Learn Integral Equations: A Guide for Self-Teaching | ODE Graduate at 19

[epr2008]

I'm 19 and i just finished ode at a community college. there are no more math classes for me to take there so I've been planning teaching myself this semester. I was good at calc and ode's so i read about integral equations and figured i would start. I am using a book by Peter Collins but he's not very good at explaining the in between steps. he always says "i'll leave this for the reader to prove." well some of these are really not easy to see... so, I started off trying to prove a lemma he introduced in the first chapter and out of nowhere i ended up with something that i think is much more enlightening i just don't know if its right, so i was looking for some input...

The lemma to be proved:
Suppose that f:[a,b]$$\rightarrow$$$$\Re$$ is continuous. Then
$$\int\limits_a^x {\int\limits_a^{x'} {f(t)dtdx' = \int\limits_a^x {(x - t)f(t)dt} } }$$

Ok so at first before i figured out what he was talking about i started out like the book did with the rhs of the equation. and i thought about it and i said well what is the integral of that? ok integration by parts...

$$\begin{array}{l} F(x) = \int\limits_a^x {(x - t)f(t)dt} \\ u = f(t) \\ du = f'(t)dt \\ dv = (x - t)dt \\ v = xt - {\textstyle{1 \over 2}}{t^2} \\ F(x) = (xt - {\textstyle{1 \over 2}}{t^2})f(t) - \int {(xt - {\textstyle{1 \over 2}}{t^2})f'(t)dt} ]\nolimits_a^x \\ \end{array}$$

Now if f is a continuously differentiable function in t then integration by parts will eventually yield a differential equation with variable coefficients of the form
$$F(x) = \sum\limits_0^\infty {{{( - 1)}^n}} {f^n}(t)\frac{{(x + t)n + 2x + t}}{{(n + 1)(n + 2)}}{t^{n + 1}} ]\nolimits_a^x$$

Now obviously had i chosen u=(x-t) and dv=f(t)dt then i would have come up with another integral but with no derivatives in the equation right? Is this somewhat along the lines of the basic theory of differential and integral equations or am i just wrong?

 

[mathman]

You are completely on the wrong track. What you need to do is look at the domain of integration in the (x',t) plane. The original integral is t first then x'. Figure out what the domain would look like if you did x' first then t. Once you do that, the answer you need immediately pops out.

 

[epr2008]

Oh i know that for that particular theorem i was on the wrong track. I figured that out problem out. i was just wondering if that was a reasonable argument?

 

[mathman]

There was nothing in the original statement which said that f had any derivatives. Even if it was analytic, I don't see how your approach gets you anywhere.

 

[epr2008]

i understand it doesn't really get me anywhere but what i was trying to do is get a feel for where integral equations come from. so i tried to relate it to a differential equation.

and yes it doesn't say anything about derivatives i was just looking at it from a different perspective... if a function is integrable on an interval shouldn't it also be differentiable on that same interval?

 

[mathman]

if a function is integrable on an interval shouldn't it also be differentiable on that same interval?

absolutely not!