Calculation of generalized Hubbell rectangular source integral

Applied Radiation and Isotopes 69 (2011) 90–93

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Calculation of generalized Hubbell rectangular source integral Jonathan Murley, Nasser Saad  Department of Mathematics and Statistics, University of Prince Edward Island Charlottetown, Prince Edward Island, Canada C1A 4P3

a r t i c l e in f o

a b s t r a c t

Article history: Received 6 August 2009 Received in revised form 23 July 2010 Accepted 26 July 2010

A simple formula for computing the generalized Hubbell radiation rectangular source integral " #   Z a,b,p, l sa b l 2 a2 x ðx þ pÞa 2 F1 a, b; g;  2 dx, H ¼ a , b, g 4p 0 x þp

Keywords: Appell hypergeometric functions F2 Hubbell radiation rectangular source integral Classical hypergeometric functions Reduction formulas Radiation field integrals

is introduced. Tables are given to compare the numerical values derived from our approximation formula with those given earlier in the literature. & 2010 Elsevier Ltd. All rights reserved.

1. Introduction In their pioneering work, Hubbell et al. (1960) obtained a series expansion for the calculation of radiation field generated by a plane isotropic rectangular source (plaque), in which the leading term is the integral Iða,bÞ ¼

s 4p

Z

b 0

  a dx arctan pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi , 2 x þ1 x2 þ 1

ð1Þ

here s is the uniform surface source strength per unit source area. In Eq. (1) the quantities a¼w/h and b¼l/h are defined in the range 0 o a r br 1, where h is the height over the a corner of a plaque of length l and width w. For the important applications of this integral in many problems in radiation field, different methods were introduced to obtain numerical values of detector response to plaque source hða,bÞ ¼

Iða,bÞ ¼ ðs=4pÞ

Z

b

0

  a dx arctan pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 x þ1 x2 þ 1

ð2Þ

For some of these methods we refer to the work of Glasser (1984), ¨ Kalla et al. (1987), Ghose et al. (1988); Galue´ et al. (1988), Gotze (1995), Kalla (1993), Timus (1993), Michieli and Maximovic (1996), Kalla and Khajah (1997, 2000), Prabha (2001), Stalker (2001), Guseinov et al. (2004), Ezure (2005), Guseinov et al. (2005), and Prabha (2007).  Corresponding author.

E-mail address: [email protected] (N. Saad). 0969-8043/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.apradiso.2010.07.018

Although h(a, b) is not expressible in simple closed form, Glasser (1984) has evaluated it in terms of Appell’s hypergeometric function F2 (see for example Slater, 1966, Chapter 8, for a study of Appell functions Fq, q¼1, 2, 3, 4). Indeed, elementary differentiation of (2), with respect to a, we have Z a Z b dy hða,bÞ ¼ dx ð3Þ 2 2 0 0 1þ x þy pffiffiffi pffiffiffi and straightforward substitutions x ¼ a u and y ¼ b v allow as to write (3) as the double-integral representation of Appell’s hypergeometric function F2 (Slater, 1966, Chapter 8, formula 8.2.3), consequently hða,bÞ ¼ abF 2 ð1; 12 , 12 ; 32 , 32; a2 ,b2 Þ:

ð4Þ

Various generalizations of Eq. (1) have been given in the literature (see for example, Kalla et al., 1987, 1991a,b, 2002; Galue´ et al., 1988, 1994; Saigo and Srivastava, 1990; Galue´, 1991, 2003; Kalla, 1993; Oner, 2007; EL-Gabali and Kalla, 1996; Galue´ and Kiryakova, 1994). More specifically, Kalla et al. (1987) introduced a generalization defined by the integral " #   Z a,b,p, l sa b l 2 a2 a H dx, ð5Þ x ðx þ pÞ F1 a, b; g;  2 ¼ 2 a, b, g 4p 0 x þp where g 4 b 40; a,b,p 4 0; 1 o l o 2a1; and 2 F1 ða, b; g; xÞ is Gauss hypergeometric function (Slater, 1966, Chapter 1). We notice that 2 3 a,b,1,0 ð6Þ H4 1 3 5 ¼ Iða,bÞ 1, , 2 2

