Approximating a linear operator equation using a generalized Fourier series: applications

Approximating a linear operator equation using a generalized Fourier series: applications T. V. HROMADKA II Research Associate, Princeton University, USA C. C. YEN Hydrologist, Williamson and Schmid, Irvine, California, USA G. F. PINDER Professor and Department Chairman of Civil Engineering, Princeton University, Princeton, NJ 08544, USA

The theory of generalized Fourier Series can be applied to the approximation of linear operator relationships. To demonstrate the computational results in using this approach several example problems where analytic solutions or quasi-analytic solutions exist are used for the testing the generalized Fourier Series technique. Applications include two-dimensional problems involving the Laplace and Poisson equations, tests for variation in results due to inner product weighting factors, and application to nonhomogeneous domain problems.

INTRODUCTION

Bf(s) equal the almost everywhere (a.e.) radial limit of f a t

INNER PRODUCTS FOR THE SOLUTION OF LINEAR OPERATOR EQUATIONS

the point s on F, with appropriate domains. The next step is to construct an operator Tmapping its domain D(T) = D(L) ¢qD(B) into L2(c1([2)) by 2

The general setting for solving a linear operator equation with boundary values by means of an inner product is as follows: let [2 be a region in R m with boundary F and denote the closure of [2 by cl(f2). Consider the Hilbert space L2(c1([2), do), which has inner product (f,g) = f f g dl~ and where do is the measure defined below. (This is a real Hilbert space. For the complex version, use the complex conjugate of the function g in the integral.) The way to construct the necessary inner product for the development of a generalized Fourier Series is to choose the measure/a correctly; that is let/~ be one measure/~x on ~2 and another measure /a2 on F. One natural choice for a plane region would be for tax to be the usual two-dimensional Lebesque measure d V on [2 and for /a2 to be the usual arc length measure ds on F. Then an inner product is given by ~

Tf(x) = Lf(x) for x in [2 (2)

Tf(s) = Bf(s) for s on F From (2), there exists a single operator T on the Hilbert space L2(c1(~2)) which incorporates both the operator L and the boundary conditions B, and which is linear if both L and B are linear. An application of this procedure using the Complex Variable Boundary Element Method (CVBEM) is given in Hromadka et al.3 In that study, L f = ~72fand Bf is the radial limit of f on F. Consider the inhomogeneous equation L f = g~ with the inhomogeneous boundary conditions B f = g2. Then define a function g on cl([2) by g = g l on

(3) (f,g)=ffgdV+fffgds ~2 v

(1)

Consider a boundary value problem consisting of an operator L defined on domain D(L) contained in L2([2) and mapping into L2(~2), and a boundary condition operator B defined on a domain D(B) in L2([2) and mapping it into L2(V). The domains of L and B have to be chosen at least so that for f i n D(L), Lfis in L2([2), and for f i n D(B), B f is in L2(I'). For example we could have L f = V2f, and

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Engineering Analysis, 1987, Vol. 4, No. 4

g=g2onI" Then if the solution exists for the operator equation

Tf = g

(4)

the solution f satisfies V 2 f = g l on [2, and f = g 2 on F in the usual sense of meaning that the radial limit of f is g2 on I'. One way to attempt to solve the equation T f = g is to look at a subspace Dn of dimension n, which is contained in D(T), and to try to minimize IITh --g[l over all the h in D n such as developed in Hromadka et al. 4

© 1987 Computational Mechanics Publications

Approximating a linear operator equation using a generalized Fourier Series: T. V. Hromadka H et al. PURPOSE OF PAPER

Y

In this paper, the application of the approximation procedure to a set of simple linear operator problems is presented. Thirteen simple but detailed example problems are used to illustrate the approximation results obtained by the method when applied to practical problems. The theoretical development of the numerical method is included in a companion p a p e r : This paper focuses on the topics of weighting factor selection and modeling sensitivity, effects of additional basis functions on computational accuracy, and the effects on modeling results due to the addition of collocation points.

SENSITIVITY OF COMPUTATIONAL RESULTS TO VARIATIONS IN THE INNER PRODUCT WEIGHTING FACTOR The inner product uses a weighting factor, e, to weight the approximation effort in satisfying the PDE and BC values by

(u'v)=elu'vdr+(1--e)ILuLvd~ I"

[2

(o.,)I

(,.,)

10,3/41 (0,I 1 2 ) ~ (0,1/4)~ (o,o)

X 0,o)

Figure 2

q~= 5.4583 + 2x + 1.75y. For comparison, the least squares method can be used to minimize the function

The effects of varying e between 0 and 1 is shown in the following simple application.

