diagram 5. Define D as follows*
(6) D * (aufl + bu^ + ctip + du^ + euz)
P2*(*a»ya) ^2f(x ,yfl)
d X e)y^ 0
such that a, b, c, d, e are constants and u « f(x,y) is the
analytical solution of the Diriohlet problem* Since all
five points are elements of R and u * f(x,y) is analytic
on R, faylor*s formula with a remainder can be used to gives
<7) ^ - u(x..y.) + h ^ 8 + §- — - J •* + o(h?),
13
(8) up • u(x0tys5 + h df(xtt,yj „2 S^CXstya)
0(h3)
(9) - «(r.,y.) - hJ^(*«'y«) • | 2 9 jY*"* • °<»>5>
(10) u_ m m(xfl>7_) - h. 11 '" +
Ox 6 i>x
a*c*..y.> »2 a2*<*«.y„>
- -*~sF*g, « fl)y
Substituting (7) through (10) into (6) gives
(11)
+ o(h5).
2) = u_(a + b + o + d + «) + (hi* - M ) 8 dx
+ t £ l i ^ ( h c . h 9 ) +
c>7 2
a2f(*.f7_) j»2
c)x S p £ (f * 1)
JLI^lA (|% +12.. i)
c>y
+ b0(h3) + c0(h3) + dO(h?) + «0(h3) •
She right side of (6) is equal to the right side of (11)
for any values of a, b, c, d, and e. Let a « ~4/h » , o
b s d a d s M 1/h » so that
a + f e + c + d + © « 0
hb » hd m 0
bo ** he w 0
.2 -2 | i j + | d - i s O 2 h2 0 + 1 8 - 1 * 0 ,
From (6) and (11) then
-*«» * °w • *i> + aq + u. -j. 3 f(xg>ya) c) f(xs>ys)
£>x« 3y' 0(h)
14
or
~^ub + u v + ur, + ua + "a + o ( h ) „ + ^ V I a i
jbr c)x o>y
She harmonic property of f(x,y) at (xg,yfl) implies
-»4u + u + u + u„ + u„ S SL_JB J L - — S + o{h) « 0.
h 2
Discarding the terns 0(h) gives the approximation
-4u +• n, + + u„ + u,_ SL * ^ P — a & m o
and
4u — u -» u — u — 11 « 0 s w p q z
and (5) been derived.
farther investigation require® careful distinction
between the analytical solution of the Dirichlet problem
and the approximate solution under consideration* Froa
this point on, let u « f(xty) represent the analytical
solution of 'the problem and let U « U(x,y) "be the approxi-
mate solution. Therefore, U = U(x,y) is defined only on
L^. If ( x8#y s)
e s* then (1) implies
(12) U(xB,y0) • g(xs,ys).
If a n d ie a type of Case 1, Step 2, then (2)
implies
(13) (3&y. + h)U(xB,ys) - V,^3C8,y» + &) - dwIJ(xs - h,yg)
-VJ(x8,ys - h) » hg(xs + dw,yg).
15
If (x ,y ,)6B, is a type of Gas® 2, Step 2, then (3) 8 & "
implies
(14) (2dwap + tidw + Mp)0(xB,ys) - d„<ipB(*a - h,ya)
-Vp U ( ls , yB "
h ) " hdp«(xB + + V '
if (xs.r8)£\ i» a type of Case 3, Step 2, then (4)
implies
(15) < V p d q + " V p + h V 4 +
- V p V ( x « " s " h ) * hdpdqg(l8 + dw'ye1
+ »yg + dp) + dwdp®^xs ~ d4,ya^*
I* (xs»ys)CHh, then (5) implies
(16) W(xs,ys) - V(xa + h,yg) - U(xfifys + h)
-U(xs - htjB) - (x0»ys - h) « 0.
Theorem 2*1. The system of linear equations result-
ing from the method of approximating the solution to the
Dirichlet problem always has a solution and this solution
is unique.
Proof* The constants of the equations are all that
is affected by ohaaging g(x,y). from matrix theory* if
a system of linear equations has one and only one solution,
then the same equations with the constants changed will
have on® and only one solution. Therefore, if the system
of equations has one and only one solution for g(x,y) « 0,
then the system of equations with g(x,y) equal to any
other function will have a unique solution.
