Abstract
The final chapter identifies thirteen projects, involving both theory and computation, that are intended to extend and test understanding of the material in earlier chapters.
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- 1.
This wonderful property (the finite difference solution is exact at every grid point) does not generalise to advection–diffusion problems in two or more space dimensions.
- 2.
For real numbers a and b the roots of the quadratic polynomial \(p(\xi ) = \xi ^2 + a\xi + b\) lie strictly within the unit circle, i.e., \(|\xi |<1\), if, and only if, \(p(1)>0\), \(p(-1)>0\) and \(p(0)<1\). These are known as the Jury conditions —for a proof see Griffiths and Higham [7, Lemma 6.10].
- 3.
A slightly different notation is adopted that uses \(x_{1}, x_{2}\) in lieu of x, y so as to facilitate the presentation of the scheme and its analysis and also to allow ready generalisation to higher space dimensions. Note also that a factor 2 has been introduced into the definition of \(r_{j}\) so as to avoid fractions occurring at a later stage.
- 4.
This is a mundane exercise compared with the celebrated article “Can one hear the shape of a drum?” by Mark Kac [10], where it transpires that the answer is no because the Laplacian operator in two dimensions can have the same eigenvalues on two different domains (drum shapes).
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© 2015 Springer International Publishing Switzerland
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Griffiths, D.F., Dold, J.W., Silvester, D.J. (2015). Projects. In: Essential Partial Differential Equations. Springer Undergraduate Mathematics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-22569-2_13
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DOI: https://doi.org/10.1007/978-3-319-22569-2_13
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