J. Murley, N. Saad / Applied Radiation and Isotopes 69 (2011) 90–93

by virtue of the identity (Slater, 1966; formula 1.5.11) x2 F1 ð1, 12 ; 32; x2 Þ ¼ arctanðxÞ:

ð7Þ

By selecting suitable values for the parameters a, b and g, Eq. (5) can be reduced to different integrals with potential applications in radiation-field problems of specific configurations of source, barrier and detector (Kalla, 1993). Such results are also useful in illumination and heat-exchange engineering Boast (1942), Fano et al. (1959), and Hubbell et al. (1960). Using a simple pffiffiffi transformation, x ¼ b u, Eq. (5) can be written as " H

#  a Z a,b,p, l sa bl þ 1 1 ðl þ 1Þ=21 b2 u ðl þ 3Þ=2ðl þ 1Þ=21 ¼ u ð1uÞ 1 a, b, g 4p 2pa 0 p 0 B B 2 F1 Ba, b; g; @

1

a2  C p C  2  C du, A b u 1  p

Theorem 1. For jxj þjyj o 1; n Z0; s, a1 , a2 A C; b1 , b2 A C\Z 0 , the Appell hypergeometric function F2 satisfies the following identity F2 ðs; a1 , a2 n; b1 , b2 ; x,yÞ ¼ F2 ðs; a1 , a2 ; b1 , b2 ; x,yÞ 

Recently, Opps et al. (2009) establish a number of new recursion formulas for the Appell hypergeometric functions F2 wherein some applications to the evaluation of some generalized radiation field integrals were discussed. The purpose of the present work is to continue our investigation of finding closed form and approximation formulas for effectively computing the radiation field integrals such as Eqs. (1) and (5). In the next section, we develop a new approximation formula to evaluate precisely and to any desire degree of accuracy the generalized Hubbell radiation rectangular source integral (9). In Section 3, numerical results and comparisons with previously reported values are presented.

2. The computation of F2 ða; b,ðk þ 1Þ=2;c,ðk þ 3Þ=2;a2 =p, 2 b =pÞ May be one of the most important cases regarding the computations of the radiation field integrals that capture the interest of many researchers is evaluating the integral Eq. (1) effectively and precisely. Some researchers were able to evaluate h(a, b) using rapidly convergent series (see the original work of Hubbell et al., 1960), further Gabutti et al. (1991) investigated h(a, b) in terms of its series expansions while numerical computations of this integral have been carried out by Hanak ¨ and Cechak (1978), Gotze (1995) developed an effective method for computing the Hubbell radiation rectangular source integral, Kalla and Khajah (1997) (see also Kalla and Khajah, 2000) used Tau Method to approximate H(a, b), Stalker (2001) used new convergent series for evaluating h(a,b) for large a and b and more recently, Ezure (2005) used Haselogrove method, Guseinov et al. (2004) used binomial expansion (see also Guseinov et al., 2005), Prabha (2006) expressed again h(a,b) using some recurrence relations (see also Prabha, 2007). For a survey of various methods in computing the Hubbell rectangular source integral Eq. (1) and its generalization, we refer to the work of Kalla et al. (2002). In this section, we given a new approximation equation that l can be used to compute H½a,b,p, a, b, g  to any degree of precision and, byproduct, we can, therefore, evaluate the Hubbell radiation rectangular source integral (1). Our approximation expression based on the following recurrence formula for F2 (see Opps et al., 2009 for detailed proof.).

n sy X

b2 k ¼ 1

F2 ðs þ 1; a1 , a2 kþ 1; b1 , b2 þ 1; x,yÞ:

ð10Þ

Writing F2 ða; b,ðl þ1Þ=2; g,ðl þ 3Þ=2; a2 =p,b2 =pÞ as F2 ða; b, ðl þ 3Þ=21; g,ðl þ3Þ=2; a2 =p,b2 =pÞ and apply the recurrence relation (10), we obtain     l þ 1 l þ 3 a2 b2 l þ 3 l þ 3 a2 b2 F 2 a; b , ; g, ;  , ; g, ;  , ¼ F2 a; b, 2 2 2 2 p p p p