X = f (k~ + k2x + kay -- ~be)2 dP F

Example problem no. 1 Let ~ = 2x2y + y2 + x + 6 where 724 = 2 + 43; on the unit square domain. Figure 1 depicts the problem domain and boundary conditions. First, define ( 1, x , y , xy, x 2) to be the set of basis functions, and use e = 0. The resulting approximation function is ~ = 2x 2. Applying the linear operator V 2 on ~, we obtain V2~ = 4. The graph of V2¢ = 2 + 4y on the unit square domain is depicted in Fig. 2. From Fig. 2, it is seen that indeed for e = O, the best approximation for 72~0 is 724 = 4 which coincides to 72~ = 4. Thus, the V2~ approximation satisfies the linear operator on the least square sense (L 2) for e = 0 for the given set of basis functions. Second, we choose ( 1, x , y ) as the set of basis functions and use e = 1. Then the resulting approximation function is

with respect to parameters kl, k2, and kaalong the boundary of the unit square domain. From Fig. I, one can write X as follows: 1

X=

f (kl+k2x--x--6)2dx x=O

,f

1

(kj + k2 + kay --2y __y2__ 7)2dy

y=O 1

+

f

(kl + k2x + k a -- 2x 2 -- x -- 7) 2 dx

x=O 1

+ f (kl+k3y--y2--6)2dY y=O

Minimizing X with respect to kl, k2, and k3, we obtain

(o~) ~~U'2xz+x+7 !ii!ililililiiiiiiili;iil;ili;iiii!;iiiiililiiiiiiii::l ii: °'') ,r-,,÷,..

iiiiiii ,iiiiii iiiiii',iiii'il• ,

jiiil iiiii i i i ! i ! i i ; i i i i i ' i i i i i ! i i i i l ) (0.0) (~'~'X'I'" ~ (I,0) Figure 1. Domain and boundary conditions for example problem no. 1

kl = 5.4583 k2 = 2.0 kz = 1.75 which verifies that the approximation function is the least square fit (L z) with respect to the problem boundary conditions.

Example problem no. 2 Let us consider a Poisson problem 724 = 2 + 2y + 12x 2 on a unit square domain with boundary conditions Cb = x 2 + yZ (Fig. 3). By choosing the following set of basis functions (x 2, x 3, x 4, y2, ya, y4, xy2, yx 2, x2ya), we obtain the approximation

Engineering Analysis, 1987, Vol. 4, No. 4

215

Approximating a linear operator equation using a generalized bburier Series." T. V. Hromadka II et al. function q)b = x2 + y2 f o r e = 1 and q ~ = x 2 + ] y a + x 4 for e = 0 with V2q~s-z= 2 + 2y + 12x 2. This verifies the two extreme values of the weighting factor e = 0 and 1 in satisfying the PDE on the domain and satisfying the BC values on the boundary, respectively.

(-I, 1.7320)

SOLVING TWO-DIMENSIONAL POTENTIAL PROBLEMS In the following are application problems and computed results in solving the Laplace equation in two dimension. Because the Laplace equation is a linear operator, the L 2 approach is used.

• !---x •

(-,,o)

;

"

"

Example problem no. 3 The flow of an idea fluid around a 90 ° bend can be expressed by the analytic function G)=Z

2

= (x 2 _ y 2 ) + 2xyi Since the state variable function ¢ = (x 2 _ y Z ) and the stream function ~b = (2xy) are of polynomial forms, the approximation functions for the state variable and stream functions are found to result in the exact solutions regardless o f the value of the weighting factor e, (0 < e < 1). Figure 4 depicts the problem domain and nodal point placement used for this application.

~

/~i~

10.S,-0.8661

(-I,-I.73205)

I_ t-

°-,

_1 -7

Figure5. Domain and nodal placement for example problem no. 4

Example problem no. 4

Y

Consider the St. Venant torsion problem for an equilateral triangular section of Fig. 5. The analytic solution for ¢ ( x , y ) is given by

~)b =xz'l'l "~

~ ( x , y ) = (x a -- 3xy2)/2a + 2a2/27

O,i)

(o,~)

Figure 5 depicts the nodal placement and boundary conditions for the case of a = 3. The L 2 approximation function is found to result in the exact solution.

