Applied Mathematics, Integral Equations

Integral Equations: Introduction (L1)

Posted by jessepfrancis on Jan 21, 2020Aug 23, 2021

We are familiar with differential equations: an equation involving one (or more) unknown function and its derivatives. Similarly, an integral equation is an equaiton in which an unknown function appears under one or more integral signs.

For example, for $a\le s\le b,a\le t\le b$ , the equations

$f(s)=\int_a^b {K(s,t)g(t)dt}$

$g(s)=f(s)+\int_a^b {K(s,t)g(t)dt}$

$g(s)=\int_a^b {K(s,t)[g(t)]^2dt}$

where the function g(s) is the unknown function while all the other functions are known, are integral equations. Note that f(s), K(s,t) can be complex valued functions of real variables s and t.

Given K(s,t), f(s) are known functions. Which of the following is/are integral equations? (Multiple options are correct)

$\int_a^b{K(s,t)\over g(t)}dt = f(s)$

True! The unknown function is inside the integral, and it’s an equation!

$\int_a^b{K(s,t)f(t)}dt = f(s)$

Oh no! There are no unknown functions!

$g(s) = f(s) +\int_a^b{K(s,t)g(t)}dt$

True! Have you found other correct answers?

$g(s) = f(s) +\int_a^b{K(s,t)f(t)}dt$

Oh no! Not an integral equation – is the unknown function inside integral?

$\int_a^b{K(s,t) g(t)}dt$

Wait! Is it even an equation?

$g(s) = \int_a^b{K(s,t) g(s)}dt$

Oh no! There are no unknown functions!

Which of the following are NOT integral equations?

$s^2 = \int_1^2(s+t)g(t)dt$

That’s an integral equation!

$s^3 = \int_4^5{t^2\over g(t)}dt$

That’s an integral equation!

$s^2 = \int_0^1t^2dt$

That’s not an integral equation! You caught it right!

$g(s) = 1 + 2\int_0^1sg(t)dt$

That’s an integral equation!

$s = \int_1^\infty \frac{3(st+s^2t^2)}{1+g(s)}dt$

That’s not an integral equation! You caught it right!

$g(s) = \int_0^2 st^2 dt$

That’s not an integral equation! You caught it right!

Published by jessepfrancis

Christian. Artist. Mathematician. Programmer. Teacher. Visit https://mathematicos.in/aboutjesse/ for my full profile. View all posts by jessepfrancis

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