**MSC2010** 34C20

### On local resolvability of a certain class of the first-order partial differential equations

#### S. N. Alekseenko^{1}, L. E. Platonova^{2}

Annotation | The Cauchy problem for a quasi-linear first order partial differential equation is studied for different cases of initial data. In the first case, the line carrying the initial data is specified parametrically; in the second case, this line is described in Cartesian coordinates and has an infinite length; in the third case, the line is specified in Cartesian coordinates and its length is finite. In each case, the local resolvability conditions are formulated for the considered quasi-linear equation and it is shown that the solution has the same smoothness as the function defining the initial conditions. To study the above problems the method of additional argument was used. Using this method, some system of integral equations is solved, and the solution of this system gives the solution of the Cauchy problem for the original equation. |
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Keywords | quasi-linear first order partial differential equation, Cauchy problem, method of an additional argument, local resolvability, integral equation |

^{1}**Sergey N. Alekseenko**,Professor of the Applied Mathematics Chair,
Nizhniy Novgorod State Technical University named after R. E. Alekseev(24 Minin St., Nizhniy Novgorod 603950, Russia), Dr.Sci.(Physies and Mathematics), ORCID:http://orcid.org/0000-0002-1455-1263, sn-alekseenko@yandex.ru^{2}**Lyubov E. Platonova**, Assistant Lecture of the Mathematical Analysis Chair, Nizhniy Novgorod State Pedagogical University named after Kozma Minin (1 Ylyanov St., Nizhniy Novgorod 604950, Russia), ORCID: http://orcid.org/0000-0003-3601-2276, fluff13@yandex.ru

**Citation**: S. N. Alekseenko, L. E. Platonova, "[On local resolvability of a certain class of the first-order partial differential equations]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,20:2 (2018) 132–147 (In Russian)

**DOI** 10.15507/2079-6900.20.201802.132-147