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On Dankwerts' transformation for two variable coupled systems

Published online by Cambridge University Press:  17 April 2009

James M. Hill  and
Alex McNabb
James M. Hill
Affiliation:
Department of Mathematics, University of Wollongong, Wollongong NSW 2500, Australia
Alex McNabb
Affiliation:
Department of Mathematics and Statistics, Massey University, Palmerston North, New Zealand
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The problem of obtaining explicit solutions to coupled linear reaction-diffusion partial differential equations is generally recognised as technically very difficult. Frequently it is possible to deduce seemingly simple expressions for Laplace or Fourier transforms of the solution but such transforms tend not to be amenable to simple inversion and usually involve, for example, square roots within a square root. Fortunately however, a general uncoupling procedure has previously been established which provides explicit integral expressions in terms of classical heat functions. Such expressions are especially useful for problems with zero boundary data but non-zero initial data. The purpose of this paper is to provide the formal details necessary to deduce corresponding uncoupling transformations for two dependent variables, which preserve zero initial data and constant boundary data. For the case of one dependent variable such a transformation is known as Dankwerts' transformation. For coupled systems the existence of a Dankwerts' transformation means that together with the existing uncoupling transformation, solutions of boundary value problems involving constant boundary data, can be decomposed, just as for the single heat equation, into a contribution from the initial condition and zero boundary data and a contribution from non-zero boundary data and zero initial condition. The problem considered is highly non-trivial and the final expressions obtained are correspondingly complicated. Nevertheless the end results are explicit and together with standard integration routines constitute a powerful solution procedure.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 41 , Issue 3 , June 1990 , pp. 355 - 369
Copyright
Copyright © Australian Mathematical Society 1990

References

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