A.A. Azamov, A.O. Begaliev. An existence theorem and an approximate solution method for the Pfaff equation with continuous coefficients ... P. 12-24

Pfaff equations with continuous coefficients are considered. A specific Cauchy problem for the Pfaff equation is transformed to an equivalent system of integral equations of a special type, which is overdetermined. It is shown that in the case of smooth coefficients the consistency of the system is equivalent to the Frobenius integrability criterion. A theorem on the existence of a solution for the obtained type of integral equations is presented. The solution is found by the Euler polygonal method, which allows one to construct an approximate solution of the Pfaff equation. An analog of Nagumo’s theorem on the uniqueness of the solution to the Cauchy problem is also given.

Keywords: Pfaff equation, integral equation, consistency of a system, Frobenius criterion, existence theorem, Euler’s broken lines, uniqueness of solution, Nagumo condition

Received May 27, 2021

Revised June 19, 2021

Accepted July 12, 2021

Funding Agency: This work was supported by the Ministry of Innovative Development of the Republic of Uzbekistan (project no. OT-F4-84).

Abdulla Azamov, Academician of AS RUz, Prof., V.I.Romanovskii Institute of Mathematics of the Uzbekistan Academy of Sciences, Tashkent, 100174 Uzbekistan, e-mail: abdulla.azamov@gmail.com

Aziz Begaliyev, PhD student, V.I.Romanovskii Institute of Mathematics of the Uzbekistan Academy of Sciences, Tashkent, 100174 Uzbekistan, e-mail: azizuzmu@mail.ru

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Cite this article as: A.A. Azamov, A.O. Begaliev. An existence theorem and an approximate solution method for the Pfaff equation with continuous coefficients, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2021, vol. 27, no. 3, pp. 12–24; Proceedings of the Steklov Institute of Mathematics (Suppl.), 2022, Vol. 317, Suppl. 1, pp. S16–S26.