Fourier Transforms

Let f(x) be piecewise smooth, and |f(x)|dx<. The Fourier transform and inverse Fourier transform of f are respectively defined as follows: F(ω)12πf(x)eiωxdx,f(x)F(ω)eiωxdω. Together, they are called a Fourier transform pair.

Fourier transforms are frequently used to solve partial differential equations defined over infinite spatial domains. So, let us use this Fourier transform pair to solve the following heat equation ut=kuxx,<x<,u(x,0)=f(x). To introduce some notation, let F and F1 stand for the Fourier transform and inverse Fourier transform operators, respectively. It follows that F(ut)=F(kuxx),12πtu(x,t)eiωxdx=k2π2x2u(x,t)eiωxdx,tU(ω,t)=kω2U(ω,t). This is an ordinary differential equation whose solution is U(ω,t)=c(ω)exp(kω2t). Applying the Fourier transform to the initial condition yields F[u(x,0)]=F[f(x)],U(ω,0)=12πf(x)eiωxdx=c(ω). To be able to apply the inverse Fourier transform to find u, we must know what the inverse Fourier transform of a Gaussian is, namely, G(ω)=exp(kω2t). Applying the definition yields F1[G(ω)]=F1[exp(kω2t)],g(x)=exp(kω2t)exp(iωx)dω=πktexp(x24kt). We must also know that convolution is defined as follows: F(ω)G(ω)=12πf(x¯)g(xx¯)dx¯. Finally, the inverse Fourier transform of U(ω,t)=exp(kω2t)c(ω) turns out to be the following: u(x,t)=12πktexp(x¯24kt)f(xx¯)dx¯.