16
With g(x,y) ss 0, tia® constants are all equal to
zero. One solution to this system is the on® with all
£©ros» To prove the uniqueness» assume there is a solution
U(xfy)» not a l l zeros, which satisfies the equations. Then
U(xfy) t 0 at some point Assume 11(^7^)<0*
If U(x »y]fc)>0» the proof is similar, Let ~M he the minimum
value of U(xi»y) on and (%»?^) aaif point o f ^ suc&
that
(17) * -«<<>•
Therefore, for all Cx»y)Cl» >
(18) tfGtjJx) = -M£U(x,y).
If (x^ti^CB^t then *
(a) (12) iaplies uC«xt7x^ * 0 wiliok contradicts (17) >
(fc) (13) implies that at least one of U(x ,yx +
- h,^), TJ(x ,y - h) is leas than tKx^y^)
which contridicts (18)f
(o) (14) implies that at least one of TJ(x - h,y^) and
11( ,7 • h) is less than contradicting
(18)i and
(d) (l'S) implies uGc^y^ - h) < u (%>?!) which contra-
dicts (3.8).
Therefore, is not an element of This leaves
*1* 1 Sh ^or there are only two possibilities to
consider:
(a*) if U(x, + h.y,) , + h), U(E. - h j , ) , and
- J>) are all e4ual, ttom fro. (16)
17
m 1 $ 7 X ) . y ( % + h9rx). fiiis can be continued
a finite number of times m to give
«»JC m ss U(*^ +• Bh)^) ,
wiser® (x^ + m h t y 3 1 ) ^ H o w e v e r , as shown above,
E(x,y) cannot equal »M at any point of and a
con traduction has been reached.
(b*) suppose that one of + h,^), 8(3^,3?^ + h),
v a x ~ 1 ^ ) , u(xlfyx - h) is not equal to at
least one of the other three. Ihen there le one
which 1® less than or equal to the others. With-
out loss of generality, assume that U(x^ + h,y^)
is that one* So that
^(*1*^1 + **) * + k»y^) +
U("x - h,y^) as + &g
DfxjL.?! - h) rn viXi + lijj) + a,
such that a 1 # ag, a^ are nonnegative and at least
one 1® positive. Substituting; these three equa-
tions into (16) gives
u ( % + h , ^ ) <tl(I1,y1),
contradicting the fact that UCx^y^) was the mini-
m m #
As each possible case was considered, the assumption that
there was a solution not all aseros led to a contradiction
and the theorem Is proved.
Consider now the convergence of the approximate solu-
tion to the analytical solution as h approaches zero.
18
Definition 2.1* In the case for which each of the Indi-
cated function values exists, let
(19) o[jr(x,y)J « 2£-4v(x#y) + v(x + h,y) + v(xty + Jx)
+ r(x - h,y) + v(x,y - h)J •
iOtUJLl* " «i MA c2 are «Wtr«y constants and all
tie indicated function values exist, then
sjV^tx.y) + c2T2(z,y)j . O1ofr1(x,7)] + OjOfVjCx.y)].
Proof* From Definition 2.1
o [ O 1 T 1 ( X , J) + o2r2(x.y)j - lg [ojTjU.y) + o^ix.j)]
+ [ c ^ x + h>y) -I- 0 2 ^ 2 • ii#y)J +
+ ©2v2(x,y + fe)j + f°iviCx - h,y) + «2T2*X ~ h'*H
+ f°lvl(x»y - h) + o2T2(x,y - h ) ] J rn o1[-AT1U,y)
+ Tx(x + h»y) + T^x^y + li) + r^x - hfy) + Tjtetjr - &)}
+ ° 2 C * » y ) + v2(x • h,y) + Vg(x,y + h) + r2(x - h,y)
+ r2(xty • h)j « + ©2$£v2(xty)J.
Lemma 2.2. Let E be a region with boundary S and
be a set of lattice points defined on RUB. If v(x,y) i®
defined on 1^ and sjV(x»y)J^O on and r(x,y)£0 on B&f
then T(xfy)>0 on 1^.
Proof. First, assume the opposite of the conclusion,
that v(x,y) <0 at some point of Since contains only
a finite number of points, there is a point (x^y^) CR^ at
which v(x,y) is a minimum, therefore,
(20) T(x1,y1) <0
19
and for each (xty^H^
(21) •(x1,y,)<Y(xty).
