Boundary Value Problem for a Nonlinear Partial Differential Equation
Ключевые слова:
Boundary value problem, seventh order nonlinear equation, adjoint spectral problem, eigenfunctions, biorthogonal systems, unique solvabilityАннотация
This paper investigates a boundary value problem for a nonlinear seventh-order partial differential equation with a nonlocal integral term. The problem is considered in a rectangular
domain and is supplemented with Samarskii–Ionkin type boundary conditions with respect to the
spatial variable and two-point boundary conditions with respect to time. Owing to the non-selfadjoint nature of the associated spatial operator, the corresponding spectral and adjoint spectral
problems are studied in detail. Explicit representations of eigenvalues and eigenfunctions are
obtained, and the biorthogonality, completeness, minimality, and Riesz basis properties of the
resulting systems are established.
By employing the Fourier method based on the biorthogonal systems of eigenfunctions, the original boundary value problem is reduced to a countable system of nonlinear integral equations. Sufficient conditions guaranteeing the existence and uniqueness of solutions are derived by means of
the method of successive approximations and the contraction mapping principle. Furthermore,
convergence of the constructed Fourier series and its derivatives is rigorously justified. As a consequence, a classical solution to the boundary value problem is obtained in an explicit series form.
The results contribute to the theory of nonlocal higher-order partial differential equations with
non-self-adjoint boundary conditions and provide an effective analytical framework for related
nonlinear mathematical models.
Библиографические ссылки
V. M. Aleksandrov, E. V. Kovalenko, Problems of continuum mechanics with mixed boundary conditions. Nauka, Moscow, 1986. [in Russian]
V. A. Chernyatin, Justification of the Fourier method in a mixed problem for partial differential
equations. Lomonosov Moscow University, Moscow, 1992. [in Russian]
O. M. Guseynov, T. J. Mamedov, ”Fundamental solution of a Cauchy and Goursat type problem
for second order hyperbolic equations and related optimal control problems”, Journal of Contemp.
Applied Mathematics 14 (1), 70–77 (2024). https://doi.org/10.5281/zenodo.10908206
S. Jain, F. Hashemi, M. Alimohammady, C. Cesarano, P. Agarwal, ”Complex solutions in magnetic
Schrodinger equations with critical nonlinear terms”, Journal of Contemp. Applied Mathematics 14
(2), 61–71 (2024). https://doi.org/10.69624/2222-5498.2024.2.61
V. A. Ilyin, ”On the solvability of mixed problems for hyperbolic and parabolic equations”, Uspekhi
Matematicheskikh Nauk 15 (2), 97–154 (1960). [in Russian]
A. R. Latifova, J. A. Osmali, ”Transformation operators for the Schr¨odinger equation with a linearly increasing potential”, Journal of Contemp. Applied Mathematics 14 (1), 22–27 (2024).
https://doi.org/10.5281/zenodo.10465123
R. S. Mamedov, Sh. I. Kuliyeva, N. Bagirli, ”Optimal control synthesis for the process described by
the third order partial differential equation”, Journal of Contemp. Applied Mathematics 13 (2),
–48 (2023). https://doi.org/10.5281/zenodo.8416972
A. I. Vagabov, ”Generalized Fourier method for solving mixed problems for nonlinear equations”,
Differential Equations 32 (1), 94–105 (1996).
S. Z. Dzhamalov, B. B. Khalkhadzhaev, S. T. Tuychiyeva, ”On the smoothness of a generalized solution of a semi-nonlocal boundary value problem for a fourth-order mixed type equation of the second kind”, Journal of Contemp. Applied Mathematics 14 (2), 108–120 (2024).
https://doi.org/10.69624/2222-5498.2024.2.108
A. Sopuev, B. S. Nuranov, ”Boundary value problems for a third order mixed parabolic-hyperbolic
equation with variable coefficients”, Lobachevskii Journal of Mathematics 45 (7), 3424–3431 (2024).
https://doi.org/10.1134/S1995080224604089
T. K. Yuldashev, ”On a boundary-value problem for a fourth-order partial integro-differential
equation with degenerate kernel”, Journal of Mathematical Sciences 245 (4), 508–523 (2020).
https://doi.org/10.1007/s10958-020-04707-2
T. K. Yuldashev, ”Determining of coefficients and the classical solvability of a nonlocal boundaryvalue problem for the Benney–Luke integro-differential equation with degenerate kernel”, Journal of
Mathematical Sciences 254 (6), 793–807 (2021). https://doi.org/10.1007/s10958-021-05341-2
T. K. Yuldashev, Yu. P. Apakov, A. Kh. Zhuraev, ”Boundary value problem for third order partial
integro-differential equation with a degenerate kernel”, Lobachevskii Journal of Mathematics 42 (6),
–1327 (2021). https://doi.org/10.1134/S1995080221060329
T. K. Yuldashev, K. Kh. Shabadikov, ”Mixed problem for a higher-order nonlinear pseudoparabolic equation”, Journal of Mathematical Sciences 254 (6), 776–787 (2021).
https://doi.org/10.1007/s10958-021-05339-w
T. K. Yuldashev, K. Kh. Shabadikov, ”Initial-value problem for a higher-order quasilinear
partial differential equation”, Journal of Mathematical Sciences 254 (6), 811–822 (2021).
https://doi.org/10.1007/s10958-021-05343-0
G. I. Barenblatt, Yu. P. Zheltov, I. N. Kochina, ”On finitness conditions in the mechanics of continuous
media. Static problems of the theory of elasticity”, Prikladnaya Matematika i Mekhanika 24 (5),
–322 (1960). [in Russian]
G. I. Barenblatt, Yu. P. Zheltov, ”Fundamental equations of filtration of homogeneous liquids in
fissured rocks”, Soviet Physics, Doklady 132 (5), 522–525 (1960).
