Home > CAPMSE > Vol. 2020 (2020) > Iss. 3

## Central Asian Problems of Modern Science and Education

#### Abstract

This work is devoted to the study of an approximate solution of the initial-boundary value problem for the second order mixed type nonhomogeneous differential equation with two degenerate lines. Similar equations have many diﬀerent applications, for example, boundary value problems for mixed type equations are applicable in various ﬁelds of the natural sciences: in problems of laser physics, in magneto hydrodynamics, in the theory of infinitesimal bindings of surfaces, in the theory of shells, in predicting the groundwater level, in plasma modeling, and in mathematical biology. In this paper, based on the idea of A.N. Tikhonov, the conditional correctness of the problem, namely, uniqueness and conditional stability theorems are proved, as well as approximate solutions that are stable on the set of correctness are constructed. In obtaining an apriori estimate of the solution of the equation, we used the logarithmic convexity method and the results of the spectral problem considered by S.G. Pyatkov. The results of the numerical solutions and the approximate solutions of the original problem were presented in the form of tables. The regularization parameter is determined from the minimum estimate of the norm of the diﬀerence between exact and approximate solutions.

#### First Page

86

#### Last Page

100

#### References

1. A.G. Kuzmin, Nonclassical equations of mixed type and their applications to gas dynamics, Izdatel’stvo Leningradskogo Universiteta, Leningrad, 1990.

2. A. I. Kozhanov Composite Type Equations and Inverse Problems. VSP, Utrecht, The Netherlands, 1999,pp.171.

3. A.M. Nakhushev, Equations of mathematical biology, Vyssh. Shkola, Moscva, 1995.

4. A. M. Nakhushev, Boundary value problem for the equation mixed type with two degenerate lines, Dokl. AN SSSR, 1966, volume 170, № 1,pp. 38-40.

5. F.I. Frankl, Selected Works on Gas Dynamics, Nauka, Moscow, 1973.

6. I.A. Makarov Potential theory for equations with two lines of degenerate. / I. A. Makarov // Differents. equations. Collection of works of mathematical departments of pedagogical institutes of the RSFSR. 1973. - Issue. 2. pp. 145-155.

7. I.N.Vekua Generalized analytical functions. / I.N. Vekua. Moscow: Fizmat giz, 1959 .pp. 628.

8. K.S. Fayazov Ill-Posed Boundary value problem for one second order mixed type equation. Uzbek. math journal. 2, 1995, pp. 89-93.

9. K.S. Fayazov, I.O. Khazhiev, Conditional stability of a boundary value problem for a system abstract diﬀerential equations of the second order with operator coeﬃcients, Uzbek Mathematical Journal, 2017:2 (2017), pp.145–155.

10. K.S. Fayazov, Y. K. Khudayberganov, An ill-posed boundary value problem for a system of equations of mixed type with two degenerate lines, Siberian Electronic Mathematical Izvestia, Volume 17, pp. 647–660 (2020), DOI 10.33048 / semi.2020.17.043. (scopus if = 0.38).

11. K.B. Sabitov, A.A. Karamova, Spectral properties of solutions of the Tricomi problem for an equation of mixed type with two lines of change in type and their applications, Izv. RAS. Ser. Mat., 2001, Volume 65, Issue 4, pp.133–150

12. L. Bers, Mathematical problems of subsonic and transonic gas dynamics, IL, Moscow, 1961. (John Wiley & Sons, New York, 1958.)

13. M. N. Kogan On magnetohydrodynamic flows of mixed type. / MN Kogan // Applied Mathematics and Mechanics. 1961. —T. 25. pp. 132-137.

14. M.M. Lavrent’ev, L.Y. Saveliev, Theory of operators and ill-posed problems, 2 nd ed., Institute of Mathematics, Novosibirsk, 2010.

15. N.V. Kislov Nonhomogeneous boundary value problems for mixed type differential-operator equations and their applications. Mat. Sat. 12 (1), 1984, pp. 19-37.

16. N.A. Larkin, V.A. Novikov, N.N. Yanenko, Nonlinear Equations of Variable Type, Nauka, Novosibirsk, 1983.

17. S.G. Pyatkov, Solvability of boundary value problems for a second-order equation of mixed type, Nonclassical partial diﬀerential equations, Collect. Sci. Works, Novosibirsk, (1988), pp. 77–90.

18. S.G. Pyatkov On the solvability of a boundary value problem for a parabolic equation with changing direction of time. Dokl. Academy of Sciences of the USSR. 285 (6), 1985, pp. 1322-1327.

19. S.P. Pulkin Investigation by equations of mixed type. Diss. PhD. n .: 01.01.02, S.P. Pulkin, Kazan: KSU, 1958.

20. V.N. Vragov, On the theory of boundary value problems for equations of mixed type in the space, Diﬀ. equations, 13:6 (1977), pp. 1098–1105.

#### Recommended Citation

Khudayberganov, Y.K.
(2020)
"APPROXIMATE SOLUTION OF AN INITIAL-BOUNDARY VALUE PROBLEM FOR AN NONHOMOGENEOUS SECOND-ORDER DIFFERENTIAL EQUATION OF MIXED TYPE WITH TWO DEGENERATE LINES,"
*Central Asian Problems of Modern Science and Education*: Vol. 2020
:
Iss.
3
, Article 7.

Available at:
https://uzjournals.edu.uz/capmse/vol2020/iss3/7