Abstract
This chapter introduces the notion of characteristics. The direction of characteristics is shown to be connected to the imposition of boundary and initial conditions that lead to well-posed problems—those that have a uniquely defined solution that depends continuously on the data. A refined classification of partial differential equations into elliptic, parabolic and hyperbolic types can then be developed.
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Notes
- 1.
The origin for t is immaterial, the intersection could be assumed to occur at \(t=t_0\), say, without affecting the resulting solution so long as we replace all occurrences of t by \(t-t_{0}\).
- 2.
In practical situations BVPs are defined on domains in \(\mathbb {R}^{2}\) and any change of variables is likely to distort the boundary of the domain and thereby complicate the imposition of boundary conditions . Such changes of variable should therefore be viewed as tools to investigate the theoretical properties of PDEs.
- 3.
The trace of a matrix A, denoted by \({{\mathrm{tr}}}A\), is the sum of its diagonal entries.
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© 2015 Springer International Publishing Switzerland
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Griffiths, D.F., Dold, J.W., Silvester, D.J. (2015). Classification of PDEs. In: Essential Partial Differential Equations. Springer Undergraduate Mathematics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-22569-2_4
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DOI: https://doi.org/10.1007/978-3-319-22569-2_4
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Publisher Name: Springer, Cham
Print ISBN: 978-3-319-22568-5
Online ISBN: 978-3-319-22569-2
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