Geodesic
On formulation of modified problems for the Euler–Darboux equation with parameters equal to $\dfrac{1}{2}$ in absolute value
Contemporary Mathematics. Fundamental Directions, Contemporary problems in mathematics and physics, Tome 65 (2019) no. 1, pp. 11-20
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

We consider the Euler–Darboux equation with parameters equal to $\dfrac{1}{2}$ in absolute value. Since the Cauchy problem in the classical formulation in ill-posed for such values of parameters, we propose formulations and solutions of modified Cauchy-type problems with the following values of parameters: a) $\alpha=\beta=\frac{1}{2},$ b) $\alpha=- \frac{1}{2},$ $\beta=- \frac{1}{2},$ c) $\alpha=\beta=- \frac{1}{2}.$ In the case а), the modified Cauchy problem is solved by the Riemann method. We use the obtained result to formulate the analog of the problem $\Delta_1$ in the first quadrant with shifted boundary-value conditions on axes and nonstandard conjunction conditions on the singularity line of the coefficients of the equation $y=x.$ The first condition is gluing normal derivatives of the solution and the second one contains limiting values of combination of the solution and its normal derivatives. The problem is reduced to a uniquely solvable system of integral equations. 
@article{CMFD_2019_65_1_a1,
     author = {M. V. Dolgopolov and I. N. Rodionova},
     title = {On formulation of modified problems for the {Euler{\textendash}Darboux} equation with parameters equal to~$\dfrac{1}{2}$ in absolute value},
     journal = {Contemporary Mathematics. Fundamental Directions},
     pages = {11--20},
     year = {2019},
     volume = {65},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CMFD_2019_65_1_a1/}
}
TY  - JOUR
AU  - M. V. Dolgopolov
AU  - I. N. Rodionova
TI  - On formulation of modified problems for the Euler–Darboux equation with parameters equal to $\dfrac{1}{2}$ in absolute value
JO  - Contemporary Mathematics. Fundamental Directions
PY  - 2019
SP  - 11
EP  - 20
VL  - 65
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/CMFD_2019_65_1_a1/
LA  - ru
ID  - CMFD_2019_65_1_a1
ER  - 
%0 Journal Article
%A M. V. Dolgopolov
%A I. N. Rodionova
%T On formulation of modified problems for the Euler–Darboux equation with parameters equal to $\dfrac{1}{2}$ in absolute value
%J Contemporary Mathematics. Fundamental Directions
%D 2019
%P 11-20
%V 65
%N 1
%U http://geodesic.mathdoc.fr/item/CMFD_2019_65_1_a1/
%G ru
%F CMFD_2019_65_1_a1
M. V. Dolgopolov; I. N. Rodionova. On formulation of modified problems for the Euler–Darboux equation with parameters equal to $\dfrac{1}{2}$ in absolute value. Contemporary Mathematics. Fundamental Directions, Contemporary problems in mathematics and physics, Tome 65 (2019) no. 1, pp. 11-20. http://geodesic.mathdoc.fr/item/CMFD_2019_65_1_a1/

[1] H. Bateman, A. Erdélyi, Higher Transcendental Functions, Russian translation, v. I, Nauka, M., 1973

[2] V. F. Volkodavov, N. Ya. Nikolaev, “On a new problem with shift in unbounded domain for the Euler–Darboux equation with positive parameters”, Mathematical Physics, KPtI, Kuybyshev, 1979, 3–9 (in Russian)

[3] V. F. Volkodavov, O. A. Repin, “Solution of a boundary-value problem with shift for a hyperbolic equation”, Differential Equations and Their Applications, KPtI, Kuybyshev, 1975, 15–21 (in Russian)

[4] V. F. Volkodavov, I. N. Rodionova, S. V. Bushkov, “Solution of a modified Cauchy problem by the Riemann method for one spatial analog of the Euler–Darboux equation with negative parameter”, Differ. Equ., 36:4 (2000), 616–619 (in Russian) | MR | Zbl

