Laplace transform isotherm .pdf - University of Hertfordshire ...

in the domain we only need to evaluate the approximate solution, whereas
the boundary element method requires us to use numerical quadrature. The
method of fundamental solutions is one of a family of methods known as
mesh-free methods (Liu 2003). These techniques are not yet widely used
but there is an increasing interest in them since they offer advantages from
the point of view of problem set-up.
2.2.9 The isotherm migration method
Generally this method is particularly useful for solving problems involving
phase change, that is a change in the material from one state to another,
for example, a change from ice to water. However, there might be situations
when rather than finding the temperature at certain points in the domain, we
would like to know the movement of lines passing through points having the
same temperature. These are known as isotherms. The isotherm migration
method enables us to do this, but first we have to re-formulate the heat
equation. The heat equation (2.2) defines the temperature u as a function
of space (x,y) and time t. For this method we need to re-write the heat
equation so that position is now a function of temperature and time. This
leads to a non-linear partial differential equation. The method was proposed
by Chernous’ko (1970) where he described the method for one-dimensional
problems and the ideas were further developed by Crank and Phahle (1973).
Since we shall be referring to the isotherm migration method throughout our
work, we shall not discuss it here, but a full explanation of its operation will
be provided further on.
2.3 Moving boundary problems
Moving boundaries occur frequently in diffusion problems. Such problems
may involve a change of state which occurs on the interface, for example,
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