Numerical solution of some classical differentialdifference equations
Authors:
George Marsaglia, Arif Zaman and John C. W. Marsaglia
Journal:
Math. Comp. 53 (1989), 191201
MSC:
Primary 65L05; Secondary 65Q05
MathSciNet review:
969490
Fulltext PDF Free Access
Abstract 
References 
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Additional Information
Abstract: For differentialdifference equations, we provide a method that gives numerical solutions accurate to hundreds or even thousands of digits. We illustrate with numerical solutions to three classical problems. With a few exceptions, previous claims of extended accuracy for these problems are found to be wrong.
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 R. Bellman & B. Kotkin, "On the numerical solution of a differentialdifference equation arising in analytic number theory," Math. Comp., v. 16, 1962, pp. 473475. MR 0148248 (26:5756)
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 [3]
 A. Buchstab, "Asymptotic estimates of a general numbertheoretic function," Mat. Sb. (N.S.), v. 44, 1937, pp. 12391246. (In Russian)
 [4]
 B. E. Blaisdell & H. Solomon, "On random sequential packing in the plane and a conjecture of Palesti," J. Appl. Probab., v. 7, 1970, pp. 667698. MR 0282389 (43:8101)
 [5]
 H. Davenport & P. ErdöS, "The distribution of quadratic and higher residues," Publ. Math. Debrecen, v. 2, 19511952, pp. 252265. MR 0055368 (14:1063h)
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 K. Dickman, "On the frequency of numbers containing prime factors of a certain relative magnitude," Ark. Mat. Astronom. Fys., v. 22A, 10, 1930, pp. 114.
 [7]
 A. Dvoretsky & H. Robbins, "On the 'parking' problem," Publ. Math. Inst. Hungar. Acad. Sci., v. 9, 1964, pp. 209226. MR 0173275 (30:3488)
 [8]
 I. S. Gradshteyn & I. M. Ryzhik, Tables of Integrals, Series, and Products, Academic Press, New York and London, 1965. MR 0197789 (33:5952)
 [9]
 D. E. Knuth, The Art of Computer Programming, Volume 2: Seminumerical Algorithms, AddisonWesley, Reading, Mass., 1981. MR 633878 (83i:68003)
 [10]
 M. Lal & P. Gillard, "Evaluation of a constant associated with a parking problem," Math. Comp., v. 28, 1974, pp. 561564. MR 0341814 (49:6560)
 [11]
 M. Lal & P. Gillard, Numerical Solution of Two DifferentialDifference Equations of Analytic Theory of Numbers, Lecture Notes in Math., vol. 109, SpringerVerlag, Berlin and New York, 1969, pp. 179187.
 [12]
 J. Van De Lune & E. Wattel, "On the numerical solution of a differentialdifference equation arising in analytic number theory," Math. Comp., v. 23, 1969, pp. 417421. MR 0247789 (40:1050)
 [13]
 David Manion, "Random spacefilling in one dimension," Publ. Math. Inst. Hungar. Acad. Sci., v. 9, 1964, pp. 143153. MR 0177435 (31:1698)
 [14]
 V. Ramaswami, "On the number of positive integers less than x and free of prime factors greater than ," Bull. Amer. Math. Soc., v. 55, 1949, pp. 11221127. MR 0031958 (11:233f)
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 Alfred Renyi, "On a onedimensional problem concerning random space filling," Publ. Math. Inst. Hungar. Acad. Sci., v. 3, 1958, pp. 109127. (In Hungarian) MR 0104284 (21:3039)
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 K. K. Norton, Numbers with Small Prime Factors and Least kth Power NonResidues, Mem. Amer. Math. Soc., vol. 106, Amer. Math. Soc., Providence, R.I., 1971, pp. 927. MR 0286739 (44:3948)
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 S. Selberg, "The number of cancelled elements in the sieve of Eratosthenes," Norsk. Mat. Tidsskr., v. 26, 1944, pp. 7984. (In Norwegian) MR 0018691 (8:317a)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198909694903
PII:
S 00255718(1989)09694903
Keywords:
Difference equations,
differentialdifference equations,
numerical solutions
Article copyright:
© Copyright 1989
American Mathematical Society
