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Solving dynamical systems with cubic trigonometric splines. (English) Zbl 1071.65174

The aim of the paper is the numerical solving using cubic trigonometric splines, of special nonlinear dynamical systems.
The first part represents an introduction for the dynamical system and the trigonometric basic splines which will be used in the sequel.
The second part is devoted to the presentation of the method and to obtain the condition which ensures the existence of the numerical solution, for \(2\times 2\) systems. The third part concerns the derivation of the error estimates in the one-dimensional case. The authors obtain that the convergence order of the spline approximations of the solution is cubic.
The fourth part contains the numerical results which are obtained for \(2\times 2\) and \(3\times 3\) systems using the presented method in comparison with those obtained with the method of rational Fourier series approximations. Thus the accuracy of the results shows the efficiency of the trigonometric splines in approximating periodical functions.

MSC:

65P10 Numerical methods for Hamiltonian systems including symplectic integrators
65L05 Numerical methods for initial value problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
37M15 Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems
65L70 Error bounds for numerical methods for ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
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