Integro-Differential Equation: Analytical Approach?

[saltydog]

I'm looking at an integro-differential equation prompted by a HW question. This is a simplified version of it:

\frac{df}{dx}=-\frac{af}{x^2}\left(\int_{x_0}^x u^2f(u)du+M\right);\quad f(x_0)=1

with:

x_0>0

and a and M positive constants.

Does anyone know if it can be approached analytically and how? I'm currently working on a modified Runge-Kutta method to solve it numerically . Is that the best approach for such equations?

 

[saltydog]

Alright, so I can rearrange to:

-\frac{x^2f^{'}}{af}=\int_{x_0}^x u^2f(u)du-M

and differentiate both side to get:

-\frac{af(x^2f^{''}+2xf^{'})-ax^2(f^{'})^2}{(af)^2}=x^2f

Simplifying I arrive at the following IVP:

xff^{''}+2ff^{'}-x(f^{'})^2+axf^3=0;\quad f(x_0)=y_0;\quad f^{'}(x_0)=-\frac{af(x_0)}{x_0^2}M

 

[saltydog]

Just want to close on this one:

I modified Runge-Kutta to integrate the developing integral at each time step and add it to the results of the calculations in an effort to solve directly, the IDE numerically for the equation:

\frac{df}{dx}=-\frac{0.5f}{x^2}\left(\int_1^x u^2f(u)du-1\right)

That is, calculate the 4 slopes that are Runge-Kutta, and for each slope calculated, integrate the solution so far determined from x=1 to x=current time step. Then factor this into the calculations for the slopes.

I next calculated numerically, the equivalent IVP:

xff^{''}+2ff^{'}-x(f^{'})^2+0.5xf^3=0;\quad f(1)=1;\quad f^{'}(1)=-0.5

and ran them both to x=20. The first plot is a superposition of both results. The agreement is close. The second plot is the error of the back substitution of the first calculation. Note how the error is largest at the beginning of the calculation due to what I suspect is insufficient knowledge of the integral during that time interval.

I'm thinking there are some IDEs where you can't simply differentiate-out the integral so easily as above and therefore, the direct numerical approach may have some utility in those cases. For example, one like:

f+\frac{df}{du}=\int_0^x f(\frac{u}{q})dq

right?

 

[schulzy]

I would like to solve integro-differential equation with Runge-Kutta, but I don't know, how can I begin it. Do you have some articles or examples, where I can understand, how I solve an integro-differential equation. I don't know what can I do with integral. Thank you for your reply.