Abstract
This chapter describes a classical technique for constructing series solutions of linear PDE problems. Classical examples like the heat equation, the wave equation and Laplace’s equations are studied in detail.
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Notes
- 1.
- 2.
The eigenvalues of S–L problems are necessarily positive, see Theorem 5.11, so \(\sqrt{\lambda }\) is a positive real number.
- 3.
The case where f is the sum of separable functions naturally arises when we consider inhomogeneous boundary conditions later in the section.
- 4.
In order to produce graphical solutions the sums in (8.46) need to be truncated to a finite number of terms. Although convergence of the series is assured—see the discussion following Definition 8.3—it is not uniform. As x approaches a discontinuity more and more terms are required in the summation (8.46a) in order to achieve any particular level of accuracy (in Fig. 8.6 we have used 100 terms).
- 5.
- 6.
We have denoted the general eigenvalue by \(\mu \) rather than \(\lambda \) in order to avoid confusion with the solutions of the corresponding one-dimensional eigenvalue problem.
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© 2015 Springer International Publishing Switzerland
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Griffiths, D.F., Dold, J.W., Silvester, D.J. (2015). Separation of Variables. In: Essential Partial Differential Equations. Springer Undergraduate Mathematics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-22569-2_8
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DOI: https://doi.org/10.1007/978-3-319-22569-2_8
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