Using (19) and tie assumption that s[vU,y)J:£o leads to
(22) *(x»y)> $ f r { * + h, y) + v(x»y + &) + v<* " *'y) + T<x'* " h)3
for all (x,y)C K^,
But tli® equation© (20) to (22) imply by teolm.iq.ue8
used in (a*) and (b*) of the proof of fheorem 2»lf that
v(x,y)< 0 at some point of Since this ie a contradic-
tion, the proof is complete.
Learnta 2 >3, Let E be a region with boundary B and let
be a set of lattice point® defined on HUB. If r^(xiy)t
Tg(x»y) are defined on and - lafrl(x*y)]l - a[T2^*»yO on
\ |^(x»y)| ,Tg(x»y) on B^* then (T^(xty)| Vg(xty)
on Bjj.
jP$go£. From the assumptions, y2(x ,y) * ^ ( x t y J ^ O on
B^. From Lemma 2„1
s/v2(x,y) - v1(x»y )] . a[rs(x,3rl] - ajVjUy)]
on \ * Since o[r2(x#y)J 4- fofcUty)]! £ Q on \ $ then
°[r2(x,y)] - ajr^xiy)] oJV2(x,y)J + ^^(x^yjJ) < 0
®u Bcfuo i |
ajr2(x,y) - r^Xty)] < 0
on By Lemma 2*2, then
v2(x*y) - ^(xtyJ^O
on R. . A
low using assumptions made.
20
r2(x,y) + r1(x,y)>v2(x,y) - Jv1(x,y)|>0
on Bh# and
a(V2<x,y) + Tj_(x, r>] * G{V2(x,y)J + ^ 0
on Hence, from Lemma 2.2,
v2(x,y) + •1(x,y>> 0
on Since v2(x,y) + v^(x»y)>Q arid v2(xty) - v Cxty)**: 0
on then
T2(x,y)2/v1(x»y)|
on and the proof Is complete.
Lemma 2.4. Let E be a region with, boundary B and L^ "be
a set of lattice point® defined on HUB. If r(x,y) is de-
fined on I»h and |g|v(xtyj]f < A1 on Jv(x»y)J^ Ag on
and r is the radius of any circle which contains HUB in
its interior,- then
on R^.
Proof. Let the equation of the circle with center
(a,lb) and radius r which contains SUB in its interior be
(x - a)2 + (y - b)2 « r2.
Define w by
w(x,y) « Lr2|l - fe" ^ ~ b?2
T + a 2 .
By direct calculation
o{w(x,y)J « -A^.
Since for any" (x,y)£RUB
21
2 . ^ 2 (* - a) ,+, (y - Z<xt
then w(x»y)> k2 on Since
on and
la[rU,y)JI •c Aj,
®|w(xiyj| cs «»A f
then
|&£r(x,y)) J ~&{w(x,y)J
on Also sine©
|v(x,y)|^A2
on and
w(x,y)> Ag
on S^f then
w(x,y)> I v(x,y) |
on B^. low, ~l&ljr(xfyjJl > ® jw(xfy)jon and w(xty)> j v(xfy) |
on which together with Lemma 2«3 implies
w(x»y)> jv(x,y)|
on 1^, or
|T(x,y)| w(x»y)^^A1r + Ag
on which concludes the proof.
Lemma 2.5. for the Dirichlet problem let u « f(x,y) be
the analytical solution and lot U(x»y) bo th© approxiiiat®
solution. If f(xiy)€£0^* on RUB and (x^»y^) is any point
of then
(2?) |t?ixify ) — f(x^>y^)|^ e + »
22
where
(24) e « max |u(x»y) -* f(x»y)| ,
lub HUB
m&.T? ( rr% ox M W M M M &
5 ?
and r is the radius of any clrole whieh. contains HUB in
its interior#
Proof. Sinoe tf(x»y) exists on only a finite number of
point® and f(x,y) is continuous on HUB, e does exist. Due
to the assumption that f(x,y)£c4 on HUB, also exists.
Let (x^y^CR^ and define Q as
Q - sji^.yj)].
