Probabilistic evolution approach for the solution of explicit autonomous ordinary differential equations. Part 2: Kernel separability, space extension, and, series solution via telescopic matrices

Abstract

This work focuses on the Kronecker power series solution of the explicit conical ODEs. This means that the Kronecker power series of the descriptive function vector of the ODEs has only zeroth, first and second Kronecker powers of the unknowns hence the only nonvanishing matrix coefficients are ${\mathbf {F}}_0, {\mathbf {F}}_1$ and ${\mathbf {F}}_2$. We focus on the cases where ${\mathbf {F}}_0$ also vanishes. These enable us to get and solve a two block term recursive ODE and the accompanying initial conditions. The resulting Kronecker power series’ kernel can be expressed as a binary product whose first factor which in square matrix type and a second factor which is in purely rectangular matrix algebraic structure. The constancy adding space extension separates the temporal behavior of the kernel in a scalar first factor while the second factor is again in rectangular matrix structure. We also show that the definition and use of rectangular eigenvalue problem takes us to constant solution of the original ODEs.