On the computation of certain integrals containing the modified Bessel function
Author:
Keith R. Lassey
Journal:
Math. Comp. 39 (1982), 625637
MSC:
Primary 65D20
MathSciNet review:
669654
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Efficient stratagems are developed for numerically evaluating one and twodimensional integrals over x, y with integrand . The integrals are expressed in terms of convergent series, which exhibit the correct limiting behavior, and which can be evaluated recursively. The performances of these stratagems are compared with numerical integration.
 [1]
Stuart
R. Brinkley Jr., Heat transfer between a fluid and a porous solid
generating heat, J. Appl. Phys. 18 (1947),
582–585. MR 0021210
(9,38a)
 [2]
H. C. Thomas, "Heterogeneous ion exchange in a flowing system," J. Amer. Chem. Soc., v. 66, 1944, pp. 16641666.
 [3]
J. E. Walter, "Ratedependent chromatographic adsorption," J. Chem. Phys., v. 13, 1945, pp. 332336.
 [4]
H. C. Thomas, "Chromatography: A problem in kinetics," Ann. New York Acad. Sci., v. 49, 1948, pp. 161182.
 [5]
S.
Goldstein, On the mathematics of exchange processes in fixed
columns. I. Mathematical solutions and asymptotic expansions, Proc.
Roy. Soc. London. Ser. A. 219 (1953), 151–171. MR 0058824
(15,429g)
 [6]
A. Ogata, Mathematics of Dispersion with a Linear Adsorption Isotherm, Prof. Pap. U. S. Geol. Surv. 411H, 1964.
 [7]
J. Hubert & A. Lenda, "A solution of the dispersionadsorption equation with linear adsorption isotherm," Nucleonika, v. 16, 1971, pp. 271278.
 [8]
F. De Smedt & P. J. Wierenga, "Mass transfer in porous media with immobile water," J. Hydrol., v. 41, 1979, pp. 5967.
 [9]
S. R. Brinkley & R. F. Brinkley, "Table of the probability of hitting a circular target," MTAC, v. 2, 1947, p. 221.
 [10]
Yudell
L. Luke, Integrals of Bessel functions, McGrawHill Book Co.,
Inc., New YorkTorontoLondon, 1962. MR 0141801
(25 #5198)
 [11]
K. R. Lassey, "The interception and retention of aerosols by vegetation: IThe formulation of a filtration model," Atmos. Environ., v. 16, 1982, pp. 1324.
 [12]
K. R. Lassey, "Conceptually simple mathematical models of filtration (and exchange) processes," Ecol. Modelling. (Submitted.)
 [13]
C. Hastings & J. P. Wong, "Analytical approximations," MTAC, v. 7, 1953, pp. 212213.
 [14]
W. J. Cody, Preliminary Report on Software for the Modified Bessel Functions of the First Kind, Argonne National Lab. Report TM357, 1980.
 [15]
D.
E. Amos, Computation of modified Bessel
functions and their ratios, Math. Comp. 28 (1974), 239–251.
MR
0333287 (48 #11612), http://dx.doi.org/10.1090/S00255718197403332877
 [16]
Walter
Gautschi and Josef
Slavik, On the computation of modified Bessel
function ratios, Math. Comp.
32 (1978), no. 143, 865–875. MR 0470267
(57 #10025), http://dx.doi.org/10.1090/S00255718197804702679
 [17]
Marietta
J. Tretter and G.
W. Walster, Further comments on the computation of
modified Bessel function ratios, Math.
Comp. 35 (1980), no. 151, 937–939. MR 572867
(81d:33005), http://dx.doi.org/10.1090/S00255718198005728673
 [18]
E.
E. Allen, Polynomial approximations to some
modified Bessel functions, Math. Tables Aids
Comput. 10 (1956),
162–164. MR 0080774
(18,300b), http://dx.doi.org/10.1090/S00255718195600807744
 [19]
F. W. J. Olver, Handbook of Mathematical Functions (M. Abramowitz and I. A. Stegun, Eds.), Chapter 9, Dover, New York, 1965.
 [20]
J. M. Blair & C. A. Edwards, Stable Rational Minimax Approximations to the Modified Bessel Functions and , Atomic Energy of Canada Ltd. Report AECL4928, 1974.
 [21]
M. B. Carver & V. J. Jones, An Evaluation of Available Quadrature Algorithms and Selection for the AECL FORTRAN Mathematical Library, Atomic Energy of Canada Ltd. Report AECL5605, 1977.
 [22]
J. Oliver, "A doublyadaptive ClenshawCurtis quadrature method," Comput. J., v. 15, 1972, pp. 141147.
 [23]
H.