Us© of finite Taylor expansions gives
(25) Gff(*1,yi)] * (xi,yi^ + $(*±97$) - h
2 D2t(x» ,y4) y3 d?t{x*»y4) 04f (E, »y,) t'd j-r^x1,y1; ^ ynx**!* mgm III I m i III! I i j F > ! Ml I l i f e Till mm < 8 p M * iiiiniinoimiini I mi i f lg i i i i « i i i A
» JhJ T* 5 ? Ox wil l inff lki I . A n n
dt(Xi>y±) hz 32 f (x±,y i ) h3 a3 f (x1 >y1 ) + f(*1t7i) - b. '."•i..nftii,,. .ft., + ii ^ 11"
OfCx^y*) h2 32f(%f3T4]
+ + f (x^y^) + h ' |
+ yj. '*tii" + ^ + f(x1 #y1) + it *—*-
* fr 3y* + fr * + fr 3 I 1
where x ^ B ^ x ^ + '^i^ E2^yi + il*xi ~ 'k<1&34<'xt*
23
yA - h<2^<y£# Since f(x,y) Is hamonie, that isf
+ ? 2X,(«^ . o, 6>x c>y
(25) becomes
S^f (B.. ty. ) 3 *f (at* fBj%) 9Or<xi»'iO • tr
i ^ L . + 3 y
<0*f(&,,7^ + 94f(3^,B4)
c) x 5 y
therefore, 2
(26) |Q| • | vCtixi.yiUM-g-* .
By direct calculation,, O^U(x^>y^)J • 0 and by use of
Lemma 2.1,
|&[u(xi,y1) • f(xi»yi)]| * lG[v(Xirf±)] - ^Cr(3c±#y1>J/
» ^[^(ac^ty^jjl * |Q{£ g t ,
low, by (24), on
(27) l^(x»y) - f(x,y)| e*
Herns®, (26) and (27) together with Lemma 2.4 imply that
' u ( xi^i > - '("l-'i)! - • +
on fi^, where r is the radius of any circle which contains
S C B in its interior and this concludes the proof,
JLtit* For the Dirichlet problem let u « f(x#y)
be the analytical solution and let U(x,y) be the approx-
imate solution. If f(x»y)£IO^ on B U B and (x^y^) is any
24
point of then
r2h2M, (28) lv(xlfy±) - f(x1,y1)| < 1 - ™ ^ + 6h
2M2,
where
(29) m lub HUB
# j «s 2t4 ^(ISI'CT and r is the radius of any circle which contains HUB in
its interior.
Prooj;, first» consider a point of (x ,y )« There
are four general oases for whioh (x.,y4 )£B. . J i) JBL
Case 1. Suppose (x^y^e^flB), then
(30) |u(xr^) - f(xyy3)j m leUyy^) - g U ^ y ^ j » 0.
Case 2. Suppose hut and
is the type discussed is Case 1, Step 2. Then using finite
Baylor expansions (5i) f(x3.y;j) -
f(x3.y3 + h) - h ~U y r 3 + h )
if aftlx^E}) + ^ ^
and for y^< \<y^ + h» a
(32) r(x3,yj - h) - f(xry3 + h) -c>y
. (2h)2 5 f(x1# )
^ ~~o?
for y - h < < y^ + h, From (32)
(33) - >. * h ) f(x1'y-1 - h ) + h> Py 2 2
25
. ( | ^ a 2 f < * r V
Substituting (33) into (31) gives
(34) f(x1»y1 + h) fixity* «* h)
f(xj,y^) « "" J \A + !i11"
2 d 2 a2£(xltB2) — #• — h """ f—
«>y c? y*
Howt
(35) f(*3.yj) - f(*3 + \,j }) - a v3 f ( x ^ V * ! *
5o2f(B3,r1)
p x
for x < x^ and
(36) f(xj - it#3T|) * f(x^ + a^jj) - (& + d^)
(b + O 2 aZf(B4,yi) + 'g J" ' "'" '%"" f Ox*
for x^ - h <B^<Xj + from (36)
„ „ - . / ' " i , ; W . . ^ „ , i , v t i ,
N V + a*) a 2 f < V i > + r r ^ f<*j - "»yj> - * a * - j p P - -
Substituting (37) into (35) gives
(38) (&. + f(xj»y^) » h f(x^ + d^y^) + d ^ x ^ - bty^)
ag(h + <*») a^fBj.y;,) ^(h + a,)2 a2f(E4,yi)
+ * . g " Px*
Use of (34) and (38) gives
Px'
26
(39) (3&w + h)f(x^y^) • + b) + V^ x3* yJ - h)
/ / 2 0 4- la £(x^ + + dwf^xj - hty^) + V 1 —T~$—
c/ y
v2w_ w \ ,2/^ . „ v ->2. 2 3' £(x1#E2) d£(h • d ^ 3
. ^ • — , ^ H -
+ d,,)2 2f(BA,yi) **» *w* (nmluilimiiuiil i«gw iii«W«*i««i|iiiia«iNM« wJfcw W .
