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;…