0
$\begingroup$

I have an ordinary integro-differential equation of the form $$y''(t)+C_1y(t)+C_2\int_{0}^{t}f(t-\tau)y''(\tau)d\tau +C_3\int_{0}^{t}f(t-\tau)y(\tau)d\tau=0$$ where $C_1$,$C_2$ and $C_3$ are constants.

I know that finding an analytical solution can be hard. How can I solve this numerically?

Any suggestion would be greatly appreciated.

$\endgroup$
1
$\begingroup$

You can try to look for an analytic solution solution using Laplace transforms or using power series. $$y(t)=\sum_{n=0}^{\infty}a_nt^n$$

After your edit Laplace transform seems good as you can solve this equation as a third order ODE for : $$Y(t)=\int_{0}^{t}y(\tau)d\tau$$ And you should have noticed that : $$\int_{0}^{t}y''(\tau)d\tau=y'(t)-y'(0)=Y''(t)-Y''(0)$$ Thus your equation becomes : $$Y'''(t)+C_1Y'(t)+C_2(Y''(t)-Y''(0))+C_3Y(t)=0$$

$\endgroup$
  • $\begingroup$ Thanks for your answer. But I'm sorry that I have missed an important term in my equation. I'll edit it now. Nevertheless your approach gave me an insight into solving the problem. Thanks again. $\endgroup$ – Rhinocerotidae Oct 20 '16 at 9:07

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.