Euler's method

Another scheme of solution is to discretize the differential equation, approximating dZ/dt = MZ by

$$Z(t+\Delta t) Z(t) + \Delta t M(t)Z(t)$$ = (246)

$$(I + \Delta t M(t)) Z(t)$$ = (247)

$$Z(t+2\Delta t) (I + \Delta t M(t+\Delta t)))(I + \Delta t M(t)) Z(t)$$ = (248)

$$Z(t+n\Delta t) \left(\prod_{i=0}^{n-1}(I + \Delta t M(t_i)) \right) Z(t)$$ = (249)

$$\left[ I + \Delta t\sum_{i=0}^{n-1}M(t_i) + \Delta t^2\sum_{i=0}^{n-1}\sum_{j<i} M(t_i)M(t_j) + \cdots\right] Z(t)$$ = (250)

$$\Omega(M,t,0)Z(0)$$ = (251)

These sums can be recognized as approximations to the integrals in Picard's method. However, this approximation is most useful in the product form, where the matrizant is sometimes called a product integral. For example, by observing the limits in the product, it is easy to conclude the rules

$$\Omega(M,t,t)$$ = I (252)

$$\Omega(M,s,t) \Omega(M,t,u) \Omega(M,s,u)$$ = (253)

$$\Omega(M,s,t)^{-1} \Omega(M,t,s)$$ = (254)

Two further rules, which follow from the definitions, are

$$\frac{\partial\Omega(M,s,t)}{\partial s} M(s)\ \Omega(M,s,t)$$ = (255)

$$\frac{\partial\Omega(M,s,t)}{\partial t} \Omega(M,s,t)\ M(t).$$ = (256)