1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Integral equation

  1. Mar 2, 2008 #1
    1. The problem statement, all variables and given/known data
    Solve for u(x):

    [tex]0 = e^{2\int u(x) dx} + u(x) e^{\int u(x) dx} - a(x)[/tex]

    2. Relevant equations

    3. The attempt at a solution
    I tried using the quadratic formula,

    [tex]e^{\int u(x) dx} = \frac{-u(x) \pm \sqrt{u^2(x) + 4a(x)}}{2}[/tex]

    , converting to log notation and differentiating, but from there I didn't know how to solve for u(x). I thought maybe I could use something on the lines of the log-definitions of the inverse trig functions. Any ideas?
  2. jcsd
  3. Mar 2, 2008 #2


    User Avatar
    Homework Helper

    what are the limits on the integration? if they are fixed then [itex]\int dx u(x)[/itex] is just a number (call it C) and
    u=(a-e^{2 C})/e^{C}
  4. Mar 2, 2008 #3
    You could convert it to a differential equation and try to solve this, however it turns out that this new differential equation is severely determined by the unknown function a(x). In order to do this, set:

    [tex]e^{\int u(x)dx}=f(x)[/tex]



    And putting this into the equation gives:


    This is a Riccati equation, which can be transformed into a linear one by transforming:



    [tex]\frac{df(x)}{dx}=-\frac{1}{[u(x)]^2}\left(\frac{du(x)}{dx}\right)^2 +\frac{1}{u(x)} \frac{d^2u(x)}{dx^2}[/tex]

    The equation becomes:

    [tex]\frac{d^2u(x)}{dx^2}-u(x)\cdot a(x)=0[/tex]

    And this one can be solved if a(x) is known. P.e. a(x)=-1 gives sin and cos functions, a(x)=x gives airy functions, a(x)=1 gives hyperbolic ones, etc.

    This might be the complete wrong idea, but I don't see it in another way.

    [Edit] Someone was faster....
  5. Mar 2, 2008 #4
    Actually, I got the integral equation by trying to solve

    [tex]\frac{d^2u(x)}{dx^2}-u(x)\cdot a(x)=0[/tex]
  6. Mar 3, 2008 #5
    foxjwill, to my knowledge there is no solution in terms of a(x). As I pointed out the solution depends so heavily on this function a(x) that you can't solve it without knowing it explicitly. The few examples I gave did show this, no? A sin or cos function compared to a hyperbolic one or even an Airy function (which is closely related to the functions of Bessel) are so different, even for the simple assumed functions of a(x) equal to -1, 1 and x. Maybe there is an explicit integral representation of the solution, but I think it will be closely related to the one you originally posted.
  7. Mar 4, 2008 #6


    User Avatar
    Homework Helper

    also, you could try using Green's functions and getting an approximation... the utility of this approach probably depends on the form of a(x). E.g., find the "homogeneous" solutions
    \frac{d^2 f}{dx^2}=0
    and the "free" green's function
    and then consider the term
    as an inhomogeneous term so that the "solution" is given by
    u(x)=f(x)+\int dx' G(x,x')a(x')u(x')

    Then, supposing u(x) is only a little different from f(x) once can develop succesive approximations for u(x) as
    u(x)\approx f(x) + \int dx' G(x,x')a(x')f(x')+\int dx' G(x,x')a(x')\int dx'' G(x',x'')a(x'')f(x'')+\ldots
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Integral equation
  1. Integral equation (Replies: 32)

  2. Integral equation (Replies: 0)

  3. Integral equation (Replies: 7)

  4. An integral equation (Replies: 9)