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Advanced Mathematics for Applications by Andrea Prosperetti

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5

Laplace Transform: Applications

The Laplace transform, the essential theory of which is covered in Chapter 11, is a useful tool for the solution of the initial-value problem for ordinary and partial differential equations and for the solution of certain types of integral equations. Since very often the transform is applied with respect to a time-like variable, we will use the letter t to indicate this variable.

The Laplace transform is applicable to functions that vanish for negative values of t (i.e., the “past”) and, as t → ∞, do not grow faster than ect for some real constant c (see §11.1). Like the Fourier sine and cosine transforms, it is a one-sided transform in the sense that it is defined over the half line 0 < t < ∞. Its inversion, ...

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