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

where βj are a set of initially unknown constants, fj are approximating
functions, usually radial basis functions, N is the number of boundary nodes
and L is the number of internal nodes. The particular solutions ûj and the
fj are related by
∇ 2 ûj = fj
After some algebraic manipulation we arrive at the expression
∇ 2 u =
N+L �
j=1
βj
� � 2
∇ ûj
which is then multiplied by the fundamental solution and integrated over the
domain. We apply Green’s second theorem, as before, but this time it must
be applied to both sides of the equation, hence the name ‘dual’ reciprocity.
Thus, as in the boundary element method, we are able to find the potential
and flux at all points on the boundary and then to find the potential at
points of interest.
The dual reciprocity method allows the solution of a variety of problems
where b may be a constant or a function of any of x, y, u and t. Naturally
the method becomes increasingly complex to use when b is a function of
more variables. The heat equation may be solved using the dual reciprocity
method and this is described in Partridge et al. (1992).
Tanaka et al. (2003) solved transient heat conduction problems in three
dimensions using a method similar to that described above, and they used
a finite difference scheme to approximate the time derivative. Each time
step related back to the previous result as a kind of new initial condition.
They noted that the time-step width was an important factor for accuracy
and stability and suggested that this was considered when setting up the
problem. They concluded that very accurate results can be obtained if
appropriate computational conditions are selected.
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