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Nonlinear initial-value problems and Schauder bases. (English) Zbl 1097.34005

The purpose of this paper is to consider an approximation of the solution to the classical initial value problem \[ x'(t)=f(t,x(t)),\quad t\in[\alpha,\alpha+\beta],\quad x(\alpha)=x_0, \] where \(f:[\alpha,\alpha+\beta]\times\mathbb R^n\to\mathbb R^n\) is a continuous function satisfying a Lipschitz-type condition with respect to the second variable. The construction of these approximations is based on a Schauder basis closely connected with the problem under consideration. The main result is illustrated by a numerical example concerning the Liénard equation.

MSC:

34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
41-XX Approximations and expansions
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References:

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