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The numerical solution of the heat conduction equation in one dimension

Published online by Cambridge University Press:  24 October 2008

M. Wadsworth  and
A. Wragg
M. Wadsworth
Affiliation:
Royal College of Advanced Technology, Salford
A. Wragg
Affiliation:
Royal College of Advanced Technology, Salford
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Abstract

The replacement of the second space derivative by finite differences reduces the simplest form of heat conduction equation to a set of first-order ordinary differential equations. These equations can be solved analytically by utilizing the spectral resolution of the matrix formed by their coefficients. For explicit boundary conditions the solution provides a direct numerical method of solving the original partial differential equation and also gives, as limiting forms, analytical solutions which are equivalent to those obtainable by using the Laplace transform. For linear implicit boundary conditions the solution again provides a direct numerical method of solving the original partial differential equation. The procedure can also be used to give an iterative method of solving non-linear equations. Numerical examples of both the direct and iterative methods are given.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 60 , Issue 4 , October 1964 , pp. 897 - 907
Copyright
Copyright © Cambridge Philosophical Society 1964

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References

REFERENCES

(1)Aitken, A. C.Proc. Roy. Soc. Edinburgh, 46 (1926), 289–305.Google Scholar
(2)Churchill, R. V.Operational mathematics (New York, 1958).Google Scholar
(3)Crank, J. and Nicolson, P.Proc. Cambridge Philos. Soc. 43 (1947), 50–67.CrossRefGoogle Scholar
(4)Lanczos, C.Applied analysis (London, 1957).Google Scholar
(5)Ledermann, W. and Reuter, G. E. H.Philos. Trans. Roy. Soc. London, Ser. A, 246 (1954), 321–369.Google Scholar
(6)Modern computing methods (H.M.S.O.; London, 1961).Google Scholar
(7)Rutherford, D. E.Proc. Roy. Soc. Edinburgh. Sect. A, 62 (1947), 229–236.Google Scholar