Integral Equations| 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.
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 21 jan 2020