High-order nonlinear initial-value problems countably determined
An iterative approach to approximate global solutions of high order initial value problems (IVPs): \[ x^{(m)}(t)= f(t, x(t), x'(t), \dots x^{(m-1)}(t) ), \quad t \in [t_0, t_0+T]\tag{1} \] with \( x^{(j)}(t_0)= x_0^j,\) \(j=0, \dots ,m-1, (1)\) 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 \( x(t) = T x(t)\) 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 \( ( \Gamma_j )_{j \geq 0}\) of piecewise linear functions associated to a sequence of nodes \( (t_j)_{j \geq 0} \) with \( t_1 = t_0+T\) dense in \([t_0, t_0+T]\) is proposed such that \( ( \varphi_k = \sum_{i \geq 0} \lambda_k^i \; \Gamma_i )\) 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.
nonlinear high order initial value problems
scientific article; zbMATH DE number 5559031
scientific article