Existence of Gauss interpolation formulas for the onedimensional heat equation
Author:
David L. Barrow
Journal:
Math. Comp. 30 (1976), 2434
MSC:
Primary 65M05
DOI:
https://doi.org/10.1090/S00255718197604135230
MathSciNet review:
0413523
Fulltext PDF Free Access
Abstract  References  Similar Articles  Additional Information
Abstract: Let $C = \{ (x(s),t(s)):a \leqslant s \leqslant b\}$ be a Jordan arc in the xt 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 onedimensional 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.

P. APPELL, "Sur lâ€™Ă©quation ${\partial ^2}z/{x^2} = \partial z/\partial t$ et la thĂ©orie de la chaleur," J. Math. Pures Appl., v. 8, 1892, pp. 187216.
 David L. Barrow and A. H. Stroud, Existence of Gauss harmonic interpolation formulas, SIAM J. Numer. Anal. 13 (1976), no. 1, 18â€“26. MR 413425, DOI https://doi.org/10.1137/0713003
 David Colton, The approximation of solutions to initial boundary value problems for parabolic equations in one space variable, Quart. Appl. Math. 33 (1975/76), no. 4, 377â€“386. MR 454348, DOI https://doi.org/10.1090/S0033569X197604543485
 David Colton, Complete families of solutions for parabolic equations with analytic coefficients, SIAM J. Math. Anal. 6 (1975), no. 6, 937â€“947. MR 385322, DOI https://doi.org/10.1137/0506082
 P. J. Davis and M. W. Wilson, Nonnegative interpolation formulas for uniformly elliptic equations, J. Approximation Theory 1 (1968), 374â€“380. MR 243236, DOI https://doi.org/10.1016/00219045%2868%29900154
 Samuel Karlin and William J. Studden, Tchebycheff systems: With applications in analysis and statistics, Pure and Applied Mathematics, Vol. XV, Interscience Publishers John Wiley & Sons, New YorkLondonSydney, 1966. MR 0204922
 J. M. Ortega and W. C. Rheinboldt, Iterative solution of nonlinear equations in several variables, Academic Press, New YorkLondon, 1970. MR 0273810
 Murray H. Protter and Hans F. Weinberger, Maximum principles in differential equations, PrenticeHall, Inc., Englewood Cliffs, N.J., 1967. MR 0219861
 P. C. Rosenbloom and D. V. Widder, Expansions in terms of heat polynomials and associated functions, Trans. Amer. Math. Soc. 92 (1959), 220â€“266. MR 107118, DOI https://doi.org/10.1090/S00029947195901071182 B. D. SHRIVER, Interpolation Formulas of Gauss Type for Approximate Solution of the nDimensional Heat Equation, Ph.D. Thesis, State University of New York at Buffalo, 1971.
 J. T. Schwartz, Nonlinear functional analysis, Gordon and Breach Science Publishers, New YorkLondonParis, 1969. Notes by H. Fattorini, R. Nirenberg and H. Porta, with an additional chapter by Hermann Karcher; Notes on Mathematics and its Applications. MR 0433481
 A. H. Stroud, Gauss harmonic interpolation formulas, Comm. ACM 17 (1974), 471â€“475. MR 0362827, DOI https://doi.org/10.1145/361082.361100
 A. H. Stroud, Some interpolation formulas for the Neumann problem for the $n$sphere, SIAM J. Numer. Anal. 12 (1975), no. 4, 593â€“604. MR 398051, DOI https://doi.org/10.1137/0712045
 A. H. Stroud and David L. Barrow, Gauss formulas for the Dirichlet problem, Bull. Amer. Math. Soc. 80 (1974), 1230â€“1232. MR 436556, DOI https://doi.org/10.1090/S00029904197413692X
 D. V. Widder, Analytic solutions of the heat equation, Duke Math. J. 29 (1962), 497â€“503. MR 157127
Retrieve articles in Mathematics of Computation with MSC: 65M05
Retrieve articles in all journals with MSC: 65M05
Additional Information
Article copyright:
© Copyright 1976
American Mathematical Society