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Numerical differentiation with change of variable

  1. Nov 2, 2012 #1
    Hi all

    I am trying to solve for an integral whose integrand is a derivative that has a change of variable inside of it.

    ∫ (dz/dx) * cos(θ) dθ between 0 and pi.

    I have a function for z(x), and also know the relation between of x and θ, but what I don't know is how to evaluate such differential-integral operation numerically with the required change of variable.

    x = c/2*(1-cos(θ) )

    dz/dx = dz/dθ * dθ/dx...... how can I evaluate that derivative NUMERICALLY in terms of θ ??

    thanks in advance
     
  2. jcsd
  3. Nov 2, 2012 #2

    HallsofIvy

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    What method are you using to evaluate dz/dx numerically in terms of x? I would suggest that you calculate the value of x for your given value of θ and find dz/dθ as (dz/dx)/(dθ/dx).
     
  4. Nov 3, 2012 #3
    I have finally solved it as I wanted to do it at the beginning, I still don't understand very much why this way is working and others i tried not, but so far i am happy with the results...

    this is the piece of code that once run in matlab solved my problem

    c = 1;
    e = 1e-3;
    Zc = @(x) -e.*x./c .* (x/c - 1);

    N = 1e5;
    xdom = linspace(0,c,N);
    th = linspace(0,pi,N);

    X= @(TH) c/2.*(1-cos(TH));
    TH = @(X) acos(1 - 2*X/c);

    Zth = Zc(X(th));

    dzdth = diff(Zth) ./ diff(th);
    dthdx = diff(TH(X(th))) ./ diff(X(th));
    dzdx = dzdth.*dthdx;

    I11 = trapz(th(2:N),dzdx)
    I22 = trapz(th(2:N),dzdx.*cos(th(2:N))
     
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