Numerical solution of the equation of heat conduction for preceding times

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The problem of determining numerically a positive solution u(x, t) of the equation ut=uxx for -T≦t≦0 from approximate values of u(x, 0)=f(x) is discussed. For a particular computational scheme estimates for the error are derived, which depend on t, T, the maximum error ɛ in the approximation for f, and on the maximum of f. It turns out that although the problem is incorrectly set in the sense ofHadamarda satisfactory numerical solution can be obtained due to the a priori assumption u≧0.


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To Mauro Picone on his 70th birth day.

This work was performed under the sponsorship of the Office of Naval Research.

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John, F. Numerical solution of the equation of heat conduction for preceding times. Annali di Matematica 40, 129–142 (1955).

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  • Heat Conduction
  • Scheme Estimate
  • Maximum Error
  • Computational Scheme
  • Preceding Time