A cardinal function method of solution of the equation
Author:
L. R. Lundin
Journal:
Math. Comp. 35 (1980), 747756
MSC:
Primary 65P05
MathSciNet review:
572852
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: The steadystate form of the KleinGordon equation is given by For solutions which are spherically symmetric, takes the form , , where r is the distance from the origin in . The function satisfies It is known that has solutions , where has exactly n zeros in , and where . In this paper, an approximation is obtained for the solution by minimizing a certain functional over a class of functions of the form It is shown that the norm of the error is as , where is positive.
 [1]
Charles
V. Coffman, Uniqueness of the ground state solution for
Δ𝑢𝑢+𝑢³=0\ and a variational
characterization of other solutions, Arch. Rational Mech. Anal.
46 (1972), 81–95. MR 0333489
(48 #11814)
 [2]
Jean
Chauvette and Frank
Stenger, The approximate solution of the nonlinear equation.
Δ𝑢=𝑢𝑢³, J. Math. Anal. Appl.
51 (1975), 229–242. MR 0373320
(51 #9520)
 [3]
Lothar
Collatz, Functional analysis and numerical mathematics,
Translated from the German by Hansjörg Oser, Academic Press, New
YorkLondon, 1966. MR 0205126
(34 #4961)
 [4]
G. W. DAREWICH & H. SCHIFF, "Particle solutions of a class of nonlinear field equations," J. Mathematical Phys., v. 8, 1967, pp. 14791482.
 [5]
R.
Finkelstein, R.
LeLevier, and M.
Ruderman, Nonlinear spinor fields, Physical Rev. (2)
83 (1951), 326–332. MR 0042204
(13,76b)
 [6]
E. HILLE, Analytic Function Theory, Vol. 2, Blaisdell, Waltham, Mass., 1962.
 [7]
IMSL Library 2, Edition 4 (Fortran V), International Mathematical and Statistical Libraries, Inc., Houston, Texas, 1974.
 [8]
L.
Lundin and F.
Stenger, Cardinaltype approximations of a function and its
derivatives, SIAM J. Math. Anal. 10 (1979),
no. 1, 139–160. MR 516759
(81c:41043), http://dx.doi.org/10.1137/0510016
 [9]
J.
McNamee, F.
Stenger, and E.
L. Whitney, Whittaker’s cardinal function in
retrospect, Math. Comp. 25 (1971), 141–154. MR 0301428
(46 #586), http://dx.doi.org/10.1090/S00255718197103014280
 [10]
Zeev
Nehari, On a nonlinear differential equation arising in nuclear
physics, Proc. Roy. Irish Acad. Sect. A 62 (1963),
117–135 (1963). MR 0165176
(29 #2465)
 [11]
P. D. ROBINSON, "Extremum principles for the equation ," J. Mathematical Phys., v. 12, 1971, pp. 2328.
 [12]
Walter
Rudin, Real and complex analysis, McGrawHill Book Co., New
YorkToronto, Ont.London, 1966. MR 0210528
(35 #1420)
 [13]
Gerald
H. Ryder, Boundary value problems for a class of nonlinear
differential equations, Pacific J. Math. 22 (1967),
477–503. MR 0219794
(36 #2873)
 [14]
H.
Schiff, A classical theory of bosons, Proc. Roy. Soc. Ser. A
269 (1962), 277–286. MR 0141455
(25 #4860)
 [15]
F. STENGER, Convergence and Error of the BubnovGalerkin Method, SIAM Conf. on Ordinary Differential Equations, Fall 1972.
 [16]
Frank
Stenger, Approximations via Whittaker’s cardinal
function, J. Approximation Theory 17 (1976),
no. 3, 222–240. MR 0481786
(58 #1885)
 [17]
F. STENGER, "Optimal convergence of minimum norm approximations in ." (Submitted.)
 [18]
J.
L. Synge, On a certain nonlinear differential equation, Proc.
Roy. Irish Acad. Sect. A 62 (1961/1962), 17–41. MR 0138893
(25 #2333)
 [19]
P.
G. Ciarlet, M.
H. Schultz, and R.
S. Varga, Numerical methods of highorder accuracy for nonlinear
boundary value problems. III. Eigenvalue problems, Numer. Math.
12 (1968), 120–133. MR 0233517
(38 #1838)
 [20]
K. YOSIDA, Functional Analysis, SpringerVerlag, New York, 1966.
 [21]
E. T. WHITTAKER, "On the functions which are represented by the expansions of the interpolation theory," Proc. Roy. Soc. Edinburgh Sect. A, v. 35, 1915, pp. 181194.
