Editing Integral equation (section)

=== 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>