Editing Integral equation (section)

==Classification and overview==
Various classification methods for integral equations exist. A few standard classifications include distinctions between linear and nonlinear; homogeneous and inhomogeneous; Fredholm and Volterra; first order, second order, and third order; and singular and regular integral equations.<ref name=":0" /> These distinctions usually rest on some fundamental property such as the consideration of the linearity of the equation or the homogeneity of the equation.<ref name=":0" /> These comments are made concrete through the following definitions and examples:

=== Linearity ===
{{Em|Linear}}: An integral equation is linear if the unknown function ''u''(''x'') and its integrals appear linearly in the equation.<ref name=":0" /> Hence, an example of a linear equation would be:<ref name=":0" /><math display="block">u(x) = f(x) + \lambda\int_{\alpha(x)}^{\beta(x)}K(x,t) \cdot u(t) \, dt</math>As a note on naming convention: i) ''u''(''x'') is called the unknown function, ii) ''f''(''x'') is called a known function, iii) ''K''(''x'',''t'') is a function of two variables and often called the [[Kernel (integral operator)|Kernel]] function, and iv) ''λ'' is an unknown factor or parameter, which plays the same role as the [[eigenvalue]] in [[linear algebra]].<ref name=":0" />

{{Em|Nonlinear}}: An integral equation is nonlinear if the unknown function ''''u''(''x'') or any of its integrals appear nonlinear in the equation.<ref name=":0" /> Hence, examples of nonlinear equations would be the equation above if we replaced ''u''(''t'') with <math>u^2(x), \, \, \cos(u(x)), \, \text{or } \,e^{u(x)}</math>, such as:<math display="block">u(x) = f(x) + \int_{\alpha(x)}^{\beta(x)}K(x,t) \cdot u^2(t) \, dt</math>Certain kinds of nonlinear integral equations have specific names.<ref name=":2" /> A selection of such equations are:<ref name=":2" />

* Nonlinear Volterra integral equations of the second kind which have the general form: <math> u(x) = f(x) + \lambda \int_a^x K(x,t) \, F(x, t, u(t)) \, dt, </math>  where ''{{mvar|F}}'' is a known function.<ref name=":2" />
* Nonlinear Fredholm integral equations of the second kind which have the general form: <math>f(x)=F\left(x, \int_a^b K(x,y,f(x),f(y)) \, dy\right)</math>.<ref name=":2" />
* A special type of nonlinear Fredholm integral equations of the second kind is given by the form: <math>f(x)=g(x)+ \int_a^b K(x,y,f(x),f(y)) \, dy</math>, which has the two special subclasses:<ref name=":2" />
** Urysohn equation: <math>f(x)=g(x)+ \int_a^{b} k(x,y,f(y)) \, dy</math>.<ref name=":2" />
** Hammerstein equation: <math>f(x)=g(x)+ \int_a^b k(x,y) \, G(y,f(y)) \, dy</math>.<ref name=":2" />

More information on the Hammerstein equation and different versions of the Hammerstein equation can be found in the Hammerstein section below.

=== Location of the unknown equation ===
{{Em|First kind}}: An integral equation is called an integral equation of the first kind if the unknown function appears only under the integral sign.<ref name=":2" /> An example would be: <math> f(x) = \int_a^b K(x,t)\,u(t)\,dt </math>.<ref name=":2" />

{{Em|Second kind}}: An integral equation is called an integral equation of the second kind if the unknown function also appears outside the integral.<ref name=":2" />

