Recollect
An operator ![L[f(x)]](https://s0.wp.com/latex.php?latex=L%5Bf%28x%29%5D&bg=ffffff&fg=6a6c6e&s=0&c=20201002)
![L[af(x)+bg(x)]=aL[f(x)]+bL[g(x)],a,b\in\mathbb R](https://s0.wp.com/latex.php?latex=L%5Baf%28x%29%2Bbg%28x%29%5D%3DaL%5Bf%28x%29%5D%2BbL%5Bg%28x%29%5D%2Ca%2Cb%5Cin%5Cmathbb+R&bg=ffffff&fg=6a6c6e&s=0&c=20201002)
Integral Operator
We can write an integral equations in terms of a linear operator, as in following examples.
Let ![L[g(s)]=g(s)-\int_a^b {K(s,t)g(t)dt}\to(1)](https://s0.wp.com/latex.php?latex=L%5Bg%28s%29%5D%3Dg%28s%29-%5Cint_a%5Eb+%7BK%28s%2Ct%29g%28t%29dt%7D%5Cto%281%29&bg=ffffff&fg=6a6c6e&s=0&c=20201002)
![L[g(s)]=f(s)](https://s0.wp.com/latex.php?latex=L%5Bg%28s%29%5D%3Df%28s%29&bg=ffffff&fg=6a6c6e&s=0&c=20201002)
Similarly, ![g(s)=\int_a^b {K(s,t)[g(t)]^2dt} \to(2)](https://s0.wp.com/latex.php?latex=g%28s%29%3D%5Cint_a%5Eb+%7BK%28s%2Ct%29%5Bg%28t%29%5D%5E2dt%7D+%5Cto%282%29&bg=ffffff&fg=6a6c6e&s=0&c=20201002)
Linear Integral Operator
An integral operator is linear if it satisfies above condition; that is,
Note equation (1) above. Clearly,
![\displaystyle L[ af(s)+bg(s) ] \newline= af(s)+bg(s) -\int_a^b {K(s,t) [af(t)+bg(t)] dt} \newline = af(s) -\int_a^b {K(s,t) af(t)dt} + bg(s) -\int_a^b {K(s,t) bg(t) dt} \newline = af(s) - a\int_a^b {K(s,t) f(t)dt} + bg(s) -b\int_a^b {K(s,t) g(t) dt} \newline = a[f(s) -\int_a^b {K(s,t) f(t)dt}] + b[g(s) -\int_a^b {K(s,t) g(t) dt}] \newline = a L[f(s)] + b L[g(s)]](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++L%5B+af%28s%29%2Bbg%28s%29+%5D+%5Cnewline%3D+++af%28s%29%2Bbg%28s%29+-%5Cint_a%5Eb+%7BK%28s%2Ct%29+%5Baf%28t%29%2Bbg%28t%29%5D+dt%7D++%5Cnewline+%3D++af%28s%29+-%5Cint_a%5Eb+%7BK%28s%2Ct%29+af%28t%29dt%7D+%2B+bg%28s%29+-%5Cint_a%5Eb+%7BK%28s%2Ct%29+bg%28t%29+dt%7D++%5Cnewline+%3D++af%28s%29+-+a%5Cint_a%5Eb+%7BK%28s%2Ct%29+f%28t%29dt%7D+%2B+bg%28s%29+-b%5Cint_a%5Eb+%7BK%28s%2Ct%29+g%28t%29+dt%7D++%5Cnewline+%3D++a%5Bf%28s%29+-%5Cint_a%5Eb+%7BK%28s%2Ct%29+f%28t%29dt%7D%5D+%2B+b%5Bg%28s%29+-%5Cint_a%5Eb+%7BK%28s%2Ct%29+g%28t%29+dt%7D%5D+++%5Cnewline+%3D++a+L%5Bf%28s%29%5D++%2B+b+L%5Bg%28s%29%5D&bg=ffffff&fg=6a6c6e&s=0&c=20201002)
Hence, that integral operator is linear. But note that operator in equation (2) is not linear, and you can prove it yourself.
In general, we will be studying equations of the form
where
- h(s), f(s) and K(s,t) are known real/complex functions,
- g(s) is the unknown function
- K(s,t) is called the kernel
is a non-zero real or complex parameter called “eigenvalue”.
- Limits, a is constant and b may be fixed or a variable.
Mathematicos 