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.

**[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**0162064**, 10.1090/S0025-5718-1963-0162064-8**[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November 1960.**[7]**F. John,*Numerical Solution of Problems which are not Well Posed in The Sense of Hadamard*, NYU, (unpublished).**[8]**F. John,*Partial Differential Equations--Lecture Notes*, NYU 1952-53.**[9]**L. Nirenberg,*On elliptic partial differential equations*, Ann.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 (3)**13**(1959), 115–162. MR**0109940****[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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DOI:
http://dx.doi.org/10.1090/S0025-5718-1963-0162064-8

Article copyright:
© Copyright 1963
American Mathematical Society