~b = ys.---.~ A;~(~)~.+2y+12x 2 ,,~-,.~)b,,l+y dR on f]

Example problem no. 5 = X

(0,0) ~)b=XZ J

(I,0)

Figure 6 depicts the nodal placement and boundary conditions for a mechanical gear problem. The resulting L 2 approximation function is = 9.518 + 0.3131x + 0.1812y + 0.7686xy + 0.2219x 2

Figure 3. Domain and boundary conditions for example problem no. 2

-- 0.2228y ~ -- 1.018x2y + 0.3397y a -- 0.2463xay + 0.2124x~y 2 + 0.2419xy 3

Y

(0,1)

q~= xZ-YZ1on ~ U F ~ : zxy

I,.

(I,I)

:

for a weighting factor of e = 0.5. Table 1 compares the results from a Complex Variable Boundary Element Method or CVBEM 6 model and the L 2 approximation function for the interior nodal points, and the defined operator relationship.

Example problem no. 6 Ideal fluid flow around a cylinder has the analytic function definition o f

f~

1 co(z) = z +Z

(o,o)

z

z

--

(~,o)

=---X

Figure 4. Domain and nodal placement for example problem no. 3

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EngineeringAnalysis, 1987, Vol. 4, No. 4

The state variable (potential) function can be expressed as ~(x,y)

= x

1+

Approximating a linear operator equation using a generalized Fourier Series: T. V. Hromadka H et al.

®

@ ,o.o

and = - 1.704 + 2 . 5 8 7 x + 1.02y - 2 . 1 1 5 x y - 1.003x 2

'°.y. /

.

®

,o.

IP

+ 1.009y 2 + 1.289x2y - - 0 . 4 3 0 9 y a + 0 , 1 4 6 1 x a y - 0 . 2 2 5 7 x 2 y 2 - 1.452xy a

''°(b

.- oo

for state variable and stream functions, respectively. Figure 7 shows the a p p r o x i m a t i o n relative error b e t w e e n a CVBEM m o d e l and the L ~ a p p r o x i m a t i o n f u n c t i o n values.

Example problem no. 7 Ideal fluid f l o w a r o u n d a cylinder in a 90 ° b e n d has the analytic solution of 6o(z) = z 2 + z -2. The state variable f u n c t i o n ¢ and stream f u n c t i o n ¢ can be expressed, respectively, as 1 q~ = ( x 2 - - Y 2 ) [ 1 + (x2 + y2)2 ]

I_

~.

~J

_

I

~

I

and 1

¢ = 2xy[1

Figure 6. Domain and boundary conditions for example problem no. 5

(x2+y~2 ]

The L 2 a p p r o x i m a t i o n functions ( e = 0.5). for the state variable f u n c t i o n q~ and t h e stream f u n c t i o n ~k are given q~ = 3 . 1 9 7 x -- 3.197y - - 1.828x 2 + 1.8283, 2 + 1.894x2y

Table 1. Comparisonof L ~and CVBEMcomputational results

- - 1.894xy ~ + 0 . 6 2 9 4 x 3 - 0 . 6 2 9 4 y a + 0 . 4 6 1 2 x 3 y

Interior nodal points x

y

CVBEM

~

0.321 0.643 0.964 1.286 1.607 0.492 0.985 1.477 1.970 2.462 0.470 0.940 1.410 0.433 0.866 1.299 1.732 2.165 0.383 0.766 1.149

0.383 0.766 1.149 1.532 1.915 0.087 0.174 0.261 0.347 0.434 0.171 0.342 0.513 0.250 0.50 0.75 1.0 1.25 0.321 0.643 0.964

9.7842 10.0391 10.2610 10.3668 10.2417 9.857 10.0992 10.2749 10.3659 10.2439 9.8476 10.0944 10.2830 9.8329 10.0839 10.2856 10.4917 10.7489 9.8119 10.0659 10.2806

9.752 10.04 10.28 10.39 10.31 9.752 10.04 10.28 10.39 10.31 9.767 10.05 10.27 9.772 10.05 10.27 10.42 10.63 9.767 10.05 10.27

-- 0 . 4 6 1 2 x y a V2~ 8.03 8.56 6.36 -5.79 -2.79 2.35 1.19 1.18 -2.98 -1.76 -3.55 -1.32 -4.50 -5.88 -2.26 -6.7 -1.39 -2.39 -4.34 -1.49 -4.87

