# Solving an integro-differential equation

I have a set of coupled integro-differential equations: $$\frac{dx_i(t)}{dt}=-x_i(t)+f_i(\mathbf{x}(t))+\sum_j{\partial_{x_j}f_i(\mathbf{x}(t))\int_{0}^{t}dt'f_j(\mathbf{x}(t'))e^{-(t-t')} }$$

The integrals, $\psi_j(t) =\int_{0}^{t}dt'f_j(\mathbf{x}(t'))e^{-(t-t')}$, are solutions of the differential equations $$\frac{d\psi_j(t)}{dt}+\psi_j(t)=f_j(\mathbf{x}(t))$$ with $\psi(0)=0$. I've been told by my supervisor that the numerical evaluation of the coupled system can be greatly sped up by using these differential equations but I don't understand how - does anyone have any ideas? I get the feeling I'm missing something obvious.

## migrated from mathematica.stackexchange.comMar 8 '17 at 17:32

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• Why don't you ask your supervisor for clarifications? – MarcoB Mar 7 '17 at 23:49
• What is $f(\mathbf{x}(t))$? Is this in some way related to $x_i(t)$? – zhk Mar 8 '17 at 5:45
• $f(\mathbf{x}_i(t))$ is a messy function of the set $\mathbf{x}$ except $x_i(t)$ – StressedOutStudent Mar 8 '17 at 12:13
• I asked my supervisor and he explained: the idea is that you solve for the system of equations ${\mathbf{x},\mathbf{\psi}}$ instead of just $\mathbf{x}$, so you have a larger system but can use standard methods. Seems obvious now -_- – StressedOutStudent Mar 8 '17 at 13:03
• Duplicate to the same unmigrated question (as of now only comments, no answer): math.stackexchange.com/questions/2176087/… – LutzL Mar 8 '17 at 17:44

$$\left[\begin{array}{cccccc} 1&-1&0&0&0&0\\ 0&1&-1&0&0&0\\ 0&0&1&-1&0&0\\ 0&0&0&1&-1&0\\ 0&0&0&0&1&-1\\ 0&0&0&0&0&1 \end{array}\right]$$ Two non-zero values per output. Corresponding integral numerical approximation matrix:
$$\left[\begin{array}{cccccc} 1&1&1&1&1&1\\ 0&1&1&1&1&1\\ 0&0&1&1&1&1\\ 0&0&0&1&1&1\\ 0&0&0&0&1&1\\ 0&0&0&0&0&1 \end{array}\right]$$