ð8Þ

which is easily compared with the single-integral representation of the Appell hypergeometric function F2 (Opps et al., 2005, formula (2.6)) to yields " #   a,b,p, l sa bl þ 1 l þ 1 l þ 3 a2 b2 ; ;  H , : ð9Þ F a ; b , g , ¼ 2 a , b, g 4p ðl þ 1Þpa 2 2 p p

91



  2ay l þ 3 l þ5 a2 b2 ; g, ;  , F2 a þ 1; b, : 2 2 l þ3 p p

By means of the identity (Slater, 1966; formula 8.3.1.3)   x , F2 ðs; a1 , b2 ; b1 , b2 ; x,yÞ ¼ ð1yÞs 2 F1 s, a1 ; b1 ; 1y

ð11Þ

ð12Þ

we may now write Eq. (11) as    a   l þ1 l þ 3 a2 b2 b2 a2 F2 a; b, ; g, ;  , ¼ 1þ F a , b ; g ; 2 1 2 2 p p p p þb2   2ay l þ 3 l þ5 a2 b2 ; g, ;  , F a þ 1; b, : ð13Þ  2 2 l þ3 2 p p Further, we may now regard F2 ða þ 1; b,ðl þ3Þ=2; g,ðl þ5Þ=2; a2 =p,b2 =pÞ as F2 ða þ1; b,ðl þ 5Þ=21; g,ðl þ 5Þ=2; a2 =p,b2 =pÞ and apply the recurrence relation (10) again to obtain !    a l þ1 l þ 3 a2 b2 b2 a2 ; g, ;  , F2 a; b, ¼ 1þ F a , b ; g ; 2 1 2 2 2 p p p pþb !  a1 2ay b2 a2 1þ F a þ1, b ; g ;  2 1 2 l þ3 p p þb   2 2 2 aða þ 1Þy l þ 5 l þ 7 a2 b2 ; g, ;  , F2 a þ2; b, : ð14Þ þ 2 2 ðl þ3Þðl þ5Þ p p After similar n  2 steps, we arrive at   l þ 1 l þ 3 a2 b2 ; g, ;  , F2 a; b, 2 2 p p    k  ak n X ðaÞk b2 b2 a2 k   ¼ ð1Þ  1þ 2 F1 a þ k, b; g; 2 3 þ l p p p þ b k¼0 2 k   ð1Þn þ 1 ðaÞn þ 1 yn þ 1 l þ 2n þ 3 l þ 2n þ 5 a2 b2   F2 a þ nþ 1; b, , þ ; g, ;  , 3þl p p 2 2 2 nþ1

ð15Þ where ðaÞk denotes the Pochhammer symbol defined, in terms of Gamma functions, by ( 1 if ðk ¼ 0; a A C\f0gÞ, Gða þkÞ ðaÞk ¼ ¼ aða þ 1Þða þ 2Þ . . . ða þ k1Þ if ðkA N; a A CÞ, GðaÞ here N being the set of positive integers. In principle, the computation of F2 ða þ n þ 1; b,ðl þ2n þ 3Þ=2; g,ðl þ 2n þ 5Þ=2; a2 =p,b2 =pÞ follows the same technique and consequently, for large n   l þ1 l þ 3 a2 b2 F2 a; b, ; g, ; , 2 2 p p

92

J. Murley, N. Saad / Applied Radiation and Isotopes 69 (2011) 90–93

!  2 k  ak ðaÞk b b2 a2  1þ : 2 F1 a þ k, b; g; 2 ð3þ l2Þk p p p þb k¼0

involved in calculating the radiation field integrals such as (1) and (5), we compare, first, our approximation formula against the   exact equation obtained earlier in computing H a,b,1,0 (Opps 1 1

n X

ð16Þ

2 , 2,1

et al., 2009, Eq. (83)), namely, for s ¼ 1, From which we now have for large n, " #  2 k  ak n a,b,p, l sa bl þ 1 X ðaÞk b b2   1þ H  a a, b, g 4p ðl þ 1Þp k ¼ 0 3 þ l p p 2 k ! a2 2 F1 a þ k, b; g; , 2 p þb

