Knowlpedia

Definition

Peano–Baker series

The time-ordered matrix series for the propagator of a nonautonomous linear ODE.

For continuous AA, the Peano–Baker series for the is

Φ(t,s)=I+∑n=1∞∫s<tn<⋯<t1<tA(t1)⋯A(tn) dtn⋯dt1(t≥s).\Phi(t,s)=I+\sum_{n=1}^\infty \int_{s<t_n<\cdots<t_1<t}A(t_1)\cdots A(t_n)\,dt_n\cdots dt_1 \quad(t\ge s).

It is obtained by repeatedly substituting in Φ(t,s)=I+∫stA(τ)Φ(τ,s) dτ\Phi(t,s)=I+\int_s^tA(\tau)\Phi(\tau,s)\,d\tau.

Convergence and order

The bounds the nn-th term by (M∣t−s∣)n/n!(M|t-s|)^n/n! on any compact interval where ∥A∥≤M\|A\|\le M. The series converges uniformly, and its integral equation identifies it with the unique propagator. Later-time factors stand on the left. Unless the coefficient matrices commute at different times, their order cannot be rearranged.