Fourier Transform: Applications

TLDR: The Fourier transform is very useful in solving a variety of linear constant coefficient ordinary and partial differential equations describing processes which take place over an infinite interval, −∞ < x < ∞.

Abstract: The Fourier transform is very useful in solving a variety of linear constant coefficient ordinary and partial differential equations describing processes which take place over an infinite interval, −∞ < x < ∞. We will provide a number of examples of this sort of application in the present section. Our first example involves a simple, time independent, equilibrium process. Example 1 We consider a stretched string, or cord, with small transverse displacement y(x), subject to an external transverse force f (x) and a transverse restoring force −κ y(x), maintained at tension τ > 0 over the interval −∞ < x < ∞ and constrained so that lim |x| → ∞ y(x) = 0. It can then be shown that y(x) satisfies τ d 2 y dx 2 − κ y(x) + f (x) = 0. Taking a 2 = κ/τ and applying the Fourier transform to both sides of this equation, using the differentiation property (twice) we have − ξ 2 + a 2 ˆ y(ξ) + 1 τ ˆ f (ξ) = 0 ⇒ ˆ y(ξ) = 1 τ ˆ f (ξ) ξ 2 + a 2. Using the convolution property of the Fourier transform we obtain y(x) = 1 τ ∞ −∞ F −1 1 ξ 2 + a 2 (r) f (x − r) dr as the solution. To make any further progress on this we need to find the function K(x) such thatˆK(ξ) ≡ (F K) (ξ) = 1 ξ 2 +a 2. It turns out that this function is K(x) =        e −ax 2a , x > 0, e ax 2a , x < 0.