Integral Equations  

In General
* Idea: Equations in which the unknown is a function which appears under an integral sign.
* History: Specific cases had been studied earlier, but the theory of integral equations started with two papers by Fredholm in 1900 and 1903; The work was later continued by Volterra, Hilbert, E Schmidt, and others.
> Online resources: see MathWorld page; Wikipedia page.

Classification According to the Form of the Equation
* Fredholm equations: First kind if the unknown function appears only inside the integral; Second kind if it appears both inside and outside the integral,

first kind:   f(x) = ab K(x, y) φ(y) dy ,
second kind:   φ(x) = f(x) + λ ab K(x, y) φ(y) dy ;

The Fredholm integral equations of the first kind are a classical example of ill-posed problem in the sense of Hadamard.
* Volterra equations: Same, but with K(x,y) = 0 for y > x (i.e., the upper limit of integration is x).
* Homogeneous: The equations above are homogeneous if f(x) = 0.
@ Fredholm equations: De Micheli & Viano IEOT(12)-a1602 [and topological information theory]; > s.a. metametarials.
@ Volterra equations: Brunner 17.

Classification According to the Kind of Kernel
* Finite rank kernels: (Also called degenerate, or separable)

K(x,y) = ∑i=1n Ai(x) Bi*(y) .

For such a kernel, the equation can be reduced to a system of linear algebraic equations.
* Hilbert-Schmidt.
* Class Cp.
* Trace-class or nuclear.

Results and Special Cases
* Fredholm alternative: @ 582, p19.
* Estimation of singular values. Use kn:= tr(K n) = ∑i=1n (λi)n, and develop some approximation schemes.
@ Special cases: Bender & Ben-Naim JPA(07)mp/06 [P(x) = ∫ab dy w(y) P(y) P(x+y) and orthogonal polynomials]; Cacciari & Moretti JPA(07) [class with applications in quantum mechanics].

References > s.a. differential equations [integro-differential].
@ General: Petrovskii 71; Cochran 72; Hochstadt 73; Muskhelishvili 77; Pipkin 91; Polyanin & Manzhirov 08 [handbook]; Wazwaz 15.
@ Types: Iovane & Ciarletta mp/03-proc [hypersingular]; Epelbaum et al a2001 [with singular potentials, and renormalization in quantum field theory].
@ Numerical solution: Atkinson 97 [second kind].
@ Related topics: Ibragimov et al ND(02)mp/01 [symmetries]; Scharnhorst JMP(03)mp/02 [Grassmann integral equations].
> Applications in physics: see mathematical physics; quantum oscillators; scattering.

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