Fredholm Integral Equation

The Fredholm Integral Equation is an integral equation whose core is the Fredholm kernel . Named after the Swedish mathematician Ivar Fredholm . Over time, the study of the Fredholm equation has grown into an independent section of functional analysis - the Fredholm theory , which studies the Fredholm kernels and Fredholm operators .

Content

General Theory

A general theory based on Fredholm equations is known as Fredholm theory . In theory, an integral transformation of a special form is considered

$\psi (s)=\int \limits _{a}^{b}\!K(s,t)\varphi (t)\,dt$ ${\ displaystyle \ psi (s) = \ int \ limits _ {a} ^ {b} \! K (s, t) \ varphi (t) \, dt}$

where is the function $K$ ${\ displaystyle K}$ is called the kernel of the equation, and the operator $A$ ${\ displaystyle A}$ defined as

$A\varphi =\int \limits _{a}^{b}\!K(s,t)\varphi (t)\,dt$ ${\ displaystyle A \ varphi = \ int \ limits _ {a} ^ {b} \! K (s, t) \ varphi (t) \, dt}$ , is called the Fredholm operator (or integral).

One of the fundamental results is the fact that the kernel K is a compact operator , otherwise known as the Fredholm operator . Compactness can be shown with uniform continuity . As an operator, a spectral theory can be applied to the kernel that studies the spectrum of eigenvalues .

The equation of the first kind

The inhomogeneous Fredholm equation of the first kind has the form:

g(t)=\int \limits _{a}^{b}\!K(t,s)f(s)\,ds

{\ displaystyle g (t) = \ int \ limits _ {a} ^ {b} \! K (t, s) f (s) \, ds}

and the problem is that for a given continuous kernel function $K(t,s)$ ${\ displaystyle K (t, s)}$ and functions $g(t)$ ${\ displaystyle g (t)}$ find function $f(s)$ ${\ displaystyle f (s)}$ .

If the kernel is a function of the difference of its arguments, i.e. $K(t,s)=K(t-s)$ ${\ displaystyle K (t, s) = K (ts)}$ , and the limits of integration $\pm \infty$ ${\ displaystyle \ pm \ infty}$ , then the right side of the equation can be rewritten as a convolution of functions $K$ ${\ displaystyle K}$ and $f$ ${\ displaystyle f}$ , and, therefore, the solution is given by the formula

f(t)={\mathcal {F}}_{\omega }^{-1}\left[{{\mathcal {F}}_{t}[g(t)](\omega ) \over {\mathcal {F}}_{t}[K(t)](\omega )}\right]=\int \limits _{-\infty }^{\infty }\!{{\mathcal {F}}_{t}[g(t)](\omega ) \over {\mathcal {F}}_{t}[K(t)](\omega )}e^{2\pi i\omega t}\,d\omega

{\ displaystyle f (t) = {\ mathcal {F}} _ {\ omega} ^ {- 1} \ left [{{\ \ mathcal {F}} _ {t} [g (t)] (\ omega) \ over {\ mathcal {F}} _ {t} [K (t)] (\ omega)} \ right] = \ int \ limits _ {- \ infty} ^ {\ infty} \! {{\ mathcal { F}} _ {t} [g (t)] (\ omega) \ over {\ mathcal {F}} _ {t} [K (t)] (\ omega)} e ^ {2 \ pi i \ omega t} \, d \ omega}

Where ${\mathcal {F}}_{t}$ ${\ displaystyle {\ mathcal {F}} _ {t}}$ and ${\mathcal {F}}_{\omega }^{-1}$ ${\ displaystyle {\ mathcal {F}} _ {\ omega} ^ {- 1}}$ - direct and inverse Fourier transforms, respectively. Necessary and sufficient conditions for the existence of a solution are determined by Picard's theorem .

Second kind equation

The inhomogeneous Fredholm equation of the second kind looks like this:

\varphi (s)=\lambda \int \limits _{a}^{b}\!K(s,t)\varphi (t)\,dt+f(s)

{\ displaystyle \ varphi (s) = \ lambda \ int \ limits _ {a} ^ {b} \! K (s, t) \ varphi (t) \, dt + f (s)}

.

The challenge is to have a core $K(t,s)$ ${\ displaystyle K (t, s)}$ and function $f(t)$ ${\ displaystyle f (t)}$ find function $\varphi (t)$ ${\ displaystyle \ varphi (t)}$ . Moreover, the existence of a solution and its multiplicity depends on the number $\lambda$ ${\ displaystyle \ lambda}$ called the characteristic number (the opposite is called an eigenvalue ). The standard solution approach uses the concept of resolvent ; a solution written in the form of a series is known as the Liouville-Neumann series .

Links

Integral Equations: Exact Solutions - From EqWorld: The World of Mathematical Equations.

Integral Equations: Solution Methods - From EqWorld: World of Mathematical Equations.