× 10-s × 1 0 -4 X 10 -4 X 10-4 X 10-a X 1 0 -4 X 10 -~ X 10 -~ × 10-3 X 10-3 X 10-4 × 10 -~ X 10-3 X 10-" X 10-~ X 10 -3 X 10-~ X 10-2 X 10-4 X 10 -3 × 10-3

and = -- 3.192 + 3 . 3 8 9 x + 3 . 3 8 9 y - - 4.31 lxy + 1.352x 2 + 1.352y 2 + 2 . 1 2 1 x 2 y + 2 . 1 2 1 x y 2 - 0 . 7 1 3 6 x a - - 0 . 7 1 3 6 y 3 -- 0 . 7 5 3 6 x 2 y 2 + 0 . 1 2 5 9 x 4 + 0 . 1 2 5 9 y 4 on the d o m a i n shown in Fig. 8. In c o m p a r i s o n o f the a p p r o x i m a t i o n to exact values it was f o u n d that the relative error is high along the c i r c u m f e r e n c e o f the cylinder. I f we i n c l u d e d some singular terms, e.g. l/x, 1]y, l[x 2, l[y 2. . . . . into the set o f basis f u n c t i o n , the a p p r o x i m a t i o n f u n c t i o n for the state variable ~ b e c o m e s = 1.892x -- 1.892y - - 1.432x 2 + 1.432y 2 + 0 . 4 4 0 9 x 2 y

-- 0.4409xy ~ 0.01721 xa

and the stream f u n c t i o n can be expressed as

~ ( x , y ) = y (1

x21y~)

0.434

x

0.434

~- y

0.1237

0.1237

x2

y2

0.01721 I - - ya

By increasing the b o u n d a r y nodal points f r o m 32 to 60 and the terms o f the set o f basis functions, we o b t a i n respectively the a p p r o x i m a t i o n f u n c t i o n s q~l = 3 . 3 2 4 x - - 3 . 3 2 4 y -- 2.162x2 + 2 . 1 6 2 y : + 0 . 3 7 2 1 x 2 y

where

w(z) = ¢ ( x , y ) + i~k(x,y) The L 2 a p p r o x i m a t i o n f u n c t i o n s (e = 0.5) are = 1.704 + 0 . 9 7 9 5 x -- 2.587y + 2 . 1 1 5 x y -- 1.009x 2 + 1.003y 2 -- 1.289xy 2 + 0 . 4 3 1 x a + 0 . 1 4 5 2 x a y + 0.2258x2y 2 -- 0.1461xy a

. 2.336 2.336 - - 0 . 3 7 2 1 x y 2 + 0 . 6 8 9 6 x ° - - 0.6896 ° -+ - x y 0.9113 + ~ x

0.9113

0.1609

y2

x3

0.1609 ~

ya

0.01315 q"

X4

0.01315

y4

Engineering Analysis, 1987, Vol. 4, No. 4

217

Approximating a linear operator equation using a generalized Fourier Series." T. V. Hromadka H et al.

_

:

,

,

. ®

®

I0

C--5 5

-5

m.__

=e ~I.0

-I0

Figure 7.

Domain, nodal placement and relative error for example problem no. 6

~

k53)

The relative e r r o r ( ~ E X A C T - - ~/t~EXACT) of the approximation function q~l is smaller than the approximation function 62.

Example problem no. 8 Figure 9 depicts the flow net for soil-water flow through a homogeneous soil as computed by the CVBEM. The L 2 approximation function of the state variable function q~ is given by = 24.0 -- 0.6549x -- 0.0989y - - 1.864xy -- 0.06852x 2 + 0.118y2 + 2.4 x 10-ax2y + 1.676 x 10-axy 2 + 1.878 x 1 0 - a x a - - 3.129 x 10-4y 3 for e = 0.5. Table 2 shows the interior nodal point values for both the CVBEM model and the L 2 approximation function.

