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.

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

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