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An Existence Theorem and an Approximate Solution Method for a Pfaff Equation with Continuous Coefficients

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Abstract

Pfaff equations with continuous coefficients are considered. A specific Cauchy problem for a 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.

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Funding

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

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Authors and Affiliations

  1. Institute of Mathematics, Academy of Sciences of the Republic of Uzbekistan, 100174, Tashkent, Uzbekistan

    A. A. Azamov & A. O. Begaliev

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  1. A. A. Azamov
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  2. A. O. Begaliev
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Correspondence to A. A. Azamov or A. O. Begaliev.

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Translated from Trudy Instituta Matematiki i Mekhaniki UrO RAN, Vol. 27, No. 3, pp. 12 - 24, 2021 https://doi.org/10.21538/0134-4889-2021-27-3-12-24.

Translated by I. Tselishcheva

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Azamov, A.A., Begaliev, A.O. An Existence Theorem and an Approximate Solution Method for a Pfaff Equation with Continuous Coefficients. Proc. Steklov Inst. Math. 317 (Suppl 1), S16–S26 (2022). https://doi.org/10.1134/S0081543822030026

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  • DOI: https://doi.org/10.1134/S0081543822030026

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