Figure 8

The L 2 techniques can be applied to other linear operators. Hromadka and Pinder s examine a linear integral equation along with two PDE problems. This section considers nonhomogeneous problems involving potential theory.

and 6z = - - 0.01232 + 3.905x -- 3.937y + 0.3525xy - - 2.892x 2 + 2.736y 2 + 2.686x2y -- 3.14xy 2 + 1.126x 2 - - 0 . 8 9 5 y 3 - - 1.017xay + 0.299x2y 2 + 0.9077xy 3 -- 0.05475x 4 + 0.0745x4y + 0.049x3y 2 -- 0.1336x2y a

218

APPLICATION TO OTHER LINEAR OPERATORS

EngineeringAnalysis, 1987, Vol. 4, No. 4

Example problem no. 9 Consider the linear operator, V2(0,

72(0 = Txx a2---~+ Tyy 02(0 ax 2 ay2 on the unit square domain with boundary condition

Approximating a linear operator equation using a generalized Fourier Series: T. K Hromadka H et at Cb = 6 + y2 + xZy (Fig. 10). The L 2 approximation functions for different Yx~ and Tyy values are shown in Table 3.

(0,1) ~'~)b:7+2xl

,w--(;bb--6+yZ+2y

~)b:6+~

-24

.O,t)

-ZO

=

(t,o)

(0,0)Cb=S:

x

Figure 10

.IIS

Example problem no. 10 - Poisson problem Consider

-I0

¢ = 6 + y 2 + 2x2y on the unit square domain (Fig. 1) with the linear operator ~2¢

.5 .4

V2¢ =

a2¢ + --

~x 2

0y ~

such that V2¢ = 2 + 4y

0 l

,,,

5

tO

15

I

I

I

The L 2 approximation functions are listed in Table 4 for the various values of the weighting factor, e.

I

Figure 9. Streamlines and potentials for soil-water flow through a homogeneous soil f o r example problem no. 8 Table 2.

Table 3.

Example problem no. 11 Consider a linear operator,

V2~ = Txx 0~¢+ 0x 2 Try ay2

CVBEM and L 2computed results/'or example no. 8

x

y

CVBEM

4 4 4 4 4 8 8 8 8 8 12 12 12 12

4 8 12 16 20 4 8 12 16 20 4 8 12 16

19.9233 20.3408 20.9110 21.5258 22.1359 15.6543 16.5817 17.8253 19.0937 20.2656 10.7715 12.5019 14.7428 16.8199

on the domain of example problem no. 10, we obtain 20.14 20.35 20.91 21.70 22.61 15.34 16.04 17.30 19.01 21.04 10.11 11.6 13.87 16.8

Vzqb = 2Txx + 2" T y y ' y The L 2 approximation functions for different values of

Txx and Tyy are listed in Table 5 for e = 0.5. The maximum relative error on the boundary is of magnitude 10-4 and on the interior is of magnitude I0 -s for the Txx ~ Tyy.

Example problem no. 12 Consider the example problem no. 10 on the problem domain of example problem no. 4 (Fig. 5), the L 2 approximation function provides the exact solution regardless the value of the weighting factor (0 < e < 1).

L 2function coefficients and relative errors ]'or example no. 9

Txx

Tyy

1

x

y

xy

x2

y2

x2y

xy2

x3

y3

Maximum relative error on boundary

1 2 1

1 1 2

5.856 5.878 5.849

0.8996 0.8277 0.8657

0.6616 0.3158 0.8637

0.2092 0.4275 0.1007

-0.8946 -0.8277 -0.8657

0.8959 1.656 0.4333

1.791 1.572 1.899

=0 -~0 ~0

=0 -~0 =0

-0.5967 -1.048 -0.3165

10-2 10-z 10-2

Maximum relative error on operator relationship 10-3 10-3 10-3

EngineeringAnalysis, 1987, Vol. 4, No. 4

219

Approximating a linear operator equation using a generalized Fourier Series: T. V. Hromadka H et al. Table 6.

Example problem no. 13 Consider

Example problem no. 13. L2approximation results

Number of nodes

~b = s i n x + y x + 10 + c o s 2 y o n a u n i t square d o m a i n w i t h a linear o p e r a t o r V2q~, s u c h that

r

a

8

16

11 + 0.999x + 0.049), + :O' -- 0.013x 2 -- 1.29)' ~ + 0x2y + Oxy 2 - 0.14x 3 + 0.53y 3

8

25

11 + 0.996x + 0.0443' + xy--O.Ollx 2 --1.28)': + Ox2y + Oxy ~--0.14x3 + 0.53), 3

16

16

11 + x + 0.047y + xy --0.016x 2 --1.293' 5 + 0.9 × l O - S x 2 y - 0.64 × 10-4x.y2-- O.14x 3 + 0.53y 3