* e)a?
from (13), since g(x^ + d^y^) « t{x^ + dw#y^)#
<«) u(*jty3) - D ( xj - ^-yj' + 3 3 7 T E u ( x3
d + w w
w+ h ® < v * j -
h ) + 3 3 7 T T f ( i i + V ] ) -
$&©», from (39) and (40)
OCZj.yj) - f(i ry 3) - j g ^ . f[0(xji7j + h)
- f(x^fyj + h)J + [d(3E3 #y^ - h) - f(xj»yj - h)J
+ [u(x^ - hty^) - £(x^ - h,y^)j - | fib2 f^ x| , 3V
~ y
2 02f(x.,S2) 32X(2,»yJ - 43i 111"» 11111»" + d (h. + d ) £ A
ay"i w w 3 i 2
c>x
so that
;J} I ^ V j 5 - f ( x r y a ) ' ^ ? s ^ H [la(xy'} + h )
f(*yyj * h>l +M xyy^ - k) - f(x 3 #y^ - h)(
+ I (x^ - Hty^) - F(x^ - h,yj)lJ
27
+ 2 ( 3 ^ tt) (2J>2 + ^ + V + ^ + h 2 + 2 V l + 4 ' »
and i t can easily be shows, that
(41) )u(Xj,y-j) - ^ [ j u ( ^ . y^ + h) - f(x^ ty^ + h)|
+ - *) - - * ) | + |u(x^ - w 3 ) . -j 3h2M2
- f(*2 • h#y^)|J + g 6
where ML is defined by (29)» ijCs
Cage 3. Suppose (x^vy^)CB^ but ( x ^ y ^ ^ ^ / I B and
is the type discussed in Case 2, Step 2. TJsiag methods
similar to those used to get (38) gives
(42) dp(h + dw) f(x^»y^) » dpfcf(xj + d ^ y ^
( (h + O l , , d J 32 f(E5 ,y.) «• d^dpfCx^ - h,y^) + " jp " ' d'Jj""' """"
(ii + a_)2<Ldv 3 2 f (Eg.y j 2 ^ 7
for Xj <Bg<Xj + d^ and x^ - h < ^ < X j j + and
(45) «,<& + dp) ' (X j . y j ) - i^fitCXyTj + 4p>
(£d (fc + d ) aZf(*1,B_) + i v a p t ( x y y i - h) + ; S ' j '
a ^ t e .+ tp)2 3 2 f (x j .E 8 ) 5 y i
for y3 <^<.3^ + dp and y^ - h<Eg<y^ > dp. So that
adding (42) and (43) giTes
(4*) [ap(ii + 4 ^ + a^tn + dp)] f ( i 3 , y 3 )
" < 2 V p + h av +
28
+ d p ) +• dwdpf(x.j - h ,y^) + d w d p f ( x j , y j - h)
( k + A w ) dP
d v <h + y 2 d A o 2 f < s g » y i )
Ox* * diF
d ^ C h + dD) ^ 2 f ( x , , E 7 ) d ^ h + d B ) 2 ^ 2 f < x , f B 8 )
* '2 0 y 2 2 0 y 2 #
Proa ( 1 4 ) , sine® g(x^ + *v»Tj) » f ( x ^ + ^ y ^ ) and
g(3tj»y^ + <lp) » *(xyyj + dp ) » t i l® :a
(45) D t e j . y j ) - K > f ( l 3 + a » ' * 3 )
+ lid^£(x^»y^ + d p ) + dwdpU(x^ - 3i,y^) + d^d^irCx^y^ - h)J
t h e r e f o r e , us ing (44) and (45) >
" ( X j . y j ) - f ( * 3 , y 3 ) « i V p + A r * h A p { & ( x 3 " h ' y 3 }
- £(x^ - b»yj)J + £u(x^ty^ - k) • f(x^#y^ - h)J
(& + O d ^ 0 2 f ( B 5 # y 1 ) (h + 4 ) 2 9 2 l ( B g , y J * # iiwriniiiiiimiiiii IN UI . ITTF". . Z „ | | L » * J * n i M i n i N i. I I R M I W M I I , R - P IT • i j | r I i M I R F I
2 Ox2 2 a x2
d p ( f e ^ d p ) 0 2 t i x ^ ) ^ (h + d p ) 2 3 2 f (X |»B Q ) J
Qy ( ) y J
| u ( x 3 , y j ) - 5 3 ^ 4 ^ — n s a j C f f e j - h «* 3 >
"* ^(*|j "* ^»yjj)l + ju(Xj»y^ — h.) «•» £ ( X j f y j — ix)|
+ (fcd , + d£ + h? + 2iidw, + d | + hd
+ + h z + 2 M p + a * ) ]
29
and it is easily shown that
(46) lUCx^y^) - ^ (|u(xj - h,y^) - f(x^ - h.y^l
3h2K9 + |U(x^,y^ - h) - - *0!) + —gr» .
