A cardinal function method of solution of the equation

Author:
L. R. Lundin

Journal:
Math. Comp. **35** (1980), 747-756

MSC:
Primary 65P05

DOI:
https://doi.org/10.1090/S0025-5718-1980-0572852-1

MathSciNet review:
572852

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Abstract: The steady-state form of the Klein-Gordon equation is given by

*r*is the distance from the origin in . The function satisfies

*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

**[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. 81-92. MR**0333489 (48:11814)****[2]**J. CHAUVETTE & F. STENGER, "The approximate solution of the nonlinear equation ,"*J. Math. Anal. Appl.*, v. 51. 1975, pp. 229-242. 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. 1479-1482.**[5]**R. FINKELSTEIN, R. LE LEVIER & M. RUDERMAN, "Nonlinear spinor fields,"*Phys. Rev.*, v. 83, 1950, pp. 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 & F. STENGER, "Cardinal type approximation of a function and its derivatives,"*SIAM J. Math. Anal.*, v. 10, 1979, pp. 139-160. MR**516759 (81c:41043)****[9]**J. McNAMEE, F. STENGER & E. L. WHITNEY, "Whittaker's cardinal function in retrospect,"*Math. Comp.*, v. 25, 1963, pp. 141-154. 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. 117-135. MR**0165176 (29:2465)****[11]**P. D. ROBINSON, "Extremum principles for the equation ,"*J. Mathematical Phys.*, v. 12, 1971, pp. 23-28.**[12]**W. RUDIN,*Real and Complex Analysis*, McGraw-Hill, 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. 477-503. MR**0219794 (36:2873)****[14]**H. SCHIFF, "A classical theory of bosons,"*Proc. Roy. Soc. Ser. A*, v. 269, 1962, pp. 277-286. MR**0141455 (25:4860)****[15]**F. STENGER,*Convergence and Error of the Bubnov-Galerkin 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. 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*, v. 62, 1961, pp. 17-41. MR**0138893 (25:2333)****[19]**P. G. CIARLET, M. H. SCHULTZ & R. S. VARGA, "Numerical methods of high-order accuracy for nonlinear problems, III. Eigenvalue problems,"*Numer. Math.*, v. 12, 1968, pp. 120-133. MR**0233517 (38:1838)****[20]**K. YOSIDA,*Functional Analysis*, Springer-Verlag, 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. 181-194.**[22]**J. M. WHITTAKER, "On the cardinal function of interpolation theory,"*Proc. Edinburgh Math. Soc. Ser.*I (2), 1927, pp. 41-46.

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DOI:
https://doi.org/10.1090/S0025-5718-1980-0572852-1

Article copyright:
© Copyright 1980
American Mathematical Society