Lösungsapproximation und Fehlerabschätzungen für ein unendliches System linearer, gewöhnlicher Differentialgleichungen mit konstanter Bandmatrix. (Approximation to the solution and error estimates for an infinite system of linear ordinary differential equations with constant band matrix) (Q1824081)

Summary: The infinite system of ordinary linear differential equations of the form \[ x_ n'(t)=\delta_{n-1}x_{n-1}(t)-\alpha_ nx_ n(t)+\beta_{n+1}x_{n+1}(t)+\gamma_{n+2}x_{n+2}(t)+f_ n(t) \] (0\(\leq \alpha_ n,\beta_ n,\gamma_ n,\delta_ n\in {\mathbb{R}})\) with homogeneous initial conditions \(x_ n(0)=0\) \((n=0,1,2,...)\) is considered. Under certain assumptions on the not necessarily limited coefficients and on the nonnegative functions \(f_ n\) it is proved that for \(N\to \infty\) and every \(t\geq 0\) the solutions \(x^ N=(x_ 0^ N,x_ 1^ N,...,x_ N^ N)\) of the in usual way truncated finite systems nondecreasingly converge componentwise to a sequence \(x=(x_ 0,x_ 1,...)\) of functions, the coordinates of which satisfy the considered initial problem. Explicit upper estimates for the nonnegative differences \(x_ n(t)-x_ n^ N(t)\) are derived.