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Towards the beginning of the twentieth century, results similar to that of Arzelà and Ascoli began to accumulate in the area of integral equations, as investigated by David Hilbert and Erhard Schmidt.
In mathematical physics, Liouville made two fundamental contributions: the Sturm–Liouville theory, which was joint work with Charles François Sturm, and is now a standard procedure to solve certain types of integral equations by developing into eigenfunctions, and the fact (also known as Liouville's theorem) that time evolution is measure preserving for a Hamiltonian system.
In statistical mechanics the Ornstein–Zernike equation (named after Leonard Ornstein and Frits Zernike) is an integral equation for defining the direct correlation function.
(b) Integral equation methods, e.g., the CHNC, an acronym for the classical-map hyper-netted-chain method, or the Fermi hyper-netted-chain method.
At Dublin, Heitler's work with H. W. Peng on radiation damping theory and the meson scattering process resulted in the Heitler-Peng integral equation.