User silmaril89 - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T19:08:05Z http://mathoverflow.net/feeds/user/15776 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/67785/linear-versus-non-linear-integral-equations linear versus non-linear integral equations silmaril89 2011-06-14T17:54:14Z 2011-06-14T19:24:17Z <p>I'm having trouble solving an integral equation. It appears to me to be a homogenous fredholm equation of the second kind. However, I'm being told that this can't be a fredholm equation, because it is non-linear. Could someone help me in trying to figure out how to classify an integral equation as linear or non-linear. Also, I'll post the equation I need to solve below, and it would be great if anyone could also give me some tips on how to try and solve it. Thank you to all who reply.</p> <p>The equation is</p> <p>$\phi(x) = (x^2 - x)\int\limits_0^1 \mathrm{d}y \frac{\phi(y)}{(y-x)^2}$</p> <p>Also, is this by chance related to an eigenvalue problem? I know that might sound like a strange question, but I've seen some people treating these as eigenvalue equations.</p> <p>By the way, I want to solve the equation for $\phi(x)$</p> http://mathoverflow.net/questions/67785/linear-versus-non-linear-integral-equations Comment by silmaril89 silmaril89 2011-06-15T17:24:57Z 2011-06-15T17:24:57Z Well, the equation I'm actually trying to solve is this, $M^2_n\phi_n(x) = [\frac{m_{1}^{2} - \beta^2}{x} + \frac{m_{2}^{2} - \beta^2}{1-x}]\phi_n(x) - \beta^2 \int\limits_0^1 \mathrm{d}y\frac{\phi_n(y)}{(y-x)^2}$. This is known as the 't Hooft equation. I'm trying to solve it for the ground state where $M_n = m_1 = m_2 = 0$, which when doing that you get the equation I presented in the problem. So, I'm sure that $(y-x)^2$ has to be there. Thanks for the input though. http://mathoverflow.net/questions/67785/linear-versus-non-linear-integral-equations Comment by silmaril89 silmaril89 2011-06-15T01:42:49Z 2011-06-15T01:42:49Z Funny that you should also mention it's designed for research-level questions though. This is a problem I'm trying to solve for a research project, but I might be in over my head... http://mathoverflow.net/questions/67785/linear-versus-non-linear-integral-equations Comment by silmaril89 silmaril89 2011-06-14T22:05:35Z 2011-06-14T22:05:35Z Thank you, I was unaware of the existence of math.stackexchange.com. http://mathoverflow.net/questions/67785/linear-versus-non-linear-integral-equations Comment by silmaril89 silmaril89 2011-06-14T20:07:09Z 2011-06-14T20:07:09Z Anybody care to comment on why I'm being down voted? I'm not necessarily opposed to it. But, I'd like to know why in order to learn how to use the site and to learn the right etiquette. http://mathoverflow.net/questions/67785/linear-versus-non-linear-integral-equations/67787#67787 Comment by silmaril89 silmaril89 2011-06-14T19:41:56Z 2011-06-14T19:41:56Z Also, it's possible that I'm leaving out something important. I believe the integral is supposed to have the letter p in front of it, which apparently stands for a finite part integral. Do you know what that means? I've never heard of it. Thanks. http://mathoverflow.net/questions/67785/linear-versus-non-linear-integral-equations/67787#67787 Comment by silmaril89 silmaril89 2011-06-14T19:16:55Z 2011-06-14T19:16:55Z Sorry, should have all the $\psi$'s as $\phi$'s http://mathoverflow.net/questions/67785/linear-versus-non-linear-integral-equations/67787#67787 Comment by silmaril89 silmaril89 2011-06-14T19:15:28Z 2011-06-14T19:15:28Z How do you know it will diverge unless $\psi(x) = 0$? I've seen that a possible way to solve this, is by a Liouville-Neumann_series, $\psi(x) = \sum\limits_{n=0}^{\infty}\lambda^n\psi_n(x)$. If so, I could probably cut it off somewhere, but I'm not sure how to find any of the values for $\lambda^n$ or $\psi_n(x)$.