Editing Integral equation (section)

=== Limits of Integration ===
<u>Fredholm</u>: An integral equation is called a [[Fredholm integral equation]] if both of the limits of integration in all integrals are fixed and constant.<ref name=":0" /> An example would be that the integral is taken over a fixed subset of <math>\mathbb{R}^n</math>.<ref name=":2" /> Hence, the following two examples are Fredholm equations:<ref name=":0" />
* Fredholm equation of the first type: <math> f(x) = \int_a^b K(x,t)\,u(t)\,dt </math>.
* Fredholm equation of the second type: <math> u(x) = f(x)+ \lambda \int_a^b K(x,t) \, u(t) \, dt. </math>
Note that we can express integral equations such as those above also using integral operator notation.<ref name=":1" /> For example, we can define the Fredholm integral operator as:<math display="block">(\mathcal{F}y)(t) := \int_{t_0}^T K(t,s) \, y(s) \, ds.</math>Hence, the above Fredholm equation of the second kind may be written compactly as:<ref name=":1" /><math display="block">y(t)=g(t)+\lambda(\mathcal{F}y)(t).</math>

{{Em|Volterra}}: An integral equation is called a [[Volterra integral equation]] if at least one of the limits of integration is a variable.<ref name=":0" /> Hence, the integral is taken over a domain varying with the variable of integration.<ref name=":2" /> Examples of Volterra equations would be:<ref name=":0" /> 
* Volterra integral equation of the first kind: <math> f(x) = \int_a^x K(x,t) \, u(t) \, dt </math>
* Volterra integral equation of the second kind: <math> u(x) = f(x) + \lambda \int_a^x K(x,t)\,u(t)\,dt. </math>
As with Fredholm equations, we can again adopt operator notation. Thus, we can define the linear Volterra integral operator <math>\mathcal{V} : C(I) \to C(I)</math>, as follows:<ref name=":2" /><math display="block">(\mathcal{V} \varphi)(t) := \int_{t_0}^t K(t,s) \, \varphi(s) \, ds</math>where <math>t \in I = [t_0 , T]</math> and ''K''(''t'',''s'') is called the kernel and must be continuous on the interval <math>D := \{(t,s) : 0 \leq s \leq t \leq T \leq \infty\}</math>.<ref name=":2" /> Hence, the Volterra integral equation of the first kind may be written as:<ref name=":2" /><math display="block">(\mathcal{V}y)(t)=g(t)</math>with <math>g(0)=0</math>. In addition, a linear Volterra integral equation of the second kind for an unknown function <math> y(t) </math> and a given continuous function <math> g(t) </math> on the interval <math> I </math> where <math> t \in I </math>:<math display="block">y(t)=g(t)+(\mathcal{V} y)(t).</math>{{Em|Volterra–Fredholm}}: In higher dimensions, integral equations such as Fredholm–Volterra integral equations (VFIE) exist.<ref name=":2" /> A VFIE has the form:<math display="block">u(t,x) = g(t,x)+(\mathcal{T}u)(t,x)</math>with <math>x \in \Omega</math> and <math>\Omega</math> being a closed bounded region in <math>\mathbb{R}^d</math> with piecewise smooth boundary.<ref name=":2" /> The Fredholm-Volterra Integral Operator <math>\mathcal{T} : C(I \times \Omega) \to C(I \times \Omega)</math> is defined as:<ref name=":2" />

<math display="block">(\mathcal{T}u)(t,x) := \int_0^t \int_\Omega K(t,s,x,\xi) \, G(u(s, \xi)) \, d\xi \, ds.</math>Note that while throughout this article, the bounds of the integral are usually written as intervals, this need not be the case.<ref name=":1" /> In general, integral equations don't always need to be defined over an interval <math>[a,b] = I</math>, but could also be defined over a curve or surface.<ref name=":1" />