16

25

16

25

11 * 0.998x + O.041y + xy --0.013x 2 -- 1.28y 2 -- 0.86 × 10-4x2y--0.43 × lO-Sxy 2-- 0.14x 3 + 0.533, 3 11 + 0.992x + 0.00086543, + 0.9995xy + 6.07 × 10-Sx 2 - 1.07y 2 + 4.62x 10-4x~y + 5.5 2x lO-4xy2--O.1837x 3+ 0.1806) '3 - 7.82x 10-~2x3y --5.48 x 10-4x2) ,2 - 7 . 8 3 2 x 10-12X.,Vs+ 1.986x4+ 1.7293 '4

V24~ = 2 - - 4 c o s 2 y - - s i n x The f u n c t i o n ~ c a n be e x p a n d e d i n t o a n i n f i n i t e series as

x2n

oo

q~=x

~ (--1) n t - y x + 10 n:O ( 2 n + 1)! -

+ {1

-

1 ~

(2y)2k'~,

or x3 ~b=ll

x5

x7

5!

7!

+x+yx----+

I-...

3! _ y 2 .j_ y 4 3

2 y 6 .~_ . . . 45 _

_

T a b l e 6 lists t h e a p p r o x i m a t i o n f u n c t i o n s for d i f f e r e n t n o d a l densities p l a c e m e n t s a n d basis f u n c t i o n sets. T h e m a x i m u m relative e r r o r a l o n g t h e b o u n d a r y is o f t h e m a g n i t u d e o f o r d e r 10 -4. In all cases, e = 0.5.

Table 4.

L2solution of Poisson problem

d 0

x 2+2x2y or y 2 + 2x2y

O
6 + y 2 + 2x2y 6 + y 2 + 2x2y

e=l

Table 5.

Example problem no. 11. L2approximation coeJyicients

Txx

Try

1

1

6 + y2 + 2x~y

2

1

6 . 0 1 4 - - 8 : 5 × 10-2x -- 0.305.,v + O.171yy + 8.5 × 10-2x 2 + 1,829y 2 + 1.829x2y -- 0.553y 3

T h e generalized F o u r i e r Series m e t h o d has b e e n applied to several linear o p e r a t o r r e l a t i o n s h i p s t o d e m o n s t r a t e t h e p r o c e d u r e s involved. Sensitivity tests are s h o w n w h i c h e x a m i n e t h e w e i g h t i n g f a c t o r used, t h e n u m b e r a n d c h o i c e o f basis f u n c t i o n s , a n d the n u m b e r a n d p l a c e m e n t o f c o l l o c a t i o n p o i n t s . Currently, the c o m p u t e r p r o g r a m used t o d e v e l o p t h e p r o v i d e d s o l u t i o n s is small (less t h a n 3 0 0 lines o f F O R T R A N c o d e ) a n d c a n be a c c o m m o d a t e d o n most personal computers. Use o f this n e w n u m e r i c a l m e t h o d a p p e a r s t o o p e n t h e d o o r to p r o v i d i n g h i g h l y a c c u r a t e s o l u t i o n s t o linear operat o r p r o b l e m s . H o w e v e r , t h e e x t e n s i o n o f t h e m e t h o d to a p p l i c a t i o n s involving n o n l i n e a r o p e r a t o r r e l a t i o n s h i p s requires f u r t h e r research. REFERENCES 1

1

2

5.993 + 3.37 x 10-2x + 0.174y - 6.75x10-~xy --3.37 x 10-2x 2 + 0.517) '5 + 2.067x2y + 0.3221y 3

2

2

6 + y2 + 2x:y

220

CONCLUSIONS

Engineering Analysis, 1987, Vol. 4, No. 4

2 3 4 5

Birkhof, G. and Lynch, R. Numerical solution of elliptic problems, SIAM Studies in Applied Math. 1984 Davis, P. J. and Rabinowitz, P. Advances in Orthonormalizing Computation, Academic Press, 1961 Hromadka II, T. V., Lai, Chintu and Yen, C. C. A complex boundary element model of flow-field problems without matrices, submitted to Engineering Analysis, 1986 Hromadka II, T. V., Pinder, G. F. and Joos, B. Approximating a linear operator equation using a generalized Fourier Series: Development, in-review, 1986 Hromadka II, T. V. The Complex Variable Boundary Element Method, Springer-Verlag, Lecture Notes in Engineering, 1984