O’Hara and Francis
J. Smith, The evaluation of definite integrals by interval
subdivision, Comput. J. 12 (1969/1970),
179–182. MR 0242369
(39 #3700)
 [24]
J. F. Hart et al., "Computer approximations," (Chapter 6.2), SIAM Series in Applied Mathematics (R. F. Drenick, H. Hochstadt and D. Gillette, Eds.), Wiley, New York, 1968.
 [1]
 S. R. Brinkley, "Heat transfer between a fluid and a porous solid generating heat," J. Appl. Phys., v. 18, 1947, pp. 582585. MR 0021210 (9:38a)
 [2]
 H. C. Thomas, "Heterogeneous ion exchange in a flowing system," J. Amer. Chem. Soc., v. 66, 1944, pp. 16641666.
 [3]
 J. E. Walter, "Ratedependent chromatographic adsorption," J. Chem. Phys., v. 13, 1945, pp. 332336.
 [4]
 H. C. Thomas, "Chromatography: A problem in kinetics," Ann. New York Acad. Sci., v. 49, 1948, pp. 161182.
 [5]
 S. Goldstein, "On the mathematics of exchange processes in fixed columns. I. Mathematical solutions and asymptotic expansions," Proc. Roy. Soc. London Ser. A, v. 219, 1953, pp. 151171. MR 0058824 (15:429g)
 [6]
 A. Ogata, Mathematics of Dispersion with a Linear Adsorption Isotherm, Prof. Pap. U. S. Geol. Surv. 411H, 1964.
 [7]
 J. Hubert & A. Lenda, "A solution of the dispersionadsorption equation with linear adsorption isotherm," Nucleonika, v. 16, 1971, pp. 271278.
 [8]
 F. De Smedt & P. J. Wierenga, "Mass transfer in porous media with immobile water," J. Hydrol., v. 41, 1979, pp. 5967.
 [9]
 S. R. Brinkley & R. F. Brinkley, "Table of the probability of hitting a circular target," MTAC, v. 2, 1947, p. 221.
 [10]
 Y. L. Luke, Integrals of Bessel Functions, Chapter 12.1, McGrawHill, New York, 1962. MR 0141801 (25:5198)
 [11]
 K. R. Lassey, "The interception and retention of aerosols by vegetation: IThe formulation of a filtration model," Atmos. Environ., v. 16, 1982, pp. 1324.
 [12]
 K. R. Lassey, "Conceptually simple mathematical models of filtration (and exchange) processes," Ecol. Modelling. (Submitted.)
 [13]
 C. Hastings & J. P. Wong, "Analytical approximations," MTAC, v. 7, 1953, pp. 212213.
 [14]
 W. J. Cody, Preliminary Report on Software for the Modified Bessel Functions of the First Kind, Argonne National Lab. Report TM357, 1980.
 [15]
 D. E. Amos, "Computation of modified Bessel functions and their ratios," Math. Comp., v. 28, 1974, pp. 239251. MR 0333287 (48:11612)
 [16]
 W. Gautschi & J. Slavik, "On the computation of modified Bessel function ratios," Math. Comp., v. 32, 1978, pp. 865875. MR 0470267 (57:10025)
 [17]
 M. J. Tretter & G. W. Walster, "Further comments on the computation of modifed Bessel function ratios," Math. Comp., v. 35, 1980, pp. 937939. MR 572867 (81d:33005)
 [18]
 E. E. Allen, "Polynomial approximations to some modified Bessel functions," MTAC, v. 10, 1956, pp. 162164. MR 0080774 (18:300b)
 [19]
 F. W. J. Olver, Handbook of Mathematical Functions (M. Abramowitz and I. A. Stegun, Eds.), Chapter 9, Dover, New York, 1965.
 [20]
 J. M. Blair & C. A. Edwards, Stable Rational Minimax Approximations to the Modified Bessel Functions and , Atomic Energy of Canada Ltd. Report AECL4928, 1974.
 [21]
 M. B. Carver & V. J. Jones, An Evaluation of Available Quadrature Algorithms and Selection for the AECL FORTRAN Mathematical Library, Atomic Energy of Canada Ltd. Report AECL5605, 1977.
 [22]
 J. Oliver, "A doublyadaptive ClenshawCurtis quadrature method," Comput. J., v. 15, 1972, pp. 141147.
 [23]
 H. O'Hara & F. J. Smith, "The evaluation of definite integrals by interval subdivision," Comput. J., v. 12, 1969, pp. 179182. MR 0242369 (39:3700)
 [24]
 J. F. Hart et al., "Computer approximations," (Chapter 6.2), SIAM Series in Applied Mathematics (R. F. Drenick, H. Hochstadt and D. Gillette, Eds.), Wiley, New York, 1968.
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
65D20
Retrieve articles in all journals
with MSC:
65D20
Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198206696546
PII:
S 00255718(1982)06696546
Keywords:
Modified Bessel functions,
integrals of Bessel functions,
recursive computation
Article copyright:
© Copyright 1982
American Mathematical Society
