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On the Cauchy problem for the one-dimensional heat equation


Author: F. Ginsberg
Journal: Math. Comp. 17 (1963), 257-269
MSC: Primary 35.78
DOI: https://doi.org/10.1090/S0025-5718-1963-0162064-8
MathSciNet review: 0162064
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Abstract: In this paper we show that the Cauchy problem for the one-dimensional heat equation, though non-well posed in the sense of Hadamard, can be solved numerically. It is shown that if we admit as solutions functions for which an a priori bound is assumed in some finite rectangle in xt space then the solution depends Hölder continuously upon the given Cauchy data. The specific numerical scheme developed also exhibits the Hölder continuity so that we are sure of a meaningful numerical method.


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  • H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, Oxford, at the Clarendon Press, 1947. MR 0022294
  • R. Courant and D. Hilbert, Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y., 1953. MR 0065391
  • F. Ginsberg, On the Cauchy problem for the one-dimensional heat equation, Math. Comp. 17 (1963), 257–269. MR 162064, DOI https://doi.org/10.1090/S0025-5718-1963-0162064-8
  • J. Hadamard, Lectures on Cauchy’s Problem in Linear Partial Differential Equations, Dover Publications, N. Y., 1952. E. Holmgren, “Sur l’extension de la methode d’integration de Riemann,” Ark. Mat. Fys., 1904. F. John, Comm. Pure Appl. Math., v. XIII, n. 4, November 1960. F. John, Numerical Solution of Problems which are not Well Posed in The Sense of Hadamard, NYU, (unpublished). F. John, Partial Differential Equations—Lecture Notes, NYU 1952-53.
  • L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 13 (1959), 115–162. MR 109940
  • Carlo Pucci, On the improperly posed Cauchy problems for parabolic equations, Symposium on the numerical treatment of partial differential equations with real characteristics: Proceedings of the Rome Symposium (28-29-30 January 1959) organized by the Provisional International Computation Centre, Libreria Eredi Virgilio Veschi, Rome, 1959, pp. 140–144. MR 0107375
  • James B. Scarborough, Numerical Mathematical Analysis, The Johns Hopkins Press, Baltimore, Md.; Oxford University Press, London, 1950. 2d ed. MR 0039361
  • J. Walsh, Interpolation and Approximation; AMS Colloquium Publications, American Mathematical Society, Rhode Island, 1956. E. Whittaker & J. Watson, Modern Analysis (Fourth Edition reprinted), Cambridge University Press, Cambridge, 1952.

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Article copyright: © Copyright 1963 American Mathematical Society