Solving Nonlinear Integral Equation with Newton Method

[sara_87]

Homework Statement

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

$$y(s)+\int^x_0{K(x,s,y(s)}ds=f(x)$$

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

Homework Equations

The Attempt at a Solution

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

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

but i am struggling on knowing what the weights w_j should be using the Newton procedure.

Thank you in advance.

 

[sara_87]

sorry, it should be
$$y_i+\sum^i_{j=0}{w_j K(x_i,x_j,y_j)}=f(x_i)$$
where y_i means y(x_i)
I can use the trapezoidal scheme and then i have:
$$y_i=f(x_i)-\frac{h}{2}(K(x_i,x_0,y_0))+h\sum^{i-1}_{j=1}{K(x_i,x_j,y_j)}+\frac{h}{2}K(x_i,x_i,y_i)$$

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 can't make y_i as the subject.