I find it hard to believe that the article on fractional differentiation only briefly mentions the Fourier transform. For functions on R^n with vanishing integral (or periodic functions on R^n with vanishing integral), the Fourier transform allows you to define arbitrary powers of the (positive) Laplacian by taking the Fourier transform, multiplying by |\xi|^\alpha, and then taking the inverse Fourier transform. If n=1, this process yields the fractional derivatives in the linked article.
Something else that's great is that this works on (compact or asymptotically Euclidean) manifolds, too! You can make sense of the Laplacian on these spaces, and then spectral theory lets you define its fractional powers. The theory of pseudodifferential operators lets you realize these powers fairly explicitly as oscillatory integrals.
If I remember the lecture I went to half a dozen years ago fractional calculus allows for some elegant solutions to cycloid curves and half infinite sheets of charge.
Something else that's great is that this works on (compact or asymptotically Euclidean) manifolds, too! You can make sense of the Laplacian on these spaces, and then spectral theory lets you define its fractional powers. The theory of pseudodifferential operators lets you realize these powers fairly explicitly as oscillatory integrals.