#### For citation:

Lukomskii S. F., Lukomskii D. S. Numerical solution of linear differential equations with discontinuous coefficients and Henstock integral. *Izvestiya of Saratov University. Mathematics. Mechanics. Informatics*, 2021, vol. 21, iss. 2, pp. 151-161. DOI: 10.18500/1816-9791-2021-21-2-151-161, EDN: WJTAJG

# Numerical solution of linear differential equations with discontinuous coefficients and Henstock integral

We consider the problem of approximate solution of linear differential equations with discontinuous coefficients. We assume that these coefficients have $f$-primitive. It means that these coefficients are Henstock integrable only. Instead of the original Cauchy problem, we consider a different problem with piecewise-constant coefficients. The sharp solution of this new problem is the approximate solution of the original Cauchy problem. We found the degree of approximation in terms of $f$-primitive for Henstock integrable coefficients. Two examples are given. In the first example, the coefficients have an infinite derivative at zero. In the second example, the coefficients have an infinite derivative at interior points.

- Ohkita M., Kobayashi Y. An application of rationalized Haar functions to solution of linear differential equations.
*IEEE Transactions on Circuit and Systems*, 1968, vol. 33, no. 9, pp. 853–862. https://doi.org/10.1109/TCS.1986.1086019 - Razzaghi M., Ordokhani Y. Solution of differential equations via rationalized Haar functions.
*Mathematics and Computers in Simulation*, 2001, vol. 56, no. 3, pp. 235–246. https://doi.org/10.1016/S0378-4754(01)00278-6 - Razzaghi M., Ordokhani Y. An application of rationalized Haar functions for variational problems.
*Applied Mathematics and Computation*, 2001, vol. 122, no. 3, pp. 353–364. https://doi.org/10.1016/S0096-3003(00)00050-3 - Gat G., Toledo R. A numerical method for solving linear differential equations via Walsh functions. In:
*Advances in Information Science and Applications. Volumes I & II. Proceedings of the 18th International Conference on Computers (part of CSCC ’14)*, 2014, pp. 334–339. - Gat G., Toledo R. Estimating the error of the numerical solution of linear differential equations with constant coefficients via Walsh polynomials.
*Acta Mathematica Academiae Paedagogicae Nyiregyhaziensis*, 2015, vol. 31, no. 2, pp. 309–330. - Lukomskii D. S., Lukomskii S. F., Terekhin P. A. Solution of Cauchy problem for equation first order via Haar functions.
*Izvestiya of Saratov University. New Series. Series: Mathematics. Mechanics. Informatics*, 2016, vol. 16, iss. 2, pp. 151–159 (in Russian). https://doi.org/10.18500/1816-9791-2016-16-2-151-159 - Bartle G. A
*Modern Theory of Integration*. Providence, AMS, 2001. 458 p. - Gordon A. T
*he Integrals of Lebesgue, Denjoy, Perron, and Henstock*. Providence, AMS, 1994. 396 p.

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