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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
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 x -- t 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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  • [1] H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, Oxford, at the Clarendon Press, 1947. MR 0022294
  • [2] R. Courant and D. Hilbert, Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y., 1953. MR 0065391
  • [3] F. Ginsberg, On the Cauchy problem for the one-dimensional heat equation, Math. Comp. 17 (1963), 257–269. MR 162064,
  • [4] J. Hadamard, Lectures on Cauchy's Problem in Linear Partial Differential Equations, Dover Publications, N. Y., 1952.
  • [5] E. Holmgren, ``Sur l'extension de la methode d'integration de Riemann,'' Ark. Mat. Fys., 1904.
  • [6] F. John, Comm. Pure Appl. Math., v. XIII, n. 4, November 1960.
  • [7] F. John, Numerical Solution of Problems which are not Well Posed in The Sense of Hadamard, NYU, (unpublished).
  • [8] F. John, Partial Differential Equations--Lecture Notes, NYU 1952-53.
  • [9] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 13 (1959), 115–162. MR 109940
  • [10] 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
  • [11] James B. Scarborough, Numerical Mathematical Analysis, The Johns Hopkins Press, Baltimore, Md.; Oxford University Press, London, 1950. 2d ed. MR 0039361
  • [12] J. Walsh, Interpolation and Approximation; AMS Colloquium Publications, American Mathematical Society, Rhode Island, 1956.
  • [13] 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