Case 4« Suppose (x^y^)^!^ but (x^y^J^B^nB and
is the type discussed in Gas® 3» Step 2. Methods similar
to those used to get (38) give
(47) v V * 1 * *V f^x3ty3^ * + dp)
„ V n 4 ^ 1 * + 32«(*h.Bo) + dwd^dpf(x^,y^ - h) +
v q p ^ ' li—2~ Oy'
dwaaa0(ii + a 0)2a 2f(x ,e10)
'2 O y 2
f•03? ^ ^ y^ dp and y^ <•» h ^ +• dp i and
(48) MJ)(dw + d(1) X(x^fy^) » hdpdqf(xj + dw,y^)
, _, , . M «^(dw + a ) a^tE, ,y ) + - d^y^) + — p w ^ ^ — ~ J r —
_ M » a * ( d w + da)2«>2f<Ei2»yj)^
X2 '
for x j < \ x < Xj + dw and x^ - + < V M d i » S
(47) and (48) gives
(49) [ V a ( h + dp) 4- hdp(dw + d^jJ^x^y^)
« (d^dpd^ + hdpd^ + hdwdp + hdwd^) f(x^,y^)
« ^ p V ( a : i + * M w d p f ( x 3 - V V
4- hdwdqtf(x3,yd + dp) • V p V ( x r y 3 " h )
30
bdpdw^dw + "V o2f(4i.yJ) h < V V 4 w + 4„)2 a2i(2il2»y1)
- 2 — 3 7 " 2 —JP 4,1,^(1 + 40) dwdpd0(la + dc)
2
+ 2 " 3yZ " 2 Jy2
i'roi (15)» since
g(Xj + dw>y^) = f(Xj + dw,yj),
gCx^,;^ + dp) * + dp),
g(*j - dq>y^) = f(xj - dqty^)t then
(50) Utij.yj)
= V p d , + M p d 4 * + Mw dq LMPV<X;J + W + hd»dqf(xi,yj + V + Mwdpf'xj - V y j '
+ dwdpdqD(xj'yj " h>]-
Subtracting (49) from (50) gives
U(x^,y^) - f(x^,y^)
1 s d! d d + lid d! + hd d +• lid 'ci w p q P q. w p V q [ v P
4 # V ' j - «
• " y S f r + V
ha»dp(*« * V * - W * i h * *»> °gf(xrIiq'
^ y
+ ^vdpda + f > ,<JioO
2 Oy2 J'
31
low
|V(xj,jj) - £(xyXj)l
- W l + " A * H r S + K , 4 , ( V A ' ^ V j - h )
- - *>l + I 2 <M pa£ + + ha£dp + 2 * ^ ^
+ +
+ * V p 4 * + + V > , ' ]
and it is easily shown that
(51) ItfUj./j) - fUj.y^liJlo^.yj - h)
- f(xj,y, - h)l + —j2 .
Henee, Oases 1, 2, 3» and 4, above, imply by results
(30), (41)» (#6), and (51) that if (x^y^^B^, then
(52) Iu(x^fyj) - tUyfj)! < (hKxyjj + h) - f(xyj^ + h)|
+ ) (*jty-j - &) - f(*^y^ - 3a)l + lu(x3 - hfyj)
/ xTi 3h%U - f(x^ - .
