Algorithms for the approximation of the generalized Hubbell rectangular source integrals

)

Radial. Phw.Chem.Vol. 43.No.5.00.497-502, 1994 c0pyright 6 1994 &vier SC&C ~td Printed in GrcatBritain.Allrishtsrcscmd 0969-806X/94 ~$6.00+0.00

Pergamon

ALGORITHMS FOR THE APPROXIMATION OF THE GENERALIZED HUBBELL RECTANGULAR SOURCE INTEGRALS LRDAGAL~,' SHYAM L. KALLA~ and C. LRUBNER' ‘Centro de Investigation de MatemPtica Aplicada (C.I.M.A.), Fact&ad de Ingenieria, Universidad de1 Zulia, Apartado de Correo No. 10.482, Maracaibo, Venezuela and 21nstitut filr Theoretische Physik, Universitit Innsbruck, TechnikerstraBe 25, A-6020 Innsbruck, Austria (Received

30 July 1992, accepted

19 November

1992)

Abstract-In this paper, we derive simply-structured, single term approximation formulae for the general&d Hubbell rectangular source integrals involving hypergeometric functions. These results are sufficiently accurate to allow a semi-quantitative assessment of the dependence of these functions on the parameters. The corresponding result for the Hubbell integral is derived as a particular case. Tables are given to compare the numerical values derived from the approximate formulae with those given earlier by Kalla et al. walla S. L., Al-Saqabi B. and Conde S. (1987a) Hadronic J. 10,221-230.1.

1.INTRODUCTION

Hubbell et al. (1960) have obtained a series expansion for the calculation of the radiation field generated by a plane isotropic rectangular source (plaque), in which the leading term is the integral

wherey>B>O;a,b,c>0;1>-l;(ifb-+co,then - 1 c L < 1); and 2F,(a, 8; y; x) is the gaussian hypergeometric function. We observe that

u,b,l,O

*

[

a >o,

b >o.

(1)

This integral has found important applications in metrology situations where the mechanical contact is not allowed. Typical examples can be found in a variety of medical, agricultural and industrial applications which make use of radiometric gauging and process control. In fact, these and other metrology devices are based on the principle of detecting radiation transmitted by a source of charged particles (electrons, neutrons) and photons (X-rays, y-rays). The integral (1) has been calculated, for selected values of a, b, by means of the everywhere convergent series as given by Hubbell et al. (1960) or by more computationally tractable, but empirical, approximations; the latter have been devised, in particular, for engineering applications (Ghose et al., 1988). Various generalisations of equation (1) have been given by several authors (Galue, 1991; Kalla et al., 1987a; Saigo and Srivastava, 1990). Kalla ef al. (1987a) have defined and studied a generalisation defined by the integral

1’3

=h(a,b)

1 By selecting suitable values for the parameters a, /l and y, equation (2) can be reduced to different integrals with potential applications in radiation-field problems of specific configurations of source, barrier and detector. In addition to their usefulness in y-ray irradiation technology (e.g. Donovan, 1961; Sivinski and Ahlstrom, 1984) design studies and civil defense radioactive fallout roof-dose predictions (Hubbell and Spencer, 1964), these results can also be useful in illumination (e.g. Boast, 1942; Walsh, 1958) and heat-exchange (e.g. Seibert, 1928; Hottel, 1942; Jakob, 1957) engineering design studies. In Gald (1991) and in Kalla et al. (1991) a generalisation of the Hubbell integral is given in the following form: 929

2

withc>O,O-1,1,>-1,andif b+co,then-1<1<2u-2p-l.Whenp=O,the function I reduces to H [

a, c, 6b, l% 7Iz1

defined by equation (2).

498

LEDA GALL

ef al.

Table I. Comparison of values of

Table 3. Comparison of values of

for some values of a, b, e calculated by the single term approximation algorithm (IS) (approximate value), and by the series expansion algorithm (Kalla ef al., 1987a, p. 227) (calculated value)

a, b, c, 0 [ I, 0.5, 1.51 for some values of o, b, e calculated by the single term approximation algorithm (24) (approximate value), and by the series expansion alnorithm fKalla et al.. 1987a. D. 227) (calculated value) H

a

b

c

0.1 0.1 0.2 0.2 0.5 0.6

0.1 0.5 0.5 0.8 0.5 0.5

0.5 1.0 2.5 2.0 1.0 1.0

Approximate value 0.1556786161 0.346031029 0.2979704472 0.5334386082 0.15456052 0.17682772

x x x x x x

lo-’ 1O-2 lo-’ lo-* 10-l 10-l

Calculated value

a

b

e

O.l5707164D-02 0.36781998D - 02 0.30671488D - 02 0.57586007D - 02 0.17188506D -01 0.20067469D - 01

0.1 0.1 0.2 0.2 0.5 0.6

0.1 0.5 0.5 0.8 0.5 0.5

0.5 1.0 2.5 2.0 1.0 1.0

In terms of Appell’s double hypergeometric function, this integral takes the form

