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High-order nonlinear initial-value problems countably determined. (English) Zbl 1169.65071

An iterative approach to approximate global solutions of high order initial value problems (IVPs):

${x}^{\left(m\right)}\left(t\right)=f\left(t,x\left(t\right),{x}^{\text{'}}\left(t\right),\cdots {x}^{\left(m-1\right)}\left(t\right)\right),\phantom{\rule{1.em}{0ex}}t\in \left[{t}_{0},{t}_{0}+T\right]\phantom{\rule{2.em}{0ex}}\left(1\right)$

with ${x}^{\left(j\right)}\left({t}_{0}\right)={x}_{0}^{j},$ $j=0,\cdots ,m-1,\left(1\right)$ is proposed. The approach extends previous results of the same authors for first order non linear IVPs [Nonlinear Anal., Theory Methods Appl. 63, No. 1 (A), 97–105 (2005; Zbl 1097.34005)].

Under continuity and uniform Lipschitz conditions on the nonlinear function $f$, the IVP (1) is equivalent to an integral equation $x\left(t\right)=Tx\left(t\right)$ with some integral operator $T$ and the problem reduces to the computation of a fixed point of $T$ in a Banach space. For the approximation of such a fixed point a suitable Faber-Schauder basis ${\left({{\Gamma }}_{j}\right)}_{j\ge 0}$ of piecewise linear functions associated to a sequence of nodes ${\left({t}_{j}\right)}_{j\ge 0}$ with ${t}_{1}={t}_{0}+T$ dense in $\left[{t}_{0},{t}_{0}+T\right]$ is proposed such that $\left({\varphi }_{k}={\sum }_{i\ge 0}{\lambda }_{k}^{i}\phantom{\rule{0.277778em}{0ex}}{{\Gamma }}_{i}\right)$ with ${\lambda }_{k}^{i}$ appropriately chosen tends to the exact solution. Several convergence results for an iterative to the solution as well as error estimates are proposed and finally the technique is applied to an IVP second order test problem to show the convergence of the method depending on the number of the nodes ${t}_{j}$ and the number of iterations.

##### MSC:
 65L05 Initial value problems for ODE (numerical methods) 34A34 Nonlinear ODE and systems, general 65L20 Stability and convergence of numerical methods for ODE 65L70 Error bounds (numerical methods for ODE)