Abstract
The problem of finding the steady state solution of the heat equation on the unit square, a basic example of the Dirichlet problem, is solved, and in the process ideas and theorems are developed, which are applicable to a broad range of mathematical problems. These important ideas include vectors spaces and linear operators, inner product spaces including \(R^N\), Fourier series and their convergence, including Fejér’s convergence theorem. The maximum principle is established for the case of the square.
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References
Weinberger HA (1965) First course in partial differential equations. Blaisdell Publishing Company, New York
Körner T (1988) Fourier analysis. Cambridge University Press, Cambridge
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Hromadka, T., Whitley, R. (2014). The Heat Equation. In: Foundations of the Complex Variable Boundary Element Method. SpringerBriefs in Applied Sciences and Technology. Springer, Cham. https://doi.org/10.1007/978-3-319-05954-9_1
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DOI: https://doi.org/10.1007/978-3-319-05954-9_1
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Publisher Name: Springer, Cham
Print ISBN: 978-3-319-05953-2
Online ISBN: 978-3-319-05954-9
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