Application to Partial Differential Equations
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.
KeywordsPartial Differential Equation Pressure Fluctuation Water Wave Point Charge Velocity Potential
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- 1.Note that ϕ is not a meromorphic function even if ψ is.Google Scholar
- 2.This problem anticipates some of the discussions of Section 11.Google Scholar
- 3.The standard reference on water waves is Stoker (1957).Google Scholar
- 4.A lucid exposition may be found in Curle & Davies (1968), Ch. 21.Google Scholar
- 5.This follows because in this case when (α±β) ≃ 0.Google Scholar
- 6.Stoker (1957), Ch. 4.Google Scholar
- 7.The result follows from Watson’s lemma.Google Scholar