2

3   a,b,p,1 2 ab 1 1 a2 b2 4 ; ,1; 1,2;  , H 1 1 5¼ pffiffiffi F2 , ,1 2 2 8p p p p 2 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! pffiffiffi " a p b2 1 1 a2 1 þ 2 F1  , ; 1;  ¼ 2 4p 2 2 p pþb   2 1 1 a 2 F1  , ; 1;  2 2 p

ð17Þ

since the two-terms asymptotic expansion of the Appell hypergeometric function F2 developed by Lo´pez and Pagola (2008) indicate that for large n, the Appell hypergeometric function   l þ 2n þ 3 l þ2n þ 5 a2 b2 ; g, ;  , F2 a þ n þ 1; b, 2 2 p p

ð18Þ

and   2 1 3  k  1=2k a,b,p,1 2 n 6X ab b2 2 k b2 4 5 6 ¼ 1þ H 1 1 pffiffiffi lim 4 , ,1 8p p n-1 k ¼ 0 ð2Þk p p 2 2 !# 1 1 a2 þ k, ; 1; 2 F1 : 2 2 2 p þb 2

on the right-hand side of Eq. (15) approach zero.

3. Numerical results and discussion

ð19Þ

In order to test our approximation formula (17) and to show that it indeed simplify much of the numerical complexity Table 1 Comparison of the values of H

h

a,b,p,1 1, 1,1 2 2

i

In Table 1, we reported our computation using Eqs. (18) and (19) for different values of a, b and p, the calculations were performed using MATHEMATICA software version 7. It should be clear that any discrepancies may appear are due to the numerical accuracy used in computing the Gauss hypergeometric functions. In Table 2, we compared our calculated values with those obtained by Guseinov and Mamedov (2005) and with those obtained by Galue´ et al. (1994). It should be clear that Eq. (17) can be used for arbitrary values of the parameters a, b, p and it is not restricted to any particular range of parameter values. In Table 3, we report our numerical computation of the Hubbell rectangular source integral (1) for s ¼ p ¼ 1 as well for

for some values of a, b and p calculated using

Eqs. (18) and (19). a

b

p

Eq. (18)

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

0.000 0.001 0.000 0.005 0.005 0.011

Table 2 The values of H

h

a,b,p,1 1 1 , ,1 2 2

219 259 222 038 794 293

i

Eq. (19) 698 518 868 075 884 885

305 891 191 567 270 774

361 975 957 902 704 813

162 573 851 293 959 334

27 0.000 219 698 3 0.001 259 518 78 0.000 222 868 0.005 038 075 5 0.005 794 884 5 0.011 293 885

305 891 191 567 270 774

361 161 975 572 957 853 902 291 704 952 813 332

92 7 9 5

integrals for s ¼ 1 and some values of a, b and p obtained from Eq. (17), Guseinov and Mamedov (2005) and Galue´ et al. (1994).

a

b

p

Eq. (17)

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

0.000 0.001 0.000 0.005 0.005 0.011

Table 3 The comparative values of H

h

a,b,p,0 1,0:5,1:5

i

219 259 222 038 794 293

698 518 868 075 884 885

305 891 191 567 270 774

361 161 975 572 957 853 902 291 704 952 813 332

92 7 9 5

Guseinov and Mamedov (2005)

Galue´ et al. (1994)

0.000 0.001 0.000 0.005 0.005 0.011

0.000 0.001 0.000 0.005 0.005 0.011

219 259 222 038 794 293

698 518 868 075 884 885

305 892 191 568 271 774

352 096 569 393 955 293

979 95 00 22 31 0

219 259 222 038 794 293

698 518 868 075 884 886

31 9 19 6 3

integral from Eq. (17), Guseinov and Mamedov (2005) and Galue´ et al. (1994).

