On solvability of one class of third order differential equations

  • B. T. Bilalov Inst. Math. and Mech. NAS Azerbaijan, Baku
  • M. I. Ismailov Baku State Univ., Azerbaijan https://orcid.org/0000-0001-5770-6619
  • Z. A. Kasumov Inst. Math. and Mech. NAS Azerbaijan, Baku
Keywords: mixed problem, Fourier method, generalized solution, Paley theorem

Abstract

UDC 517.9

One-dimensional mixed problem for one class of third order partial differential equation with nonlinear right-hand side is considered. The concept of generalized solution for this problem is introduced. By the Fourier method, the problem of existence and uniqueness of generalized solution for this problem is reduced to the problem of solvability of the countable system of nonlinear integro-differential equations. Using Bellman's inequality, the uniqueness of generalized solution is proved. Under some conditions on initial functions and the right-hand side of the equation, the existence theorem for the generalized solution is proved using the method of successive approximations.

References

A. Ashyralyev, K. Belakroum, On the stability of nonlocal boundary value problem for a third order PDE, AIP Conf. Proc., 2183, Article 070012 (2019), https://doi.ord/10.1063/1.5136174

Sh. Amirov, A. I. Kozhanov, A mixed problem for a class of strongly nonlinear higher-order equations of Sobolev type, Dokl. Math., 88, № 1, 446 – 448 (2013) (in Russian), https://doi.org/10.1134/s1064562413040236 DOI: https://doi.org/10.1134/S1064562413040236

Yu. P. Apakov, B. Yu. Irgashev, Boundary-value problem for a degenerate high-odd-order equation, Ukr. Math. J.,66, № 10, 1475 – 1490 (2015), https://doi.org/10.1007/s11253-015-1039-7 DOI: https://doi.org/10.1007/s11253-015-1039-7

C. Latrous, A. Memou, A three-point boundary value problem with an integral condition for a third-order partial differential equation, Abstr. and Appl. Anal., 2005, № 1, 33 – 43 (2005), https://doi.org/10.1155/AAA.2005.33 DOI: https://doi.org/10.1155/AAA.2005.33

Y. Apakov, S. Rutkauskas, On a boundary value problem to third order PDE with multiple characteristics, Nonlinear Anal. Model. and Control, 16, № 3, 255 – 269 (2011).

M. Kudu, I. Amirali, Method of lines for third order partial differential equations, J. Appl. Math. and Phys., 2, № 2, 33 – 36 (2014).

A. Ashyralyev, D. Arjmand, M. Koksal, Taylor’s decomposition on four points for solving third-order linear timevarying systems, J. Franklin Inst. Eng. and Appl. Math., 346, 651 – 662 (2009), https://doi.org/10.1016/j.jfranklin.2009.02.017 DOI: https://doi.org/10.1016/j.jfranklin.2009.02.017

A. Ashyralyev, D. Arjmand, A note on the Taylor’s decomposition on four points for a third-order differential equation, Appl. Math. and Comput., 188, № 2, 1483 – 1490 (2007), https://doi.org/10.1016/j.amc.2006.11.017 DOI: https://doi.org/10.1016/j.amc.2006.11.017

Kh. Belakroum, A. Ashyralyev, A. Guezane-Lakoud, A note on the nonlocal boundary value problem for a third order partial differential equation, AIP Conf. Proc., 1759, Article 020021 (2016), http://dx.doi.org/10.1063/1.4959635.

A. Ashyralyev, P. E. Sobolevskii, New difference schemes for partial differential equations, Birkh¨auser, Basel etc. (2004), https://doi.org/10.1007/978-3-0348-7922-4 DOI: https://doi.org/10.1007/978-3-0348-7922-4

Kh. Belakroum, A.Ashyralyev, A. Guezane-Lakoud, A note on the nonlocal boundary value problem for a third order partial differential equation, Filomat, 32, № 3, 801 – 808 (2018), https://doi.org/10.2298/fil1803801b DOI: https://doi.org/10.2298/FIL1803801B

A. Ashyralyev, Kh. Belakroum, A. Guezane-Lakoud, Stability of boundary-value problems for third-order partial differential equations, Electron. J. Different. Equat., 53, 1 – 11 (2017). DOI: https://doi.org/10.1063/1.4959635

J. Nagumo, S. Arimoto, S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. JRE, 50, № 10, 2061 – 2070 (1962).

A. Reiko, H. Yojiro, On global solutions for mixed problem of a semilinear differential equation, Proc. Japan Acad., 39, № 10, 721 – 725 (1963).

M. Jamaguti, The asymptotic behavior of the solution of a semilinear partial differential equation related to an active pulse transmission line, Proc. Japan Acad., 39, № 10, 726 – 730 (1963).

J. M. Greenberg, R. C. MacCamy, V. J. Mizel, On the existence, uniqueness and stability of solutions of the equation $σ‎' (u_x) u_{xx} + λ u_{xtx} = ρ_0 u_{tt}$, J. Math. and Mech., 17, № 7, 707 – 728 (1968).

