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Existence of Gauss interpolation formulas for the one-dimensional heat equation


Author: David L. Barrow
Journal: Math. Comp. 30 (1976), 24-34
MSC: Primary 65M05
DOI: https://doi.org/10.1090/S0025-5718-1976-0413523-0
MathSciNet review: 0413523
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Abstract: Let $ C = \{ (x(s),t(s)):a \leqslant s \leqslant b\} $ be a Jordan arc in the x-t plane satisfying $ (x(a),t(a)) = (a,{t_ \ast}),(x(b),t(b)) = (b,{t_\ast})$, and $ t(s) < {t_\ast}$ when $ a < s < b$. Let $ a < {x_\ast} < b$. We prove the existence of Gauss interpolation formulas for C and the point $ ({x_\ast},{t_\ast})$, for solutions u of the one-dimensional heat equation, $ {u_t} = {u_{xx}}$. Such formulas approximate $ u({x_\ast},{t_\ast})$ in terms of a linear combination of its values on C. The formulas are characterized by the requirement that they are exact for as many basis functions (the heat polynomials) as possible.


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  • [1] P. APPELL, "Sur l'équation $ {\partial ^2}z/{x^2} = \partial z/\partial t$ et la théorie de la chaleur," J. Math. Pures Appl., v. 8, 1892, pp. 187-216.
  • [2] D. L. BARROW & A. H. STROUD, "Existence of Gauss harmonic interpolation formulas," SIAM J. Numer. Anal. (To appear.) MR 0413425 (54:1539)
  • [3] D. COLTON, "The approximation of solutions to initial-boundary value problems for parabolic equations in one space variable," Quart. Appl. Math. (To appear.) MR 0454348 (56:12599)
  • [4] D. COLTON, "Complete families of solutions for parabolic equations with analytic coefficients," SIAM J. Math. Anal. (To appear.) MR 0385322 (52:6186)
  • [5] P. J. DAVIS &. M. W. WILSON, "Nonnegative interpolation formulas for uniformly elliptic equations," J. Approximation Theory, v. 1, 1968, pp. 374-380. MR 39 #4559. MR 0243236 (39:4559)
  • [6] S. KARLIN & W. J. STUDDEN, Tchebycheff Systems: With Applications in Analysis and Statistics, Pure and Appl. Math., vol. 15, Interscience, New York, 1966. MR 34 #4757. MR 0204922 (34:4757)
  • [7] J. M. ORTEGA & W. C. RHEINBOLDT, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York and London, 1970. MR 42 #8686. MR 0273810 (42:8686)
  • [8] M. H. PROTTER & H. F. WEINBERGER, Maximum Principles in Differential Equations, Prentice-Hall, Englewood Cliffs, N. J., 1967. MR 36 #2935. MR 0219861 (36:2935)
  • [9] P. C. ROSENBLOOM & D. V. WIDDER, "Expansions in terms of heat polynomials and associated functions," Trans. Amer. Math. Soc., v. 92, 1959, pp. 220-266. MR 21 #5845. MR 0107118 (21:5845)
  • [10] B. D. SHRIVER, Interpolation Formulas of Gauss Type for Approximate Solution of the n-Dimensional Heat Equation, Ph.D. Thesis, State University of New York at Buffalo, 1971.
  • [11] J. T. SCHWARTZ, Nonlinear Functional Analysis, Gordon and Breach, New York, 1969. MR 0433481 (55:6457)
  • [12] A. H. STROUD, "Gauss harmonic interpolation formulas," Comm. Assoc. Comput. Mach., v. 17, 1974, pp. 471-475. MR 0362827 (50:15265)
  • [13] A. H. STROUD, "Some interpolation formulas for the Neumann problem for the nsphere," SIAM J. Numer. Anal. (To appear.) MR 0398051 (53:1906)
  • [14] A. H. STROUD & D. L. BARROW, "Gauss formulas for the Dirichlet problem," Bull. Amer. Math. Soc., v. 80, 1974, p. 1230. MR 0436556 (55:9499)
  • [15] D. V. WIDDER, "Analytic solutions of the heat equation," Duke Math. J., v. 29, 1962, pp. 497-503. MR 28 #364. MR 0157127 (28:364)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1976-0413523-0
Article copyright: © Copyright 1976 American Mathematical Society

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