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The Quarterly Journal of Mechanics and Applied Mathematics 1957 10(2):220-231; doi:10.1093/qjmam/10.2.220
© 1957 by Oxford University Press
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TWO METHODS FOR THE NUMERICAL SOLUTION OF MOVING-BOUNDARY PROBLEMS IN DIFFUSION AND HEAT FLOW

J. CRANK

( Courtaulds' Research Laboratory )

Two methods are described for the numerical treatment of heat-flow problems in which a transformation boundary moves through the medium. There is a corresponding problem in diffusion through a medium containing fixed sites on which some diffusing substance is instantaneously and permanently immobilized. In the first method the problem is transformed by a change of variable from one involving the simple heat-conduction equation with an awkward moving boundary to an eigenvalue problem in which the equation is slightly more complicated but the boundary is fixed. In the second method, Lagrangian interpolation formulae are used to develop finite-difference approximations to space derivatives of temperature based on values at points unequally spaced in the direction of heat flow. This enables the course of the transformation boundary to be followed.


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