{{Em|Third kind}}: An integral equation is called an integral equation of the third kind if it is a linear Integral equation of the following form:<ref name=":2" /><math display="block"> g(t)u(t) + \lambda \int_a^b K(t,x)u(x)  \, dx = f(t) </math>where ''g''(''t'') vanishes at least once in the interval [''a'',''b'']<ref>{{Cite journal |last1=Bart |first1=G. R. |last2=Warnock |first2=R. L. |date=November 1973 |title=Linear Integral Equations of the Third Kind |url=http://epubs.siam.org/doi/10.1137/0504053 |journal=SIAM Journal on Mathematical Analysis |language=en |volume=4 |issue=4 |pages=609–622 |doi=10.1137/0504053 |issn=0036-1410|url-access=subscription }}</ref><ref>{{Cite journal |last=Shulaia |first=D. |date=2017-12-01 |title=Integral equations of the third kind for the case of piecewise monotone coefficients |journal=Transactions of A. Razmadze Mathematical Institute |language=en |volume=171 |issue=3 |pages=396–410 |doi=10.1016/j.trmi.2017.05.002 |issn=2346-8092|doi-access=free }}</ref> or where ''g''(''t'') vanishes at a finite number of points in (''a'',''b'').<ref>{{Cite journal |last=Sukavanam |first=N. |date=1984-05-01 |title=A Fredholm-type theory for third-kind linear integral equations |journal=Journal of Mathematical Analysis and Applications |language=en |volume=100 |issue=2 |pages=478–485 |doi=10.1016/0022-247X(84)90096-9 |issn=0022-247X|doi-access=free }}</ref>

=== 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" />

=== Homogeneity ===
{{Em|Homogeneous}}: An integral equation is called homogeneous if the known function <math>f</math> is identically zero.<ref name=":0" />

{{Em|Inhomogeneous}}: An integral equation is called inhomogeneous if the known function <math>f</math> is nonzero.<ref name=":0" />

=== Regularity ===
{{Em|Regular}}: An integral equation is called regular if the integrals used are all proper integrals.<ref name=":1" />

{{Em|Singular}} or {{Em|weakly singular}}: An integral equation is called singular or weakly singular if the integral is an improper integral.<ref name=":1" /> This could be either because at least one of the limits of integration is infinite or the kernel becomes unbounded, meaning infinite, on at least one point in the interval or domain over which is being integrated.<ref name=":0" />

Examples include:<ref name=":0" /><math display="block">F(\lambda) = \int_{-\infty}^\infty e^{-i\lambda x} u(x) \, dx</math><math display="block">L[u(x)] = \int_0^\infty e^{-\lambda x} u(x) \, dx</math>These two integral equations are the Fourier transform and the Laplace transform of ''u''(''x''), respectively, with both being Fredholm equations of the first kind with kernel <math>K(x,t)=e^{-i\lambda x}</math> and <math>K(x,t)=e^{-\lambda x}</math>, respectively.<ref name=":0" /> Another example of a singular integral equation in which the kernel becomes unbounded is:<ref name=":0" /> <math display="block">x^2= \int_0^x \frac{1}{\sqrt{x-t}} \, u(t) \, dt.</math>This equation is a special form of the more general weakly singular Volterra integral equation of the first kind, called Abel's integral equation:<ref name=":1" /> <math display="block">g(x)=\int_a^{x} \frac{f(y)}{\sqrt{x-y}} \, dy</math>{{Em|Strongly singular}}: An integral equation is called strongly singular if the integral is defined by a special regularisation, for example, by the Cauchy principal value.<ref name=":1" />

=== Integro-differential equations ===
An [[Integro-differential equation|Integro-differential]] equation, as the name suggests, combines differential and integral operators into one equation.<ref name=":0" /> There are many version including the Volterra integro-differential equation and delay type equations as defined below.<ref name=":2" /> For example, using the Volterra operator as defined above, the Volterra integro-differential equation may be written as:<ref name=":2" /><math display="block">y'(t)=f(t, y(t))+(V_\alpha y)(t)</math>For delay problems, we can define the delay integral operator <math>(\mathcal{W}_{\theta , \alpha} y)</math> as:<ref name=":2" /><math display="block">(\mathcal{W}_{\theta , \alpha} y)(t) := \int_{\theta(t)}^t (t-s)^{-\alpha} \cdot k_2(t,s,y(s), y'(s)) \, ds  </math>where the delay integro-differential equation may be expressed as:<ref name=":2" />
<math display="block">y'(t)=f(t, y(t), y(\theta (t)))+(\mathcal{W}_{\theta , \alpha} y)(t).</math>