Aooroximate value 0.1566316 0.3470936 0.2994303 0.5367042 0.1660033 1.195406

x 1O-2 x IO-* x lo-* x IO-* X 10-l x 10-l

Calculated value O.l5707164D-02 0.36781998D - 02 0.30671488D -02 0.57586007D - 02 0.17188506D -01 0.20067469D - 01

One immediate special case of equation obtained when p = 1 = q and m = 2,

(7) is

I[&I$J7P] ua = z

r(Y) r(lor(Y

- B)

For selected values of the parameters, the integrals H, I and S may be calculated by means of the series

X

is a generalised hypergeometric function.

expansions given in (Kalla et al., 1987a, 1991; Gal&, 1991; Saigo and Srivastava, 1990). However, sometimes we are not interested in the exact values of these integrals, but we need to have an idea about their approximate behaviour for selected ranges of the parameters involved. In the present work we derive some simplystructured, single term approximations for the functions S and I. They are sufficiently accurate to allow a semi-quantitative assessment of the dependence of these functions on the parameters, which is usually an important first step in physical applications. As a particular case, we obtain two single term approximation algorithms for the H-integral. Similar approximation algorithms can be developed for other hypergeometric integrals. Tables for the values of H for selected parameters (Tables l-4) as calculated from the approximation formula (15) and formula (24) are given (a = 1) and compared with the exact values given earlier by Kalla et al. (1987a). We observe that the results (15) and (24) provide a relatively good approximation.

Table 2. Comparison of values of

Table 4. Comparison of values of

x tfl- ‘(x2 + c + a*t)-” dt dx

(5)

(6) On the other hand, Saigo and Srivastava (1990) have studied a family of integrals of the form,

where min{a, b, c} > 0, 1 E ( - 1, 1) and

H[ z,G:5f

H[~:~OY5I 11 for some values of (I, b, c calculated by the single term approximation algorithm (15) (approximate value), and by the series expansion algorithm (Kalla et al., 1987a, p. 228) (calculated value) (I

b

e

0.1 0.1 0.2 0.2 0.5 0.5

0.2 0.5 0.2 1.0 0.5 1.0

0.5 0.5 2.0 2.0 0.5 2.5

Approximate value 0.2149360285 0.1000115312 0.221320267 0.4470669393 0.4835865327 0.10138137

x x x x x x

lo-’ lo-* 10-l lo-* lo-* 10-l

I]

for some values of o, b, c calculated by the single term approximation algorithm (24) (approximate value), and by the series expansion algorithm (Kalla er al., 1987a, p. 228) (calculated value)

Calculated value

11

0.21969831D -03 0.12595189D -02 0.22286819D - 03 0.50380756D - 02 0.57948843D - 02 0. I 1293886D - 01

0.1 0.1 0.2 0.2 0.5 0.5

b

P

0.2 0.5 0.2 I.0 0.5 1.0

0.5 0.5 2.0 2.0 0.5 2.5

Aaoroximate value 0.217256 0.1129731 0.2237091 0.4518925 0.6151821 0.106863

x x x x x x

lo-’ lo-’ IO-’ lo-* lo-* 10-l

Calculated value 0.21969831D 0.12595189D 0.22286819D 0.50380756D 0.57948843D 0.11293886D

-03 -02 - 03 - 02 - 02 -01

Algorithms for the generalized Hubbell integrals 2. SINGLE

TERM APPROXIMATION

FOR THE

499

In view of the symmetry property of the hypergeometric function, equation (13) can alternatively be written as,

From equation (S), we have

I

X

IS

P-‘(1 -t)‘+-‘(b”y

+

c +

a2t)-=dt

(9)

0

where B(/3, y - /I) is the B function. Since (b’y+c+o’i)-c(l+~y)(l+~r)

