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]**Charles V. Coffman,*Uniqueness of the ground state solution for Δ𝑢-𝑢+𝑢³=0 and a variational characterization of other solutions*, Arch. Rational Mech. Anal.**46**(1972), 81–95. MR**0333489**, https://doi.org/10.1007/BF00250684**[2]**Jean Chauvette and Frank Stenger,*The approximate solution of the nonlinear equation. Δ𝑢=𝑢-𝑢³*, J. Math. Anal. Appl.**51**(1975), 229–242. MR**0373320**, https://doi.org/10.1016/0022-247X(75)90155-9**[3]**Lothar Collatz,*Functional analysis and numerical mathematics*, Translated from the German by Hansjörg Oser, Academic Press, New York-London, 1966. MR**0205126****[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. LeLevier, and M. Ruderman,*Non-linear spinor fields*, Physical Rev. (2)**83**(1951), 326–332. MR**0042204****[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,*Cardinal-type approximations of a function and its derivatives*, SIAM J. Math. Anal.**10**(1979), no. 1, 139–160. MR**516759**, https://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**, https://doi.org/10.1090/S0025-5718-1971-0301428-0**[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****[11]**P. D. ROBINSON, "Extremum principles for the equation ,"*J. Mathematical Phys.*, v. 12, 1971, pp. 23-28.**[12]**Walter Rudin,*Real and complex analysis*, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR**0210528****[13]**Gerald H. Ryder,*Boundary value problems for a class of nonlinear differential equations*, Pacific J. Math.**22**(1967), 477–503. MR**0219794****[14]**H. Schiff,*A classical theory of bosons*, Proc. Roy. Soc. Ser. A**269**(1962), 277–286. MR**0141455****[15]**F. STENGER,*Convergence and Error of the Bubnov-Galerkin 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****[17]**F. STENGER, "Optimal convergence of minimum norm approximations in ." (Submitted.)**[18]**J. L. Synge,*On a certain non-linear differential equation*, Proc. Roy. Irish Acad. Sect. A**62**(1961/1962), 17–41. MR**0138893****[19]**P. G. Ciarlet, M. H. Schultz, and R. S. Varga,*Numerical methods of high-order accuracy for nonlinear boundary value problems. III. Eigenvalue problems*, Numer. Math.**12**(1968), 120–133. MR**0233517**, https://doi.org/10.1007/BF02173406**[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