Abstract
In the earlier chapter we have seen how the derivative operator is represented by a matrix. In this chapter we apply this idea to the solution of heat conduction problems in one dimension. First, one-domain problems are discussed, then two-domain problems are discussed.
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Notes
- 1.
The reader ought to be able to figure out, without any computation, the value of \(\widehat{u}_{3}\) that he should get from the solution of (5).
- 2.
Notice that the matrix \(\left|\begin{array}{cc} D_{00}&D_{0N}\\ D_{N0}&D_{ NN} \end{array}\right|\) is somewhat particular since it reads as \(A=\left|\begin{array}{cc} -a&-b\\ b&a \end{array}\right|\) whose square is simply given by \(A^2=\left(a^2-b^2\right)I\), \(I\) being the identity matrix. Computing then \(A^{-1}\) is straightforward, which leads to the expressions (55) and (56) for \(C_{0v}\) and \(C_{Nv}\).
References
Isaacson E, Keller H (1966) Analysis of numerical methods. Wiley, New York
Macaraeg M, Streett C (1986) Improvements in spectral collocation discretization through a multiple domain technique. Appl Numer Math 2:95–108
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© 2012 Springer-Verlag Berlin Heidelberg
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Guo, W., Labrosse, G., Narayanan, R. (2012). Steady One-Dimensional (1D) Heat Conduction Problems. In: The Application of the Chebyshev-Spectral Method in Transport Phenomena. Lecture Notes in Applied and Computational Mechanics, vol 68. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34088-8_3
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DOI: https://doi.org/10.1007/978-3-642-34088-8_3
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