E. N. Bereslavskii, ”Effect of evaporation or infiltration on the free surface of groundwater in certain
problems of underground hydromechanics”, American Journal of Applied Mathematics and Statistics
, (5), 159–163 (2017). https://doi.org/10.12691/ajams-5-5-1
G. I. Barenblatt, V. M. Yentov, V. M. Ryzhik, Movement of liquids and gases in natural reservoirs.
Nedra, Moscow, 1984. [in Russian]
K. S. Basniev, I. N. Kochina, V. M. Maksimov, Underground hydromechanics. Nedra, Moscow, 1993.
[in Russian]
P. Ya. Polubarinova–Kochina, The theory of groundwater movement. Nauka, Moscow, 1977. [in Russian]
G. A. Sviridyuk, D. E. Shafranov, ”The Cauchy problem for the Barenblatt–Zheltov–Kochina equation on a smooth manifold”, Vestnik Chelyabinskogo Gosudarstvennogo Universiteta 9, 171–177
(2003). [in Russian]
M. A. Sagadeeva, F. L. Hasan, ”Bounded solutions of Barenblatt–Zheltov–Kochina model in Quasi–
Sobolev spaces”, Bulletin of the South Ural State University. Seria Mathematical Modelling, Programming and Computer Software 8 (4), 138–144 (2015).
H. K. Musaev, H. K. Nabieva, ”Estimates for the abstract Boussinesq equations”, Journal of Contemp. Applied Mathematics 13 (2), 30–42 (2023). https://doi.org/10.5281/zenodo.8378410
N. T. Orumbayeva, M. T. Kosmakova, T. D. Tokmagambetova, A. M. Manat, ”Solutions of boundary value problems for loaded hyperbolic type equations”, Bulletin of the Karaganda University.
Mathematics 118 (2), 177–188 (2025).
T. K. Yuldashev, ”Determination of the coefficient in nonlocal problem for a Boussinesq type integrodifferential equation with degenerate kernel”, Vladikavkazskii Matematicheskii Zhurnal 21 (2), 67–84
(2019). [in Russian] https://doi.org/10.23671/VNC.2019.2.32118
T. K. Yuldashev, ”Inverse boundary-value problem for an integro-differential Boussinesq-type equation with degenerate kernel”, Journal of Mathematical Science (N.Y.) 250 (5), 847–858 (2020).
N. S. Imanbaev, ”On stability of basis property of root vectors system of the Sturm–
Liouville operator with an integral perturbation of conditions in nonstrongly regular Samarskii–
Ionkin type problems”, International J. Differential Equat. 2015 (641481), 1–6 (2015).
http://dx.doi.org/10.1155/2015/641481
N. S. Imanbaev, M. A. Sadybekov, ”Characteristic determinant of the spectral problem for the ordinary differential operator with the boundary load”, AIP Conference Proceedings 1611, 261–265
(2014).
N. S. Imanbaev, ”Distribution of eigenvalues of a third-order differential operator with strongly regular boundary conditions”, AIP Conference Proceedings 1997 (020027), (2018).
D. E. Uzaqbaeva, T. K. Yuldashev, J. A. Otarova, ”Initial-boundary-value problem for inhomogeneous
mixed type fourth-order differential equation with power degeneration”, Uzbek Mathematical Journal
(2), 327–336 (2025). https://doi.org/10.29229/uzmj.2025-2-31
T. K. Yuldashev, B. J. Kadirkulov, M. M. Babayev, ”Mixed problem for a nonlinear differential equation of the seventh order with identification source”, Baku Mathematical Journal 4 (2), 209–230
(2025). https://doi.org/10.32010/j.bmj.2025.15
T. K. Yuldashev, M. M. Babayev, ”On solvability of a mixed problem for a fifth order linear partial
differential equations with parameter”, Lobachevskii Journal of Mathematics 45 (10), 5313–5327
(2024). https://doi.org/10.1134/S1995080224606039
T. K. Yuldashev, Kh. R. Mamedov, M. M. Babayev, ”Dependence on the parameters of the solution of a mixed problem for a nonlinear integro-differential equation of the fifth order with
a degenerate kernel”, Journal of Contemporary Applied Mathematics 14 (2), 38–60 (2024).
https://doi.org/10.69624/2222-5498.2024.2.38
T. K. Yuldashev, M. M. Babayev, ”Boundary value problem for a seventh order nonhomogeneous
partial differential equation”, Journal of Osh University ”Differential Equations” 1 (1), 39–52 (2026).
N. K. Bari, ”Biorthogonal systems and bases in Hilbert space”, Uchyonie Zapiski Moscovscogo Gosudarstvennogo Universiteta 148 (4), 69–107 (1951). [in Russian]
I. S. Gokhberg, M. G. Krein, Introduction to the theory of linear non-selfadjoint operators. Nauka,
Moscow, 1965. [in Russian]
G. M. Keselman, ”On non-conditional convergence of expansions by eigenfunctions of some differential operators”, Izvestiya vuzov. Matematika No. 2, 82–93 (1969). [in Russian]
T. K. Yuldashev, B. J. Kadirkulov, A. T. Ramazanova, Zh. Zh. Shermamatov, ”Nonlinear optimal
control problem in a two-point boundary regime for a pseudoparabolic equation with Samarskii–
Ionkin type conditions”, Nanosystems : Physics, Chemistry, Mathematics 16 (5), 563–576 (2025).
