Annali di Matematica Pura ed Applicata

, Volume 66, Issue 1, pp 155–165 | Cite as

A Cauchy problem for the heat equation

  • J. R. Cannon


Let u(x, t) satisfy the heat equation in 0<x<1, 0<t≤T. Let u(x, 0)=0 for 0<x<1 and let |u(0, t)|<ε, | ux(0, t) |<ε, and | u(1, t) |<M for 0≤t≤T. Then,\(\left| {u\left( {x, t} \right)} \right|< M_{1^{1 - \beta \left( x \right)\varepsilon \beta \left( x \right)} } \), where M1 and β(x) are given explicitly by simple formulas. The application of the a priori bound to obtain error estimates for a numerical solution of the Cauchy problem for the heat equation with u(x, 0)=h(x), u(0, t)=f(t), and ux(0, t)=g(t) is discussed.


Error Estimate Cauchy Problem Heat Equation Simple Formula Obtain Error Estimate 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Behnhe, H. andF. Sommer,Theorie der Analytischen Funktionen einer Komplexen Veränderlichen, Springer-Verlag, 1962, p. 128.Google Scholar
  2. [2]
    Cannon, J. R.,A priori estimate for the continuation of the solution of the heat equation in the space variable, to appear.Google Scholar
  3. [3]
    Carleman, T.,Fonctions Quasi Analytiques, Paris, Gauthier-Villars, 1926, pp. 3–5.Google Scholar
  4. [4]
    Douglas, Jim, Jr.,Mathematical programming and integral equations, Symposium, Provisional International Computation Centre, Birkhauser Verlag, Basel, 1960.Google Scholar
  5. [5]
    Pucci, Carlo,Alcune limitazioni per le soluzioni di equazioni paraboliche, Annali di Matematica pura ed applicata, Serie IV, Tomo XLVIII, 1959, pp. 161–172.Google Scholar
  6. [6]
    Pucci, Carlo,Nuove ricerche sul problema di Cauchy, Memoria Acc. delle Scienze di Torino, 1954.Google Scholar
  7. [7]
    Ginsberg, F.,On the Cauchy problem for the one-dimensional heat equation, Math. Comp. 17 (1963), 257–269.CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Nicola Zanichelli Editore 1964

Authors and Affiliations

  • J. R. Cannon
    • 1
  1. 1.Brookhaven National Laboratory, Upton, L. I.New York

Personalised recommendations