Next: sum of coefficients
Up: the matrizant
Previous: Picard's method
Another scheme of solution is to discretize the differential equation, approximating
dZ/dt = MZ by
![$\displaystyle Z(t+\Delta t)$](img411.gif) |
= |
![$\displaystyle Z(t) + \Delta t M(t)Z(t)$](img412.gif) |
(246) |
|
= |
![$\displaystyle (I + \Delta t M(t)) Z(t)$](img413.gif) |
(247) |
![$\displaystyle Z(t+2\Delta t)$](img414.gif) |
= |
![$\displaystyle (I + \Delta t M(t+\Delta t)))(I + \Delta t M(t)) Z(t)$](img415.gif) |
(248) |
![$\displaystyle Z(t+n\Delta t)$](img416.gif) |
= |
![$\displaystyle \left(\prod_{i=0}^{n-1}(I + \Delta t M(t_i))
\right) Z(t)$](img417.gif) |
(249) |
|
= |
![$\displaystyle \left[ I + \Delta t\sum_{i=0}^{n-1}M(t_i) +
\Delta t^2\sum_{i=0}^{n-1}\sum_{j<i} M(t_i)M(t_j) + \cdots\right] Z(t)$](img418.gif) |
(250) |
|
= |
![$\displaystyle \Omega(M,t,0)Z(0)$](img406.gif) |
(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
![$\displaystyle \Omega(M,t,t)$](img419.gif) |
= |
I |
(252) |
![$\displaystyle \Omega(M,s,t) \Omega(M,t,u)$](img420.gif) |
= |
![$\displaystyle \Omega(M,s,u)$](img421.gif) |
(253) |
![$\displaystyle \Omega(M,s,t)^{-1}$](img422.gif) |
= |
![$\displaystyle \Omega(M,t,s)$](img423.gif) |
(254) |
Two further rules, which follow from the definitions, are
Next: sum of coefficients
Up: the matrizant
Previous: Picard's method
Microcomputadoras
2001-04-05