# Integro-differential equation with convolution

Given a $$\mathcal{C^\infty}$$ matrix-valued function $$f$$ from $$\mathbb{R}^+$$ to $$\mathbb{R}^{n,n}$$, I'd like to solve the following integro-differential equation:

$$\ddot x(t) + \int_0^t f(\tau) \dot x(t-\tau)\,\mathrm{d}t + x(t) = 0$$

for some initial conditions $$x(0)=x_0$$, $$\dot x(0)= v_0$$ and $$x$$ a vector. I wrote a numerical solver and I would like to verify the results by comparing them to reference solutions. When $$f\equiv 0$$, it converges. Would you have any idea of further verifications I could perform to ensure my solver converges to the right solution?

I am also interested in references (as in this unanswered question) dealing with analytic solutions and / or numerical methods.

• $\mathrm{d}t$ should be $\mathrm{d}\tau$... – pluton Mar 10 at 21:06