Abstract
The use of the Fourier transform to obtain a form of solution to a partial differential equation (together with associated boundary conditions) is a very general technique. For simple problems, the integral representation obtained as the solution will be amenable to exact analysis; more often the method converts the original problem to the technical matter of evaluating a difficult integral. Numerical methods may be necessary in general, although asymptotic and other useful information can often be obtained directly by appropriate methods. We illustrate some of the more simple problems in this section, leaving applications involving mixed boundary values, Green’s functions, and transforms in several variables until later.
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Footnotes
Note that ϕ is not a meromorphic function even if ψ is.
This problem anticipates some of the discussions of Section 11.
The standard reference on water waves is Stoker (1957).
A lucid exposition may be found in Curle & Davies (1968), Ch. 21.
This follows because in this case when (α±β) ≃ 0.
Stoker (1957), Ch. 4.
The result follows from Watson’s lemma.
This problem is considered by K. K. Puri, J. Eng. Math. (1970), 4, 283.
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© 1985 Springer Science+Business Media New York
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Davies, B. (1985). Application to Partial Differential Equations. In: Integral Transforms and their Applications. Applied Mathematical Sciences, vol 25. Springer, New York, NY. https://doi.org/10.1007/978-1-4899-2691-3_8
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DOI: https://doi.org/10.1007/978-1-4899-2691-3_8
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-96080-7
Online ISBN: 978-1-4899-2691-3
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