Consider now the three points (x^fy^ + h), (x^,y^ «• h), and
(xj - la»y )« If gill three points are elements of then
(52) implies
I^Cx^yj) - f(x^ty^)| £-2j| + — y S -
If two points are elements of and one is an element of
\ then (52) and (23) imply
32
, \ , , * , . 0 r 2 h 2 3 h 2 M 0
(53) |u (x^ ,y^ ) - f ( x j » y ^ ) | ^ - | + ^ + —55— + g ^ •
I f on© po in t I s an element of and two are elements o f
Rh t then (52) and (23) imply
t ^ A l ^h%€ (54) | U ( X j , y j ) - f ( x j # y 3 ) | + | + yg £ + — j - l «
I f a l l three po in ts are elements o f then (23) impl ies
tha t
( 5 5> * 2 ^ • » * ! « .
I n any case, (53)» (54) , and (55) imply tha t
, . £ , lrv, . . v, > 3« 3r2h2KJL ItoPvu (56) M x y ^ ) - f ( x r y 3 ) | i -2| + ^ 4 + ^
But (X|»y^) i s an a r b i t r a r y po in t o f @0 tha t (56)
imp l ies
3 r V M . 3 f c V * 5S + — j r "
and
(5?) + .
Subs t i t u t i ng (57) i n t o (23) gives
!u(3t1»y i) - f ( x 1 , y i ) | £ 4- 6h2K2
f o r and the lemma i s proved#
JfePyfR ,£.•,£• I f u a f ( x f y ) i s the so l u t i on of the
M r i o h l e t problem and U(x f y ) i s the approximate so lu t i on
and f ( x , y)CC^ on HUB, then tT(Xfy) converge© to f ( x # y )
as h converges t o zero*
33
Proo£» At points of the proof follows directly from
(28). At points of the proof follows directly from (23)
and (57).
Hote that if |U(x»y) - f(x,y)| is called the error in
the approximate solution U(x,y)# then (28) is an error
bound. Howeverf (28) has very little value as an error
bound since M2 and are known to exist, but, in general,
cannot be evaluated.
CHAPTER III
SOLUTION OP THE SET OF
LINEAR EQUATIONS
The final step in the procedure for finding the approx-
imate solution to a Diriohlet problem is the solution of
the set of linear equations which can he accomplished by
many methods. These methods are usually divided into direct
and iterative. The matrix notation for this set of linear
equations is AX « 0, where A * (a^»a^) is the I x I matrix
composed of the coefficients of the H unknowns in the system
I of linear equations 2 5 * x4 « e., 1^i£lf, X is the
3*1 a*3 3 i
column vector of the 1 unknowns x^, Xg, • «. » x^ and C is
the oolumn vector of the constants, c1# c2, • , c^, of
the equations. Direct methods, of which Gaussian elimination
together with its many variants are typical, yield the exact
answer in a finite? number of operations if there is no
round-off error. Usually the procedure for a direct method
is complicated and nonrepetitious. Iterative methods con-
sist of the repetition of a simple procedure, hut usually
only give the exact answer as a limit of a sequence, even
in the absence of round-off error.
Iterative methods are preferred for solving large
spars© systems AX » C since they usually take full advantage
34
35
of the numerous zeros of A in storage and in operation.
They minimise round-off error trouble because they tend to
be self-correcting, which is not ordinarily a characteristic
of direct methods#
In any iterative method, to solve the nonsingular sys-m
tern AX * 0 (the solution of which is A C ) , a sequence of
column vectors ••• is defined with the
hop® that as The Gauss-Seidel method is one such method. In this method the termt composed
(V) (V)
of p x2 » ••• » » of the sequence of column vectors
is calculated with the use of the following equation.
» - s - JcE If i £H. 1 J»1 ai,i 3 j«i+l ai,i * aifi
fhis indicates that a^^ for l£l£N cannot he zero. She
first column vector in the sequence is a guess to the
solution made by the user* 9?he sequence produced by the
0auss~Seidel method does not always converge. Proof will
now be considered that every set of linear equations asso-
ciated with the procedure for the approximate solution to
a Mriohlet problem can be solved by the Gauss-Seidel
method.
Definition 3.1. The S x I matrix A « (&±,a^) has
diagonal dominance if
I*1-1' -
5*1
36
for all liiil with strict inequality for at least one i.
