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Find the root of integral equation

  1. Dec 15, 2013 #1
    1. The problem statement, all variables and given/known data
    Hi everyone.
    I have encountered a weird equation while doing some research and I have no idea how to solve it.
    The equation goes like this

    ∫ dR / (1+ c*r) ^ (a/r) = d, limits of integration are from 0 to Rmax,
    where Rmax ^2 = ^2 - α^2, where u is a constant value of r.
    and r^2 = R^2 + α^2,
    c is a constant and d is my independent variable.

    2. Relevant equations

    3. The attempt at a solution
    I want to do a C function that takes one value of d and returns a value of α. I thought of doing
    ∫ (1 / (1+ c*r) ^ (a/r) ) - d = 0 and try to find the root, or do minimization. The problem is that α also appears on the limits of integration, and even if it wasn't there, I still don't know how to do either.
    Also, for any value of d, there is a value of α, but r changes from r = α to r= u.

    I would appreciate any help. Thanks
     
  2. jcsd
  3. Dec 16, 2013 #2

    haruspex

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    Even though alpha appears both inside the integral and in the limit, you can differentiate wrt alpha. That allows you to use standard iterative methods: given a trial alpha, find the resulting d, calculate the error and use the derivative to estimate an adjustment to alpha.
    ... and it will have the standard pitfalls.
     
  4. Dec 17, 2013 #3
    Could you refer me to sources where they explain more on the iterative methods that can do this sort of problem? I have no experience in finding a numerical value for an unknown that appears inside an integral (and inside the limits for that matter)
     
  5. Dec 17, 2013 #4

    haruspex

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    A basic method when you know the derivative function is Newton-Raphson. See e.g. http://en.wikipedia.org/wiki/Newton's_method.
    You understand how to expand ##\frac d{dx}\int^{f(x)}g(x, z).dz##, right?
     
  6. Dec 17, 2013 #5
    Is that just the fundamental theorem of calculus?

    And also, is there any method to find the root that does not involve a first trial of alpha?
    Thanks
     
  7. Dec 17, 2013 #6

    haruspex

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    It's a little more complicated because x is in two places.
    No.
     
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