Abstract
This chapter introduces the notions of boundary and initial value problems. Some operator notation is developed in order to represent boundary and initial value problems in a compact manner. Familiarity with this notation is essential for understanding the presentation in later chapters. An initial classification of partial differential equations is then developed.
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Notes
- 1.
The outward normal direction on \(y=0\) is in the direction of \(\vec {n} = (0, -1)\).
- 2.
Nonlinear boundary conditions such as \( u_n={\text {e}}^u\) on \({\partial \varOmega }\) are certainly possible, but will not be considered here.
- 3.
The examples of ill-posed problems that we shall give are clear cut without the need to specify precisely which norms are used.
- 4.
This proof is deferred to Chap. 7.
- 5.
Note that we cannot conclude that this problem is well posed since we would need to consider all possible choices of the data in order to make that claim.
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© 2015 Springer International Publishing Switzerland
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Griffiths, D.F., Dold, J.W., Silvester, D.J. (2015). Boundary and Initial Data. In: Essential Partial Differential Equations. Springer Undergraduate Mathematics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-22569-2_2
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DOI: https://doi.org/10.1007/978-3-319-22569-2_2
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Publisher Name: Springer, Cham
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Online ISBN: 978-3-319-22569-2
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