Truncation approximants to probabilistic evolution of ordinary differential equations under initial conditions via bidiagonal evolution matrices: simple case

In this work, we investigate the probabilistic evolution approach (PEA) to ordinary differential equations whose evolution matrices are composed of only two diagonals under certain initial value impositions. We have been able to develop analytic expressions for truncation approximants which can be generated by using finite left uppermost square blocks in the denumerable infinite number of PEA equations and their infinite limits. What we have revealed is the fact that the truncation approximants converge for initial value parameter, values residing at most in a disk centered at the expansion point and excluding the nearest zero(es). The numerical implementations validate this formation.