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*)
= ∫_{a}^{b}
*K*(*x*, *y*) *φ*(*y*) d*y* ,

second kind: *φ*(*x*) = *f*(*x*)
+ *λ*
∫_{a}^{b}
*K*(*x*, *y*) *φ*(*y*) d*y* ;

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=1}^{n}
*A*_{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__

*

**Results and Special Cases**

* __Fredholm alternative__: @ 582, p19.

* __Estimation of singular
values__. Use *k*_{n}:=
tr(*K*^{ n})
= ∑_{i=1}^{n}
(*λ*_{i})^{n},
and develop some approximation schemes.

@ __Special cases__: Bender & Ben-Naim JPA(07)mp/06 [*P*(*x*)
= ∫_{a}^{b}
d*y* *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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send feedback and suggestions to bombelli at olemiss.edu – modified 21 jan 2020