[5] V. F. Volkodavov, V. A. Spitsyn, Yu. I. Fedorov, “Boundary-value problems for one system of equations in rigid-plastic media”, Differential Equations (Mathematical Physics), 236, Ped. Univ., Kuybyshev, 1980, 36–45 (in Russian)

[6] V. M. Dolgopolov, M. V. Dolgopolov, I. N. Rodionova, “Construction of special classes of solutions for some hyperbolic differential equations”, Rep. Russ. Acad. Sci., 429:5 (2009), 583–589 (in Russian) | MR | Zbl

[7] V. M. Dolgopolov, M. V. Dolgopolov, I. N. Rodionova, “On delta-problems for generalized Euler–Darboux equation”, Nat. Univ. Uzbekistan, Tashkent, 2017, 203–204 (in Russian)

[8] V. M. Dolgopolov, I. N. Rodionova, “Modified Cauchy problem for one third-order hyperbolic problem in three-dimensional space”, Bull. Samara State Tech. Univ. Ser. Phys.-Math. Sci., 1:18 (2009), 41–46 (in Russian) | DOI

[9] M. V. Dolgopolov, I. N. Rodionova, “Problems for equations of hyperbolic type on a plane and in a three-dimensional space with conjugation conditions on a characteristic”, Bull. Russ. Acad. Sci. Ser. Math., 75:4 (2011), 21–28 (in Russian) | DOI | MR | Zbl

[10] V. M. Dolgopolov, I. N. Rodionova, “Extremal properties of special classes of solutions for one equation of hyperbolic type”, Math. Notes, 92:4 (2012), 533–540 (in Russian) | DOI | MR | Zbl

[11] M. V. Dolgopolov, I. N. Rodionova, V. M. Dolgopolov, “On one nonlocal problem for the Euler–Darboux equation”, Bull. Samara State Tech. Univ. Ser. Phys.-Math. Sci., 20:2 (2016), 259–275 (in Russian) | DOI | Zbl

[12] L. D. Landau, E. M. Lifshits, Mechanics of Continua, Gostekhizdat, M., 1953 (in Russian) | MR

[13] A. M. Nakhushev, “New boundary-value problem for one degenerating hyperbolic equation”, Rep. Acad. Sci. USSR, 187:4 (1969), 736–739 (in Russian) | Zbl

[14] A. M. Nakhushev, “On some new boundary-value problems for hyperbolic equations and mixed-type equations”, Differ. Equ., 5:1 (1969), 44–59 (in Russian) | MR | Zbl

[15] Z. A. Nakhusheva, Nonlocal Boundary-Value Problems for Differential Equation of Main Types and Mixed Type, Kabardino-Balkarskiy nauchnyy tsentr RAN, Nal'chik, 2012 (in Russian)

[16] S. G. Samko, A. A. Kilbas, O. I. Marichev, Integrals and Derivatives of Fractional Order and Some Their Applications, Nauka i tekhnika, Minsk, 1987 (in Russian)

[17] M. M. Smirnov, Degenerating Hyperbolic Equations, Vysheyshaya shkola, Minsk, 1977 (in Russian)

[18] M. M. Smirnov, Equations of Mixed Type, Vysshaya shkola, M., 1985 (in Russian) | MR

[19] V. V. Sokolovskiy, Mechanics of Continua, Fizmatgiz, M., 1960 (in Russian)

[20] K. P. Stanyukovich, Theory of Unsteady Motions of a Gas, Byuro novoy tekhniki, M., 1948 (in Russian)

[21] S. A. Chaplygin, Collected Works, v. 2, On Gas Jets, Gostekhizdat, M.–L., 1948 (in Russian) | MR

[22] F. I. Frankl', Selected Works on Gas Dynamics, Nauka, M., 1973 (in Russian)

[23] I. N. Rodionova, V. M. Dolgopolov, M. V. Dolgopolov, “Delta-problems for the generalized Euler–Darboux equation”, Bull. Samara State Tech. Univ. Ser. Phys.-Math. Sci., 21:3 (2017), 417–422 | DOI | Zbl