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: *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 integral equations__: De Micheli & Viano IEOT(12)-a1602 [and topological information theory].

**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].

@ __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 22
feb 2016