J. M. Greenberg, On the existence, uniqueness and stability of solutions of the equation $ρ_{0}{X}_{tt}=E({ X}_{x}) {X}_{xx} + λ { X}_{xxt}$, J. Math. Anal. and Appl., 25, 575 – 591 (1969), https://doi.org/10.1016/0022-247X(69)90257-1 DOI: https://doi.org/10.1016/0022-247X(69)90257-1

J. M. Greenberg, R. C. MacCamy, On the exponential stability solutions of the equation $E(u_{x}) u_{xx} + λ u_{xtx} = ρ u_{tt}$, J. Math. Anal. and Appl., 31, № 2, 406 – 417 (1970), https://doi.org/10.1016/0022-247X(70)90034-X DOI: https://doi.org/10.1016/0022-247X(70)90034-X

P. L. Davis, On the existence, uniqueness and stability of solutions of a nonlinear functional defferential equation, J. Math. Anal. and Appl., 34, № 1, 128 – 140 (1971), https://doi.org/10.1016/0022-247X(71)90163-6 DOI: https://doi.org/10.1016/0022-247X(71)90163-6

A. I. Kozhanov, Mixed problem for some classes of third order nonlinear equations, Mat. Sb., 118, № 4, 504 – 522 (1982) (in Russian).

S. A. Gabov, G. Y. Malysheva, A. G. Sveshnikov, On one equation of composite type related to the oscillations of stratified fluid, Differents. Uravneniya, 19, № 7, 1171 – 1180 (1983) (in Russian).

B. A. Iskenderov, V. Q. Sardarov, Mixed problem for Boussinesq equation in cylindrical dominain and behavior of its solution at $t to+infty$, Trans. Nat. Akad. Sci. Azerb., 22, № 4, 117 – 134 (2002).

A. B. Aliev, A. A. Kazimov, The asymptotic behavior of weak solution of Cauchy problem for a class Sobolev type semilinear equation, Trans. Nat. Akad. Sci. Azerb. Ser. Phys.-Techn. and Math. Sci., 26, № 4, 23 – 30 (2006).

T. K. Yuldashev, On an optimal control problem for a nonlinear pseudohyperbolic equation, Model. Anal. Inform. Sist., 20, № 5, 78 – 89 (2013).

G. A. Rasulova, Study of mixed problem for one class of third order quasilinear differential equations, Differents. Uravneniya, 3, № 9, 1578 – 1591 (1967) (in Russian).

A. I. Kozhanov, Mixed problem for one class of nonclassical equations, Differents. Uravneniya, 15, № 2, 272 – 280 (1979) (in Russian).

O. A. Ladyzhenskaya, Mixed problem for hyperbolic equation, Gostekhizdat, Moscow (1953) (in Russian).

G. I. Laptev, On one third order quasilinear partial differential equation, Differents. Uravneniya, 24, №7, 1270 – 1272 (1988) (in Russian).

P. L. Davis, A quasilinear hyperbolic and related third-order equations, J. Math. Anal. and Appl., 51, № 3, 596 – 606 (1975), https://doi.org/10.1016/0022-247X(75)90110-9 DOI: https://doi.org/10.1016/0022-247X(75)90110-9

R. S. Zhamalov, Directional derivative problem for one third order equation, Kraevyye Zadachi dlya Neklassicheskikh Uravnenii Matematicheskoi Fiziki, 17, 115 – 116 (1989) (in Russian).

K. I. Khudaverdiyev, A. A. Veliyev, Study of one-dimensional mixed problem for one class of third order

pseudohyperbolic equations with nonlinear operator right-hand side, Chashioglu, Baku (2010) (in Russian).

Z. I. Khalilov, A new method for solving the equations of vibrations of elastic system, Izv. Azerb. Branch AS USSR, № 4, 168 – 169 (1942).

V. A. Ilyin, On solvability of mixed problem for hyperbolic and parabolic equations, Uspekhi Mat. Nauk, 15, № 2, 97 – 154 (1960) (in Russian), https://doi.org/10.1070/RM1960v015n02ABEH004217 DOI: https://doi.org/10.1070/RM1960v015n02ABEH004217

K. I. Khudaverdiyev, On generalized solutions of one-dimensional mixed problem for a class of quasilinear differential equations, Azerbaĭdžan. Gos. Univ. Učen. Zap. Ser. Fiz.-Mat. Nauk, 1965, № 4, 29 – 42 (1965) (in Russian).

V. A. Chernyatin, Justification of Fourier method for mixed problem for partial differential equations, Izd-vo Mosk. Un-ta (1991) (in Russian).

B. T. Bilalov, Z. G. Guseynov, $scr K$-Bessel and $scr K$-Hilbert systems. $scr K$-bases, Dokl. RAN, 429, № 3, 298 – 300 (2009), https://doi.org/10.1134/S1064562409060118 DOI: https://doi.org/10.1134/S1064562409060118

B. T. Bilalov, Bases and tensor product, Trans. Nat. Acad. Sci. Azerb., 25, № 4, 15 – 20 (2005).

B. T. Bilalov, Some questions of approximation, Elm, Baku (2016).

A. Zygmund, Trigonometric series, vols. 1, 2, Mir, Moscow (1965).

S. Kachmazh, G. Steinhous, Theory of orthogonal series, GIFML, Moscow (1958).

Published
22.03.2021
How to Cite
Bilalov, B. T., M. I. Ismailov, and Z. A. Kasumov. “On Solvability of One Class of Third Order Differential Equations”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, no. 3, Mar. 2021, pp. 314 -28, doi:10.37863/umzh.v73i3.195.
Section
Research articles