 [22]
J. M. WHITTAKER, "On the cardinal function of interpolation theory," Proc. Edinburgh Math. Soc. Ser. I (2), 1927, pp. 4146.
 [1]
 C. V. COFFMAN, "Uniqueness of the ground state solution for and a variational characterization of other solutions," Arch. Rational Mech. Anal., v. 46, 1972, pp. 8192. MR 0333489 (48:11814)
 [2]
 J. CHAUVETTE & F. STENGER, "The approximate solution of the nonlinear equation ," J. Math. Anal. Appl., v. 51. 1975, pp. 229242. MR 0373320 (51:9520)
 [3]
 L. COLLAT Z, Functional Analysis and Numerical Mathematics, Academic Press, New York, 1966. MR 0205126 (34:4961)
 [4]
 G. W. DAREWICH & H. SCHIFF, "Particle solutions of a class of nonlinear field equations," J. Mathematical Phys., v. 8, 1967, pp. 14791482.
 [5]
 R. FINKELSTEIN, R. LE LEVIER & M. RUDERMAN, "Nonlinear spinor fields," Phys. Rev., v. 83, 1950, pp. 326332. MR 0042204 (13:76b)
 [6]
 E. HILLE, Analytic Function Theory, Vol. 2, Blaisdell, Waltham, Mass., 1962.
 [7]
 IMSL Library 2, Edition 4 (Fortran V), International Mathematical and Statistical Libraries, Inc., Houston, Texas, 1974.
 [8]
 L. LUNDIN & F. STENGER, "Cardinal type approximation of a function and its derivatives," SIAM J. Math. Anal., v. 10, 1979, pp. 139160. MR 516759 (81c:41043)
 [9]
 J. McNAMEE, F. STENGER & E. L. WHITNEY, "Whittaker's cardinal function in retrospect," Math. Comp., v. 25, 1963, pp. 141154. MR 0301428 (46:586)
 [10]
 Z. NEHARI, "On a nonlinear differential equation arising in nuclear physics," Proc. Roy. Irish Acad. Sect. A, v. 62, 1963, pp. 117135. MR 0165176 (29:2465)
 [11]
 P. D. ROBINSON, "Extremum principles for the equation ," J. Mathematical Phys., v. 12, 1971, pp. 2328.
 [12]
 W. RUDIN, Real and Complex Analysis, McGrawHill, New York, 1966. MR 0210528 (35:1420)
 [13]
 G. H. RYDER, "Boundary value problems for a class of nonlinear differential equations," Pacific J. Math., v. 22, 1967, pp. 477503. MR 0219794 (36:2873)
 [14]
 H. SCHIFF, "A classical theory of bosons," Proc. Roy. Soc. Ser. A, v. 269, 1962, pp. 277286. MR 0141455 (25:4860)
 [15]
 F. STENGER, Convergence and Error of the BubnovGalerkin Method, SIAM Conf. on Ordinary Differential Equations, Fall 1972.
 [16]
 F. STENGER, "Approximation via Whittaker's cardinal function," J. Approximation Theory, v. 17, 1976, pp. 222240. MR 0481786 (58:1885)
 [17]
 F. STENGER, "Optimal convergence of minimum norm approximations in ." (Submitted.)
 [18]
 J. L. SYNGE, "On a certain nonlinear differential equation," Proc. Roy. Irish Acad. Sect. A, v. 62, 1961, pp. 1741. MR 0138893 (25:2333)
 [19]
 P. G. CIARLET, M. H. SCHULTZ & R. S. VARGA, "Numerical methods of highorder accuracy for nonlinear problems, III. Eigenvalue problems," Numer. Math., v. 12, 1968, pp. 120133. MR 0233517 (38:1838)
 [20]
 K. YOSIDA, Functional Analysis, SpringerVerlag, New York, 1966.
 [21]
 E. T. WHITTAKER, "On the functions which are represented by the expansions of the interpolation theory," Proc. Roy. Soc. Edinburgh Sect. A, v. 35, 1915, pp. 181194.
 [22]
 J. M. WHITTAKER, "On the cardinal function of interpolation theory," Proc. Edinburgh Math. Soc. Ser. I (2), 1927, pp. 4146.
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
65P05
Retrieve articles in all journals
with MSC:
65P05
Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198005728521
PII:
S 00255718(1980)05728521
Article copyright:
© Copyright 1980
American Mathematical Society
