Abstract
In the present work, an initial-boundary value problem for a high even order partial differential equation with a Bessel operator is considered and the existence, uniqueness and stability of the solution of the considered problem have been investigated.
REFERENCES
A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics (Nauka, Moscow, 1972; Dover, New York, 2011).
B. G. Korenev, Issues of Calculation of Beams and Slabs on an Elastic Foundation (Stroiizdat, Moscow, 1954) [in Russian].
E. V. Makhover, ‘‘Bending of a plate of variable thickness with a sharp edge,’’ Uch. Zap. LGP Gertsena 17 (2), 28–39 (1957).
E. V. Makhover, ‘‘On the natural frequency spectrum of a plate with a sharp edge,’’ Uch. Zap. LGP Gertsena 197, 113–118 (1958).
M. S. Salakhitdinov and D. Amanov, ‘‘Solvability and spectral properties of a self-adjoint problem for a fourth-order equation,’’ Uzb. Mat. Zh., No. 3, 72–77 (2005).
M. S. Azizov, ‘‘A boundary problem for the fourth order equation with a singular coefficient in a rectangular region,’’ Lobachevskii J. Math. 41, 1043–1050 (2020). https://doi.org/10.1134/S1995080220060050
A. K. Urinov and M. S. Azizov, ‘‘Boundary problem for the loaded partial differential equations of fourth order,’’ Lobachevskii J. Math. 42, 621–631 (2021). https://doi.org/10.1134/S1995080221030197
K. B. Sabitov and O. V. Fadeeva, ‘‘Initial-boundary value problem for the equation of forced vibrations of a cantilever beam,’’ Vestn. Samar. Tekh. Univ., Ser. Fiz.-Mat. Nauki 25, 51–66 (2021). https://doi.org/10.14498/vsgtu1845
A. K. Urinov and M. S. Azizov, ‘‘Boundary value problems for a fourth order partial differential equation with an unknown right-hand part,’’ Lobachevskii J. Math. 42, 632–640 (2021). https://doi.org/10.1134/S1995080221030203
K. B. Sabitov, ‘‘The Dirichlet problem for higher-order partial differential equations,’’ Math. Notes 97, 255–267 (2015). https://doi.org/10.1134/S0001434615010277
Sh. G. Kasimov and U. S. Madrakhimov, ‘‘Initial boundary value problem for the beam vibration equation in the multidimensional case,’’ Differ. Equat. 55, 1336–1348 (2019). https://doi.org/10.1134/S0012266119100094
A. K. Urinov and M. S. Azizov, ‘‘An initial boundary value problem for a partial differential equation of higher even order with a Bessel operator,’’ Vestn. Samar. Tekh. Univ., Ser. Fiz.-Mat. Nauki 26, 273–292 (2022). https://doi.org/10.14498/vsgtu1893
A. K. Urinov and M. S. Azizov, ‘‘On the solvability of nonlocal initial-boundary value problems for a partial differential equation of high even order,’’ Vestn. Udmurt. Univ., Mat. Mekh. Komp’yut. Nauki 32, 240–255 (2022). https://doi.org/10.35634/vm220206
B. Yu. Irgashev, ‘‘On a boundary value problem for a high order mixed type equation,’’ Sib. Electron. Math. Rep. 17, 899–912 (2020). https://doi.org/10.33048/semi.2020.17.066
T. K. Yuldashev, ‘‘Solvability of a boundary value problem for a differential equation of the Boussinesq type,’’ Differ. Equat. 54, 1384–1393 (2018).
T. K. Yuldashev, ‘‘Nonlocal inverse problem for a pseudohyperbolic-pseudoelliptic type integro-differential equations,’’ Axioms 9, 45-1–21 (2020). https://doi.org/10.3390/axioms9020045
T. K. Yuldashev, ‘‘Inverse problem for a nonlinear Benney–Luke type integro-differential equations with degenerate kernel,’’ Russ. Math. 60 (8), 53–60 (2016).
T. K. Yuldashev, ‘‘Nonlinear optimal control of thermal processes in a nonlinear inverse problem,’’ Lobachevskii J. Math. 41, 124–136 (2020).
T. K. Yuldashev and B. J. Kadirkulov, ‘‘Nonlocal problem for a mixed type fourth-order differential equation with Hilfer fractional operator,’’ Ural Math. J. 6, 153–167 (2020).
T. K. Yuldashev and B. J. Kadirkulov, ‘‘Inverse boundary value problem for a fractional differential equations of mixed type with integral redefinition conditions,’’ Lobachevskii J. Math. 42, 649–662 (2021).
K. B. Baikuziev and B. S. Kalanov, ‘‘On the solvability of a mixed problem for a higher-order equation that degenerates on the boundary of a domain,’’ in Boundary Value Problems for Differential Equations (Riga, 1972), Vol. 2, pp. 40–54 [in Russian].
K. B. Baikuziev, ‘‘A mixed problem for a single higher-order equation degenerating at the boundary of a domain,’’ Differ. Uravn. 20, 7–14 (1984).
B. Yu. Irgashev, ‘‘Boundary value problem for one degenerating equation high order with junior members,’’ Tr. Inst. Mat., No. 6, 23–29 (2019).
Yu. P. Apakov and B. Yu. Irgashev, ‘‘Boundary-value problem for a degenerate high-odd-order equation,’’ Ukr. Math. J. 66, 1318–1331 (2014). https://doi.org/10.1007/s11253-015-1039-7
A. K. Urinov and M. S. Azizov, ‘‘On the solvability of an initial-boundary value problem for a high even order partial differential equation degenerating on the domain boundary,’’ J. Appl. Ind. Math. 17, 414–426 (2023). https://doi.org/10.1134/S1990478923020199
A. K. Urinov and D. A. Usmonov, ‘‘An initial-boundary problem for a hyperbolic equation with three lines of degenerating of the second kind,’’ Vestn. Samar. Tekh. Univ., Ser. Fiz.-Mat. Nauki 26, 672–693 (2022). https://doi.org/10.14498/vsgtu1962
A. K. Urinov and D. A. Usmonov, ‘‘On one problem for a fourth-order mixed-type equation that degenerates inside and on the boundary of a domain,’’ Vestn. Udmurt. Univ., Mat. Mekh. Komp. Nauki 33, 312–328 (2023). https://doi.org/10.35634/vm23020
I. A. Kipriyanov, Singular Elliptic Boundary Value Problems (Nauka, Moscow 1997) [in Russian].
M. A. Naimark, Linear Differential Operators (Nauka, Moscow, 1969) [in Russian].
S. G. Mikhlin, Lectures on Linear Integral Equations (Fizmatlit, Moscow, 1959) [in Russian].
G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge Univ. Press, Cambridge, 1995).
H. Bateman and A. Erdelyi, Higher Transcendental Functions, I (McGraw-Hill, New York, 1953).
Funding
This work was supported by ongoing institutional funding. No additional grants to carry out or direct this particular research were obtained.
Author information
Authors and Affiliations
Corresponding authors
Ethics declarations
The authors of this work declare that they have no conflicts of interest.
Additional information
Publisher’s Note.
Pleiades Publishing remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
(Submitted by T. K. Yuldashev)
Rights and permissions
About this article
Cite this article
Urinov, A.K., Azizov, M.S. Initial-Boundary Value Problem for a Degenerate High Even-Order Partial Differential Equation with the Bessel Operator. Lobachevskii J Math 45, 864–874 (2024). https://doi.org/10.1134/S1995080224600158
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1995080224600158