a

b

p

Eq. (17)

0.1 0.1 0.2 0.2 0.5 0.6 0.8 1.0 0.5 0.8

0.1 0.5 0.5 0.8 0.5 0.5 0.6 0.8 2.0 2.6

0.5 1.0 2.5 2.0 1.0 1.0 2.8 4.2 5.4 7.5

0.001 0.003 0.003 0.005 0.017 0.020 0.012 0.013 0.012 0.017

570 678 067 758 188 067 248 484 012 255

716 199 148 600 506 469 693 796 547 112

369 808 756 701 077 440 964 561 384 588

171 686 681 331 523 266 534 181 049 23 496 68 171 94 457 52 013 146 899 273

Guseinov and Mamedov (2005)

Galue´ et al. (1994)

0.001 0.003 0.003 0.005 0.017 0.020 0.012 0.013 0.012 0.017

0.001 0.003 0.003 0.005 0.017 0.020

570 678 067 758 188 067 248 484 012 255

716 199 148 600 506 469 693 796 547 128

369 808 756 701 077 441 963 561 385 853

157 778 507 357 717 718 979 864 362 560

32 47 53 22 6 5 3 6 9 6

570 678 067 758 188 067

716 199 148 600 506 469

4 8 8 7

J. Murley, N. Saad / Applied Radiation and Isotopes 69 (2011) 90–93

some other values of p using equation (17), or simply 2 3  k  1k a,b,p,0 n sab X k! b2 b2 3 1þ H4 1 3 5  1, , 4pp k ¼ 0 2 k p p 2 2 ! 1 3 a2 2 F1 k þ 1, ; ; : 2 2 p þ b2

ð20Þ

In the same table, we also compared our results with the earlier numerical values obtained of Guseinov and Mamedov (2005) and Galue´ et al. (1994). It is worth noting an important feature of the approximation formula (17) is that it is self-adjusting; that is, if jHn þ 1 Hn j5E,

ð21Þ

where E is the desire accuracy then n should be increased to reach the required accuracy. Here, Hn refer to the right-hand side of Eq. (17).

Acknowledgment The present investigation was supported, in part, by the Natural Sciences and Engineering Research Council of Canada under Grant GP249507. References Boast, W.B., 1942. Illuminating Engineering. McGraw-Hill, New York. EL-Gabali, M., Kalla, S.L., 1996. Some generalized radiation field integrals. Comput. Math. Appl. 32, 121–128. Ezure, H., 2005. Confirmation of accuracy of Haselgrove method using Hubbell radiation rectangular source integral. J. Nucl. Sci. Tech. 42, 384–389. Fano, U., Spencer, L.V., Berger, M.J., 1959. Penetration and diffusion of X-rays. Encyclopedia of Physics II, vol. 38. Springer, Berlin, pp. 671. Galue´, L., 1991. Una generalizacio´n de la integral de Hubbell. Rev. Te´c. Ing. Univ. Zulia 14, 153–160. Galue´, L., Kalla, S.L., Leubner, C., 1988. Algorithms for the approximation of the Hubbell rectangular source integral and its generalization. Radiat. Phys. Chem. 43, 497–502. Galue´, L., 2003. Unification of generalized radiation field integrals involving confluent hypergeometric function. Radiat. Phys. Chem. 66, 269–273. Galue´, L., Kalla, S.L., Leubner, C., 1994. Algorithms for the approximation of the Hubbell rectangular source integral. Radiat. Phys. Chem. 43, 497–502. Glasser, M.L., 1984. Solution to problem 83-6 in the problems and solutions section. SIAM Rev. 26, 2. Gabutti, B., Kalla, S.L., Hubbell, J.H., 1991. Some expansions related to the Hubbell rectangular source integral. J. Comput. Appl. Math. 37, 273–285. Galue´, L., Kiryakova, V., 1994. Further results on a family of generalized integrals. Radiat. Phys. Chem. 43, 573–579.