(10) If in equation (14) p = 0, we obtain a single term approximation for the function H[u, b, c, I/a, fi, y], namely,

if a2b2/(2c) 4 1, the result (9) can be written as:

X

HEH

s

[

dr

a, 6, a, Ac, y11

0

I

(11)

X s

II

The most primitive fashion to obtain a single term approximation would be to set (1 + x/r)z- ’ equal to unity throughout the interval of integration. A superior approach is to set (1 +x/rye’

N (1 -x)-(‘-‘)”

(12)

which in the neighbourhood of x = 0 is a better approximation to (1 + x,/ry'-' than just unity (Kalla et af., 1987b). Using equation (12) and the definition of the /3 function, we obtain from equation (11) the single term approximation for I:

with a2b2/(2c) < 1. The corresponding integral is

formula

r &,b)=z&

for

the

Hubbell

; T(b2+1) 0

(16)

with a2b2/2 < 1. Following a similar procedure, we now obtain an approximation formula for a more general function. 3. SINGLE TERM APPROXIMATION

oa

=--

FOR THE

In this section, we derive a new single term approximation for SE’)[a, b, c, A/a, (cr,), (&,)] as defined by equation (7). Using the result of Prudnikov er al. (1990),

b”+’

872 cz

(1 x1- ‘(a - x)~- ‘,F,((a,); (6,); wx) dx (13)

with a2b2/(2c) < 1.

s0

a,Rea,RefI>O;p,
(17)

LEDAGALIJ~et al.

500 we can write

1

1

l

+sp=lIff

4~ Ya, A - a) %a,, I%-aA

ss I

X

I

o

. . .. . .

o

(P-2)

g-1(1

_vp-“-‘v;l-l(l

_1)2)81-.2-‘t);3-‘(l

(1

)(

Wa2, &-ad

_u,y2-al-l~;2--1

_~p.3-‘.

__p_,)~.D-ap-l-I

. .v”p_-I-‘(l

P

I

~(xm+c+amuu,u2v,~~~u,_,)-~~do,_,~~~dv,du2dv,dvdr. Making the transformation

y = (x/b)‘“, we obtain,

+sp,=E!z

1

l

47~ m B(a,

BI -a)

WI,

1

f12 -

al)

W&2,

Bs -

a2)

1 X

B(a,,,~~_a~_l~~o’S,I~;-;~Sf~~~~+~~~~~-~ B(a3,B4-ad”’

x

(yp+C)4-‘g-l(1

x

#yy’(l

_D)BI-DI-l~fl-yl

_I)j)k-a3-l..

_q)c2-.!--10~2--(1

.$P_-,I_‘(1

_-t)z)B3-42-l

-_o p_,yP-c-I-’

(18)

x~bm+c+amvv,v2o,~~~o,_,)~u~dvp_,~~~dosdv2dv,dt)dy. Let q=l-wi,i=0,1,2

,...,

p-l,(uo=u),

so that,

1 1 l 47~ m B(a, B, - a) B(a,, P2 - a,) B(a2, B3- a2)

SIS&!!!b”

_w)pI-IwB~-=-l(l

_W,~~-~w~-~~-~(l

x

($p+c)“p-yl

x

wpw’(l

x

(yb”+ c + am(l - w)(l - wI)(l - w2)(l - w3).. .

._wj).‘-‘wp-‘3-‘..

.(l

_w2)“-~

_wp_,)“-l-Iw~_~app-I-l

x(l-~,_,))-~~dw~_,~~~dw~dw~dw,dwdy. We now make the approximation (yb”+c

+a’“(1 -w)(l

-w,)(l

(19)

(see Appendix) - w2)(1 -w3)...(1

-wP_,))

~c(l+~y)(l+3(l-~w)(l-~w,)(l-~w2)~~~(l-~wP-l)

(20)

subject to the conditions: o<+ Then we can write S&$9’EQa

xwBI-u-l(l

_b”+l 4n mc”

-%

_w,)oII-a~~(FIP+(I”~-lw~-.I-l(~

_wj)ol’-E~~~/c+~~-lw~-‘3-l..

