10 Fourier and Laplace Transform Methods
On a bounded interval, separation of variables expands the solution in a Fourier series. On the whole line the discrete frequencies \(k\pi /L\) fill out a continuum, and the series becomes the Fourier transform. Its power for linear equations with constant coefficients is that it turns differentiation into multiplication: a partial differential equation in \((x, t)\) becomes an ordinary differential equation in \(t\) for each frequency, which can be solved explicitly. The Laplace transform does the same in time, and is the tool for problems driven by boundary data.
10.1 The Fourier transform
10.2 Solving equations on the line
10.3 The Laplace transform
10.4 Transforms in software
10.5 Practice Problems
10.2 Solving equations on the line
10.3 The Laplace transform
10.4 Transforms in software
10.5 Practice Problems
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