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  • Mathematical Proceedings of the Cambridge Philosophical Society, Volume 43, Issue 1
  • January 1947, pp. 50-67

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

  • J. Crank (a1) and P. Nicolson (a2)
  • DOI:
  • Published online: 24 October 2008

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


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)D. R. Hartree and J. R. Womersley Proc. Roy. Soc. A, 161 (1937), 353.

N. R. Eyres , D. R. Hartree and others. Philos. Trans. A, 240 (1946), 1.

(3)L. F. Richardson Philos. Trans. A, 210 (1910), 307.

(4)L. F. Richardson Philos. Trans. A, 226 (1927), 299.

(7)A. N. Lowan Amer. J. Math. 56, no. 3 (1934), 396.

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