Numerical solution for an integral equation?

[Asban]

Hello,

I have been encountered an integral equation that I need to solve\evaluate numericly and I didn't find anything like it in my search yet.

The equation:

\frac{d^2 F(t)}{dt^2}=const*\int_{t'}\frac{sin(F(t)-F(t'))}{(t-t')^{2k}}dt'

If it helps there is a specific case, when K=1 there is an analitical solution: F(t)=2arctan(\frac{t}{t_0}).

For now, two mainly things will help me:

1: What is the exact name of category of this integral equation?
2: What is the name of the numerical solution that solve it, or solve something similar to this equation and where should I start looking?

Thank you
Ofek

 

[Mandelbroth]

Hello,

I have been encountered an integral equation that I need to solve\evaluate numericly and I didn't find anything like it in my search yet.

The equation:

$$\frac{d^2 F(t)}{dt^2}=const*\int_{t'}\frac{sin(F(t)-F(t'))}{(t-t')^{2k}}dt'$$

I don't understand your notation for the integral...?

 

[Asban]

If I understant your question, the range is from 0 to infinity for all practical purposes but in general is suppose to be indefinite integral.

 

[Mandelbroth]

If I understant your question, the range is from 0 to infinity for all practical purposes but in general is suppose to be indefinite integral.

Is there a reason you put the t-prime on the bottom part of the integral, then?

Also, does your t-prime imply the derivative of t, or is that a separate variable? Your notation is confusing.

 

[Asban]

t and t' are different parameters, so as you see in the right hand side t acts as a constant in the integral,
but still in the right hand side there is a 2nd derivative of F with respect to the parameter t.

I need to evaluate numericly the function F(t) that solve this equation.

 

[DEvens]

If you were to sort out what that integral means, you might have some better luck. You can't do an integral numerically unless it has specific limits. So what you mean by putting the t' by the integral sign has to be made clear.

After that, round up the usual suspects. Look at things like a Laplace transform to try to get rid of the 2nd derivative. Look at taking some derivative of the entire equation to convert it to a differential equation without any integrals. Look at things like Gaussian quadrature to convert the integral to a set of valuations of F(t) with appropriate coefficients. Look up ways to numerically solve differential equations. You could start with a book like _Numerical Recipes in C_ (or Fortran if you prefer) for introductory methods.
Dan