Newton–Raphson method - Finite difference method

Hi

I am trying to solve a nonlinear differential equation with the use of the finite difference method and the Newton-Raphson method. Is there any one that knows where I can find some literature about the subject?

I am familiar with the use of the finite difference method, when solving linear differential equations. It is the Newton-Raphson method when using the finite difference method that is new for me.

Hello Excom,

finite difference methods (simple one-step methods such as Euler, Trapezoid, Midpoint, or more complex multi-step methods like the Adams' families, or non-linear methods such as Runge-Kutta, etc etc) can ALL be used to solve both linear and non-linear ordinary differential equations (obviously depending on the kind of differential system there are methods that will perform better than others..) but they're all used for solving a general IVP of the form:

$$\left. \begin{array}{l} \frac {dy} {dx} = f(x,y) \\ y( x_{0} ) = y_{0} \end{array} \right\} \mbox{ze IVP :p}$$

(which may be a scalar equation or a system of equations), regardless of whether f is linear or not.

Newton-Raphson is for solving non-linear algebraic equations, not differential equations. You will have to use Newton-Raphson (or any other technique for solving non-linear equations) within your finite difference method if the said method is implicit, that is, to solve for the current time-step of the solution as a function of the values at previous time-steps. For example, Adams-Moulton methods are implicit so you will have to solve a non-linear algebraic equation (or system of equations) at each time-step. . . but Forward Euler or Trapezoid or even Runge-Kutta or Adams-Bashforth are all explicit difference methods, and there's no need to solve non-linear equations within the method, so no need for Newton-Raphson :)

If you're still interested in Newton-Raphson, there are loads of resources on the net, just search on google:) eg one link I found:
http://www.math.ubc.ca/~clarkson/newtonmethod.pdf" [Broken]

Hope I could be of help, good luck with your non-linear differential equation! xD

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In my case, you can approximate the denominator term,
f'(x)
with a forward, backward, or central difference.

So, just to elaborate, if you have,
f(x) = x^2
and,
f(x+h) = (x+h)^2
f(x-h) = (x-h)^2
for some small h (gridspace)

Then, using Central Difference your Newton-Raphson equation becomes,
x[i+1] = x - f(x)/f'(x)
= x - f(x) / ( (f(x+h)-f(x-h) )/(2*h) )
= x - x^2/( ( (x+h)^2 - (x-h)^2 ) / (2*h) )

For other example, e.g. f(x) = x^4 + x^3 + x + 5, I'm getting faster convergence via the finite difference version of f'(x) than using the analytical version of it.

I have not encounter any reference for this, but I don't see anything wrong with this.

All the best! :)

Take a look at this I dont know if it can help you :
www.firavia.com/newton.pdf[/URL]

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