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Newton method

  1. Nov 7, 2009 #1
    1. The problem statement, all variables and given/known data

    If I have a non linear integral equation of the form:

    [tex]y(s)+\int^x_0{K(x,s,y(s)}ds=f(x)[/tex]

    and i want to find a way to solve this numerically using the newton method

    2. Relevant equations



    3. The attempt at a solution

    after discretizing, and using the quadrature rule, i have:

    [tex]y(s_i)+\sum^i_{j=0}{w_j K(x_i,s_j,y(s_j)}ds=f(x_i)[/tex]

    but i am struggling on knowing what the weights wj should be using the newton procedure.

    Thank you in advance.
     
  2. jcsd
  3. Nov 8, 2009 #2
    sorry, it should be
    [tex]y_i+\sum^i_{j=0}{w_j K(x_i,x_j,y_j)}=f(x_i)[/tex]
    where y_i means y(x_i)
    I can use the trapezoidal scheme and then i have:
    [tex]y_i=f(x_i)-\frac{h}{2}(K(x_i,x_0,y_0))+h\sum^{i-1}_{j=1}{K(x_i,x_j,y_j)}+\frac{h}{2}K(x_i,x_i,y_i)[/tex]

    but how can this be solved when i need to know y_i to get y_i since y_i is in the last term of the right hand side?
    since i cant make y_i as the subject.
     
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