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A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type

  • J. Crank (a1) and P. Nicolson (a2)
Extract

This paper is concerned with methods of evaluating numerical solutions of the non-linear partial differential equation

where

subject to the boundary conditions

A, k, q are known constants.

Equation (1) is of the type which arises in problems of heat flow when there is an internal generation of heat within the medium; if the heat is due to a chemical reaction proceeding at each point at a rate depending upon the local temperature, the rate of heat generation is often defined by an equation such as (2).

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(1)Hartree, D. R. and Womersley, J. R.Proc. Roy. Soc. A, 161 (1937), 353.
(2)Jackson, R., Sarjant, R. J., Wagstaff, J. B., Eyres, N. R., Hartree, D. R. and Ingham, J. The Iron and Steel Institute, 1944, Paper No. 15/1944 of the Alloy Steels Research Committee.
Eyres, N. R., Hartree, D. R. and others. Philos. Trans. A, 240 (1946), 1.
(3)Richardson, L. F.Philos. Trans. A, 210 (1910), 307.
(4)Richardson, L. F.Philos. Trans. A, 226 (1927), 299.
(5) A.M.P. Memo. No. 131. IM: Title ‘On the solution of certain boundary problems’.
(6)Levy, and Baggott, . Numerical Studies in Differential Equations, Chapter iv.
(7)Lowan, A. N.Amer. J. Math. 56, no. 3 (1934), 396.
(8)Bamford, C. H., Crank, J. and Malan, D. H.Proc. Cambridge Phil. Soc. 42 (1946), 166.
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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