Geodesic
Solution of Cauchy problem for equation first order via Haar functions
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 16 (2016) no. 2, pp. 151-159.

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In this article we consider a Cauchy problem for the first order differential equation and are looking for its numerical solution. For this aim we represent the derivative of the solution as Haar decomposition. We also obtain estimates of approximate solution. The method is computationally simple and applications are demonstrated through illustrative examples. These examples show that in some cases the error of the proposed method is much less, than in second order Runge–Kutta method. 
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D. S. Lukomskii; S. F. Lukomskii; P. A. Terekhin. Solution of Cauchy problem for equation first order via Haar functions. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 16 (2016) no. 2, pp. 151-159. http://geodesic.mathdoc.fr/item/ISU_2016_16_2_a4/

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[5] Lukomskii D. S., Terekhin P. A., “An error estimate for the Cauchy problem by using compression systems and shifts”, Proceedings of the Mathematical Centre named N. I. Lobachevsky, 51, Kazan Matnematical Society, Kazan, 2015, 295–297 (in Russian)