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I have these three differential equations in which I need to solve numerically:

$$ \frac{dn_0}{dt}= -n_0(t)W_{01}(t) + n_1(t)K_{10} $$

$$ \frac{dn_1}{dt}= -n_1(t)W_{12}(t) - n_1(t)K_{10} + n_2(t)K_{21} + n_0(t)W_{01}(t) $$

$$ \frac{dn_2}{dt}= n_1(t)W_{12}(t) - n_2(t)K_{21} $$

such that

$$ n_0(0)=1 $$ $$ n_0(N)=0 $$ $$ n_1(0)=0 $$ $$ n_1(N)=1 $$ $$ n_2(0)=0 $$ $$ n_2(N)=0 $$

Using the central finite difference formula:

$$\frac{n_{0}(t + \Delta t) - n_{0}(t - \Delta t)}{2\Delta t}=-n_0(t)W_{01}(t) + n_1(t)K_{10}$$

$$\frac{n_{1}(t + \Delta t) - n_{1}(t - \Delta t)}{2\Delta t}=-n_1(t)W_{12}(t) - n_1(t)K_{10} + n_2(t)K_{21} + n_0(t)W_{01}(t)$$

$$\frac{n_{2}(t + \Delta t) - n_{2}(t - \Delta t)}{2\Delta t}=n_1(t)W_{12}(t) - n_2(t)K_{21} $$

How do I determine the functions $n0$, $n1$ and $n2$ knowing that $n0 + n1 + n2 =1$, and that the three equations are coupled?

And I could not understand how to calculate the derivatives, how can I determine their value with the finite difference method without knowing the functions?

Can someone please help me?

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