Equations in which the unknown function appears under an integral sign — a complementary framework to differential equations, often arising as their boundary-integral reformulations (Green's-function method). Classical taxonomy (Volterra…
integral-equations
Fredholm alternative: compact K implies finite-dim eigenspace or unique solve
Fredholm 1903 (Acta Math. 27:365) for a compact integral operator (Kφ)(x) = ∫_a^b K(x,y)·φ(y) dy with square-integrable kernel: the…
Volterra 2nd-kind: φ(x) = g(x) + λ∫_a^x K(x,y)φ(y) dy always solvable
Volterra 1896 — the 2nd-kind Volterra integral equation φ(x) = g(x) + λ·∫_a^x K(x,y)·φ(y) dy (variable upper limit, i.e. causal kernel)…
Neumann series: (I − λK)⁻¹ = Σ λⁿ Kⁿ for |λ|·‖K‖ < 1
Neumann 1877 series expansion of the resolvent operator (I − λK)⁻¹ for a bounded linear operator K with |λ|·‖K‖ < 1: (I − λK)⁻¹ =…
Fredholm K(x,y)=xy on [0,1]: eigenvalue λ=3 exact
Exact symbolic solution of the 2nd-kind Fredholm eigenvalue equation φ(x) = λ·∫_0^1 x·y·φ(y) dy with rank-1 degenerate kernel K(x,y) = x·y…
Volterra φ(x) = 1 + ∫_0^x φ(y)dy ⇒ φ(x) = exp(x) exact
Exact symbolic solution of the 2nd-kind Volterra equation φ(x) = 1 + ∫_0^x φ(y) dy (kernel K(x,y) ≡ 1, driving term g(x) ≡ 1). …
Neumann resolvent for K=c const: R = c/(1-λc), Σ geometric exact
Exact closed-form resolvent for the simplest non-trivial Fredholm kernel K(x, y) ≡ c (constant) on [0, 1]. Iterated kernels: K¹(x, y) = c;…
Fredholm alternative theorem
For compact operator K on Hilbert space H: equation (I - K)x = y has unique solution iff homogeneous (I - K)z = 0 has only trivial…
Abel integral equation (weak singular)
Abel 1826 tautochrone problem: ∫_0^x (x-t)^{-α} f(t) dt = g(x) inverted via fractional-integration. Solution f(x) = (sin πα/π) d/dx ∫_0^x…
Wiener-Hopf method
Wiener-Hopf 1931: solves convolution-integral-equations on half-line ∫_0^∞ K(x-t)f(t)dt = g(x) via factorisation in complex plane. …
Nyström method (quadrature)
Nyström 1928: discretise integral equation ∫K(x,t)f(t)dt = g(x) via quadrature rule → linear system for f at nodes. Convergence O(h^p)…
Neumann series (iterative inversion)
(I - K)^{-1} = Σ K^n converges if ||K|| < 1. Successive-approximation solution to integral equations. Foundation of perturbation theory +…
Green's function (integral form)
G(x,y) is fundamental-solution: L_x G(x,y) = δ(x-y). Solution to L u = f given by u(x) = ∫G(x,y)f(y)dy. Encodes operator-inverse as…
Fredholm alternative (1903)
Fredholm 1903: for compact operator K on Banach space, either (I-K)x = y has unique solution for all y, or homogeneous (I-K)x = 0 has…
Volterra integral equation (second kind)
Volterra 1896: f(x) = phi(x) + integral_0^x K(x,t) phi(t) dt; iterative Neumann series converges for Lipschitz kernel; basis of…
Hammerstein equation (nonlinear integral)
Hammerstein 1930: phi(x) = integral K(x,t) f(t, phi(t)) dt; nonlinear integral equation; existence via Schauder fixed-point; applications…
Wiener-Hopf equation (half-line)
Wiener-Hopf 1931: integral equation on half-line with translation-invariant kernel K(x-y); factorization of symbol K-hat in Fourier-Laplace…
Singular integral (Cauchy principal value)
Cauchy 1827 PV integral; Riesz-Privalov 1916 boundary-values of holomorphic functions; Calderon-Zygmund 1952 singular-integral L^p theory.
Abel integral equation (fractional)
Abel 1826 tautochrone problem: phi(x) = integral_0^x f(t)/sqrt(x-t) dt; fractional half-derivative; used in tomography (Abel inversion) +…
Fredholm theorem (1903)
I Fredholm 1903 first kind + second kind; modern compact-operator spectral-theory + Hilbert-Schmidt.
Galerkin (1915)
B Galerkin 1915 + Bubnov 1913 Galerkin-method for IE; modern boundary-element-method numerical implementation.
Collocation method
Modern collocation IE-method; spectral + Chebyshev + adaptive; Atkinson 1997 'Numerical Solution of Integral Equations'.
Nyström method (1928)
E Nyström 1928 quadrature-based IE-method; modern fast-multipole + skeletonization for large-scale.
Radiative transfer (Chandrasekhar 1950)
S Chandrasekhar 1950 (Nobel 1983) Radiative-Transfer; modern atmosphere + ocean radiation + neutron-transport.
Hilbert-Schmidt (1907)
D Hilbert + E Schmidt 1907 IE-eigenvalue theory; modern compact-self-adjoint-operator spectral decomposition.