93

Ghose, A.M., Bradley, D.A., Hubbell, J.H., 1988. Radiation field from a rectangular plaque source: critique of some engineering approximations. Appl. Radiat. Isot. 3, 421–427. ¨ Gotze, F., 1995. An effective method for computing the Hubbell rectangular integral. Radiat. Phys. Chem. 46, 329–331. ¨ ner, F., Mamedov, B.N., 2004. Evaluation of the Hubbell radiation Guseinov, I.I., O rectangular source integrals using binomial coefficients. Radiat. Phys. Chem. 69, 109–112. Guseinov, I.I., Mamedov, B.N., 2005. Calculation of the generalized Hubbell rectangular source integrals using binomial coefficients. Appl. Math. Comput. 161, 285–292. Guseinov, I.I., Mamedov, B.N., Ekerno˘glu, A.S., 2005. A new algorithm for accurate and fast evaluation of the Hubbell radiation rectangular source integral. Radiat. Phys. Chem. 74, 261–263. Hanak, V., Cechak, T., 1978. Radiation field from a rectangular source. Dadernd Energ. 24, 94. Hubbell, J.H., Bach, R.L., Lamkin, J.C., 1960. Radiation field from a rectangular source. J. Res. Natl. Bur. Stand. 64C, 121–138. Kalla, S.L., 1993. The Hubbell rectangular source integral and its generalizations. Radiat. Phys. Chem. 41, 775–781. Kalla, S.L., Al-Saqabi, B., Conde, S., 1987. Some results related to radiation field problems. Hadronic J. 10, 221–230. Kalla, S.L., Al-Shammery, A.H., Khajah, H.G., 2002. Development of the Hubbell rectangular source integral. Acta Appl. Math. 74, 35–55. Kalla, S.L., Khajah, H.G., 1997. Tau method approximation of some integrals related to radiation field problems. Comput. Math. Appl. 33, 21–27. Kalla, S.L., Khajah, H.G., 2000. Tau method approximation of the Hubbell rectangular source integral. Radiat. Phys. Chem. 59, 17–21. Kalla, S.L., Galue´, L., Kiryakova, V., 1991a. Some expansion related to a family of generalized radiation integrals. Math. Balkanic 5, 190–202. Kalla, S.L., Leubner, C., Al-Saqabi, B., 1991b. Some results for the radiation flux in a rectangular straight duct. Nucl. Sci. J. 28, 403–410. Lo´pez, J.L., Pagola, P., 2008. A simplification of the Laplace’s method for double integrals. Application to the second Appell function. Electron. Trans. Numer. Anal. (ETNA) 30, 224–236. Michieli, I., Maximovic, A., 1996. Legendre expansion related to the Hubbell rectangular source integral. Radiat. Phys. Chem. 47, 779–784. Oner, F., 2007. On the evaluation of rectangular plane-extended sources and their associated radiation fields. Appl. Radiat. Isot. 65, 1121–1124. Opps, S.B., Saad, N., Srivastava, H.M., 2005. Some reduction and transformation formulas for the Appell hypergeometric function F2. J. Math. Anal. Appl. 302, 180–195. Opps, S.B., Saad, N., Srivastava, H.M., 2009. Recursion formulas for Appell’s hypergeometric function F2 with some applications to radiation field problems. Appl. Math. Comput. 207, 545–558. Prabha, H., 2001. Computation of rectangular source integral by rational parameter polynomial method. Radiat. Phys. Chem. 62, 203–206. Prabha, H., 2007. Rectangular source integral and recurrence relations. Radiat. Phys. Chem. 76, 687–690. Saigo, M., Srivastava, R., 1990. Some new results for radiation-field problems. Fukuoka Univ. Sci. Rep. 20, 1–13. Slater, L.J., 1966. Generalized Hypergeometric Functions. Cambridge University Press, Cambridge. Stalker, J., 2001. On Hubbell’s rectangular source integral. Radiat. Phys. Chem. 62, 311–315. Timus, D.M., 1993. Some further results on the Hubbell rectangular source integral. J. Comput. Appl. Math. 45, 331–336.