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Mathematics of Computation

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