Editing Integral equation (section)

=== 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.