Integral Equations

In General
* 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.

Classification According to the Form of the Equation
* Fredholm equations:

first kind:   f(x) = _a^b K(x, y) (y) dy ,
second kind:   (x) = f(x) + _a^b K(x, y) (y) dy .

* 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.

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

K(x,y) = _i=1ⁿA_i(x) B_i*(y) .

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

Results and Special Cases
* Fredholm alternative: @ 582, p19.
* Estimation of singular values. Use k_n:= tr(K ⁿ) = _i=1ⁿ (_i)ⁿ, and develop some approximation schemes.
@ Special cases: Bender & Ben-Naim JPA(07)mp/06 [P(x) = _a^b 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: Cochran 72; Hochstadt 73; Muskhelishvili 77; Petrovskii; Pipkin 91.
@ Types: Iovane & Ciarletta mp/03-in [hypersingular].
@ 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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Send feedback and suggestions to bombelli at olemiss.edu – Modified 11 jun 2008