A ``sinc-Galerkin'' method of solution of boundary value problems

Author:
Frank Stenger

Journal:
Math. Comp. **33** (1979), 85-109

MSC:
Primary 65L10; Secondary 65N30

MathSciNet review:
514812

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Abstract: This paper illustrates the application of a "Sinc-Galerkin" method to the approximate solution of linear and nonlinear second order ordinary differential equations, and to the approximate solution of some linear elliptic and parabolic partial differential equations in the plane. The method is based on approximating functions and their derivatives by use of the Whittaker cardinal function. The DE is reduced to a system of algebraic equations via new accurate explicit approximations of the inner products, the evaluation of which does not require any numerical integration. Using *n* function evaluations, the error in the final approximation to the solution of the DE is , where *c* is independent of *n*, and *d* denotes the dimension of the region on which the DE is defined. This rate of convergence is optimal in the class of *n*-point methods which assume that the solution is analytic in the interior of the interval, and which ignore possible singularities of the solution at the endpoints of the interval.

**[1]**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****[2]**Y. H. CHIU,*An Integral Equation Method of Solution of*, Ph. D. thesis, University of Utah, 1976.**[3]**G. H. Golub and C. Reinsch,*Handbook Series Linear Algebra: Singular value decomposition and least squares solutions*, Numer. Math.**14**(1970), no. 5, 403–420. MR**1553974**, 10.1007/BF02163027**[4]**Ulf Grenander and Gabor Szegö,*Toeplitz forms and their applications*, California Monographs in Mathematical Sciences, University of California Press, Berkeley-Los Angeles, 1958. MR**0094840****[5]**R. V. L. HARTLEY, "The transmission of information,"*Bell System Tech. J.*, v. 7, 1928, pp. 535-560.**[6]**L. LUNDIN,*A Cardinal Function Method of Solution of*, Ph. D. thesis, University of Utah, 1975.**[7]**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**, 10.1137/0510016**[8]**J. McNamee, F. Stenger, and E. L. Whitney,*Whittaker’s cardinal function in retrospect*, Math. Comp.**25**(1971), 141–154. MR**0301428**, 10.1090/S0025-5718-1971-0301428-0**[9]**H. NYQUIST, "Certain topics in telegraph transmission theory,"*Trans. Amer. Inst. Elec. Engrg.*, v. 47, 1928, pp. 617-644.**[10]**W. PETRICK, J. SCHWING & F. STENGER, "An algorithm for the electromagnetic scattering due to an axially symmetric body with an impedance boundary condition,"*J. Math. Anal. Appl*. (To appear.)**[11]**J. SCHWING,*Eigensolutions of Potential Theory Problems in*, Ph. D. thesis, University of Utah, 1976.**[12]**C. E. Shannon,*A mathematical theory of communication*, Bell System Tech. J.**27**(1948), 379–423, 623–656. MR**0026286****[13]**Frank Stenger,*Approximations via Whittaker’s cardinal function*, J. Approximation Theory**17**(1976), no. 3, 222–240. MR**0481786****[14]**Frank Stenger,*Kronecker product extensions of linear operators*, SIAM J. Numer. Anal.**5**(1968), 422–435. MR**0235711****[15]**Frank Stenger,*Optimal convergence of minimum norm approximations in 𝐻_{𝑝}*, Numer. Math.**29**(1977/78), no. 4, 345–362. MR**0483329****[16]**E. T. WHITTAKER, "On the functions which are represented by the expansions of the interpolation theory,"*Proc. Roy. Soc. Edinburgh*, v. 35, 1915, pp. 181-194.**[17]**J. M. WHITTAKER,*Interpolatory Function Theory*, Cambridge, London, 1935.

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DOI:
http://dx.doi.org/10.1090/S0025-5718-1979-0514812-4

Article copyright:
© Copyright 1979
American Mathematical Society