Definition 3.2. 3!he I x If matrix A m ( a ^ ) is
reducible if the set of integers £~1» 2, ... 9 bJ is the
" t o 0 f t w 0 8 e t B 3 8,111 1 B»011 t h a t ai,j - 0 f o r
all i in S and 3 in 3?*
Ufa® following theorem can be found in most book© which
diecuss iterative methods for solving linear simultaneous
equations.
Theorem 3.1. If the M x $ matrix A has diagonal
dominance and is not reducible, then the Gauss-Seidel
method converges for any initial vector approximation X ^ #
Sheorem 5.2. The G-auss-Seidel method converges for
any set of linear equations produced by the procedure of
approximating the solution of the Dirichlet problem with
any initial vector approximation .
Proof, First» A must be shown to have diagonal domi-
nance. Consider row i of A which is associated with the
equation used to determine u^ as described in Steps 2 and
3 of Chapter II* If then four cases exist,
fhe manner in which the matrix A was constructed causes
the coefficient of uA to be the diagonal element ai
and the coefficients of u^, up, u f uB, if they exist, are ai,w* ai#p*
ai»q» ai»z* respectively# From Case 1, Step 2,
(34w + h>ui - dwup - 4 A - V z - hu2-
so that the only nonzero elements of row i of A are
37
(3dw + h), - d^, - dw, and - dw for which
+ n| > | - I + I - dw I + / - dw |
or
K , l l >
m
]?rom Case 2, Step 2,
(2d d + hd + hd )Ui - d d a - d d u » hd u«, + hd u«« v w p w p' i w p q. w p z p 2' Y 3 1
so that the only nonzero elements of row i of A ar«
(2d.,dr. + hd__ + hd„)f - d-d. and - d-.d for whioh
w p w p w p w p
I 2 V p + + Mpl>l- V p l + V p 1
or
l ai.i J >
am
From Case 3, Step 2,
( V p d q + hiwdp + " V * + M p V u i " V p V k
• hdpdqu2, + hdv/dqu3, + hdwdpu4,
so that the only nongero elements of row i of A are
( V p 1 ! + h V p + " V , + & < W m a " W i f o r whio11
1 W o . + h V p + hV(i + Mp dq 1 ? | - V p d , 1
01*
? j5'al.a1-
}*i
38
If is on B» then
ui m 8^xi,yi^
so that the only nonzero element of row i is 1 and
|l|>0 or
H
•ai,i' >
3**
if "^en
4ni - "w - up - uq - u2 " 0
so that the only nonzero ©laments of row i of A are 4, - 1,
- 1, - 1# and - 1 for which
|#| l| + 1 * it + I » It + I* 11
or
If
K . J - j g K . j ) -
m
There must he at least one lattice point which is an
element of B^, so that strict inequality does hold for
at least one i and therefore A has diagonal dominance.
Consider now whether A is reducible or not# By the
way A was constructed all diagonal elements are nonzero,
ai*i ^ 8 0 "^at n o ©lament i of 3 can also he an element
of f if A is reducible. If two lattice points (x^»y^) and
(x^»y^) are adjacent then u± is used in the calculation of
u^ and is used in the calculation of u^» So that row i
has a nonzero element in column j and row j has a nonzero
39
element in column i. Then if A is reducible, one of S and
$ must contain both i and J since # 0 and * 0»
flie assumption was made earlier that any two lattice points
can be connected by a path moving along lattice points from
on® point to an adjacent point. Shua the above procedure
can be repeated a finite number of times to show that any
two numbers i»3 must both be in on© of the two set® 8 and
T. therefore, if one of the integers £l, 2, ... , is in
8, then all of them must bt in 3 and likewise for f« Thus#
A is not reducible since 3 and f must both be nonempty for
A to be reducible# By Theorem 3*1 the §suss-Seidel method
converges for matrix A and the proof is complete#
Proof is given in many books that the maximum and
minimum value® of the solution of the Diriohlet problem
occur on the boundary B* Thus, an initial vector approxi-
mation can be best mad# by considering values on 3.
BXBLIGGMPHir
Porsythe, George E, and Wasow, Wolfgang a,» iouatioftSi
Greenspan, Donald,
Varga.^Hichard^S,, i S t i M S s I e w Jersey,
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