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 Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.]: Year: Volume: Issue: Page: Find

 Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2016, Volume 20, Number 2, Pages 259–275 (Mi vsgtu1487)

Differential Equations and Mathematical Physics

On one nonlocal problem for the Euler–Darboux equation

M. V. Dolgopolov, I. N. Rodionova, V. M. Dolgopolov

Samara National Research University, Samara, 443086, Russian Federation

Abstract: The boundary value problem with displacement is determined for the generalized Euler–Darboux equation in the field representing the first quadrant. This problem, unlike previous productions, specifies two conditions, connect integrals and fractional derivatives from the values of the sought solution in the boundary points. On the line of singularity of the coefficients of the equations the matching conditions continuous with respect to the solution and its normal derivation are considered. The authors took for the basis of solving the earlier obtained by themselves the Cauchy problem solution of the special class due to the integral representations of one of the specified functions acquired simple form both for positive and for negative values of Euler–Darboux equation parameter. The nonlocal problem set by the authors is reduced to the system of Volterra integral equations with unpacked operators, the only solution which is given explicitly in the corresponding class of functions. From the above the uniqueness of the solution of nonlocal problem follows. The existence is proved by the direct verification. This reasoning allowed us to obtain the solution of nonlocal problem in the explicit form both for the positive and for the negative values of Euler–Darboux equation parameter.

Keywords: integral equations system, boundary value problem, partial differential equation
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DOI: https://doi.org/10.14498/vsgtu1487

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Bibliographic databases:

UDC: 517.956.3
MSC: 35L10, 35Q05
Original article submitted 20/III/2016
revision submitted – 18/V/2016

Citation: M. V. Dolgopolov, I. N. Rodionova, V. M. Dolgopolov, “On one nonlocal problem for the Euler–Darboux equation”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 20:2 (2016), 259–275

Citation in format AMSBIB
\Bibitem{DolRodDol16} \by M.~V.~Dolgopolov, I.~N.~Rodionova, V.~M.~Dolgopolov \paper On one nonlocal problem for the Euler--Darboux equation \jour Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.] \yr 2016 \vol 20 \issue 2 \pages 259--275 \mathnet{http://mi.mathnet.ru/vsgtu1487} \crossref{https://doi.org/10.14498/vsgtu1487} \zmath{https://zbmath.org/?q=an:06964486} \elib{http://elibrary.ru/item.asp?id=27126230} 

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This publication is cited in the following articles:
1. I. N. Rodionova, V. M. Dolgopolov, M. V. Dolgopolov, “Delta-problems for the generalized Euler–Darboux equation”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 21:3 (2017), 417–422
2. M. V. Dolgopolov, I. N. Rodionova, “O postanovke vidoizmenennykh zadach dlya uravneniya Eilera–Darbu v sluchae parametrov, ravnykh po modulyu $\dfrac{1}{2}$”, Sovremennye problemy matematiki i fiziki, SMFN, 65, no. 1, Rossiiskii universitet druzhby narodov, M., 2019, 11–20
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