1

1

B(a, BI - a) &x1, B2- aI 1

_w2)“2-4(dc+4-l

x

wp-“‘-l(l

.

x

(1 - w~_,)4-~-r~~a”‘c+~~-‘~~_~~~-~-‘dwp_,~~~dw~dw2dw,dw

dy,

(21)

Algorithms for the generalized Hubbell integrals in the form:

where we have used equation (12), i.e,

HEH

a, 6, G 1 1

[ a, B?Y

aa b”+’ N--8xcb-B(c

+a2)+

From the definition of the Beta function, we thus find

for 0 < a2/c B 1 (or a2/c B 1) and (ab)2/c2 * 1, wi the corresponding formula for the Hubbell integt being

X B(a,

8, -a)

X B(a,,B2-~,)

for 0 < a2 g 1 (or a2 B 1) and (ab)2 4 1. Acknowledgemenr-The authors would like to tha CONDES-Universidad de1 Zulia for financial support. REFERENCES

B(a,-,,&--a,-,)

’ cm

which can be rewritten as,

ua b”+’ N--

4n mc”-4

(c +a”)-+

for 0 < am/c < 1 (or am/c % 1) and (ab)“/c2 6 1. Letting p = 1 and m = 2 in equation (23), we , obtain with a0 = a, a, = fi, /?I = y, another single term approximation for the function H [

a, 6, c, a, B, Y11

Boast W. B. (1942)Illuminating Engineering. McGraw-H New York. Donovan J. L. (1961) Three-plaque, two-pass irradiatl Nucfeonics 19, 104-I 10. Galu6 L. (1991) Una generalixacion de la integral Hubbell. Rev. T.&. Zng., Univ. Z&a. 14, 153-160. Ghose A. M., Bradley D. A. and Hubbell J. H. (191 Radiation field from a rectangular plaque source: Critiq of some engineering approximations. Appl. Radiat. Is 39, 421-427. Hottel H. C. (1942) Radiant heat transmission. In HC Transmission (Edited by McAdems W. H.), Chap. McGraw-Hill, New York. Hubbell J. H. and Spencer L. V. (1964) Shielding agair gamma rays, neutrons, and electrons from nuch weapons. A review and bibliography. National Bureau Standards Monograph 69. Hubbell J. H., Bach. R. L. and Lamkin J. C. (19( Radiation field from a rectangular source. J. Res. Na Bur. Stand. 64C, 121-138. Jakob K. (1957) Heat radiation through a non-absorbi medium. In Heat Transfer, Chap. 31. Wiley, New Yol Kalla S. L., Al-Saqabi B. and Conde S. (1987a) Some resu related to radiation-field problems. Hadronic J. 221-230. Kalla S. L., Galu6 L. and Kiryakova V. (1991) SOI expansions related to a family of generalized radiati integrals. Matematica Balkanica, New Ser. 5, 190-20: Kalla S. L., Leubner C. and Hubbell J. H. (1987b) Furtl results on generalized elliptic-type integrals. Appl. Ana 2!5, 269-274.

LEDA GAL&

502

Prudnikov A. P., Brychkov Yu. A. and Marichev 0. I. (1990) Integrals and Series: More Special Functions, Vol. 3. Gordon and Breach, New York. Saigo M. and Srivastava R. (1990) Some new results for radiation-field problems. Fukuoka Univ. Sci. Rep. 20, I-13.

et al.

Seihert 0. (1928) Die Warmeaufnahme der hestrahlten Kesselheiztlache. Arch. Wiirme. 9, 180-188. Sivinski J. and Ahlstrom S. (1984) Design and economic considerations for a cesium-137 sludge irradiator. Radial. Pbys. Chem. 24, 191-201. Walsh, J. W. T. (1958) Principles of Photometry. In Photometry, Chap. 5. Constable, London.

APPENDIX We analyse the factor tyb” + c + am(l - w)(l - w,)(l - w*)(l - wj). . . (1 -

wp_,))

of equation (19). For p = 1 we have ybm+c+am(l-w)=yb”+c+a”-a”w=c

~+ry+?-cw C

. C

C

>

If (ab)“/c’ << 1,

l+T-zw

) (l+b”y c )(1+go(-g&w). =c

so that, ybm+c+am(l-w)=c(l+yy)(l+%)(l-&w).

For p = 2, yb”+c+am(l-w)(l-w,)=c

l+~y+~-$w-~w,+~ww, C

C

C

If (ab)‘“/c2 B 1, l+c-a”w-~w,+a”ww, C

Now, if 0 < am/c < 1 or am/c b 1, then

etc.

C

C

C

C

>