×

zbMATH — the first resource for mathematics

On nonlocal boundary value problem for the equation of motion of a homogeneous elastic beam with pinned-pinned ends. (English) Zbl 1406.35395
Summary: In the current paper, in the domain \(D=\{(t,x): t\in(0,T), x\in(0,L)\}\) we investigate the boundary value problem for the equation of motion of a homogeneous elastic beam \[ u_{tt}(t,x)+a^{2}u_{xxxx}(t,x)+b u_{xx}(t,x)+c u(t,x)=0, \] where \(a,b,c \in \mathbb{R}\), \(b^2<4a^2c\), with nonlocal two-point conditions \[ u(0,x)-u(T, x)=\phi(x), \quad u_{t}(0, x)-u_{t}(T, x)=\psi(x) \] and local boundary conditions \(u(t, 0)=u(t, L)=u_{xx}(t, 0)=u_{xx}(t, L)=0\). Solvability of this problem is connected with the problem of small denominators, whose estimation from below is based on the application of the metric approach. For almost all (with respect to Lebesgue measure) parameters of the problem, we establish conditions for the solvability of the problem in the Sobolev space. In particular, if \(\phi\in\mathbf{H}_{q+\rho+2}\) and \(\psi \in\mathbf{H}_{q+\rho}\), where \(\rho>2\), then for almost all (with respect to Lebesgue measure in \(\mathbb{R}\)) numbers \(a\) exists a unique solution \(u\in\mathbf{C}^{2}([0,T];\mathbf{H}_{q})\) of the problem considered.

MSC:
35Q74 PDEs in connection with mechanics of deformable solids
35L25 Higher-order hyperbolic equations
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
74B05 Classical linear elasticity
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Azizbayov E., Mehraliyev Y. A boundary value problem for equation of homogeneous bar with periodic conditions. Amer. J. Appl. Math. Stat. 2015, 3 (6), 252-256. · Zbl 1265.35343
[2] Azizbayov E., Mehraliyev Y. A time-nonlocal boundary value problem for equation of homogeneous bar motion. Bull. Kyiv National Univ. Ser. Math. Mekh. 2012, 27, 20-23. · Zbl 1265.35343
[3] Bors I., Milchis T. Dynamic response of beams on elastic foundation with axial load. Acta Tech. Napocensis: Civil Eng. & Arch. 2013, 56 (1), 67-81.
[4] Chandra T.K. The Borel-Cantelli Lemma. In: Ahmad Sh. SpringerBriefs in Statistics, 2. Springer, India, 2012. · Zbl 1264.60003
[5] Goi T.P., Ptashnik B.I. Nonlocal boundary-value problems for systems of linear partial differential equations with variable coefficients. Ukrainian Math. J. 1997, 49 (11), 1659-1670. doi: 10.1007/BF02487504 (translation of Ukrain. Mat. Zh. 1997, 49 (11), 1478-1487 (in Ukrainian)). · Zbl 0933.35046
[6] Goi T.P., Ptashnik B.I. Problem with nonlocal conditions for weakly nonlinear hyperbolic equations. Ukrainian Math. J. 1997, 49 (2), 204-215. doi: 10.1007/BF02486436 (translation of Ukrain. Mat. Zh. 1997, 49 (2), 186-195. (in Ukrainian)) · Zbl 0936.35110
[7] Il’kiv V.S., Nytrebych Z.M., Pukach Ya. Boundary-value problems with integral conditions for a system of Lame equations in the space of almost periodic functions. Electron. J. Differential Equations 2016, 2016 (304), 1-12.
[8] Il’kiv V.S., Ptashnyk B.I. Problems for partial differential equations with nonlocal conditions. Metric approach to the problem of small denominators. Ukrainian Math. J. 2006, 58 (12), 1847-1875. doi: 10.1007/s11253-006-0172-8 (translation of Ukrain. Mat. Zh. 2006, 58 (12), 1624-1650. (in Ukrainian)) · Zbl 1114.35001
[9] Kalenyuk P.I., Kohut I.V., Nytrebych Z.M. Problem with nonlocal two-point condition in time for a homogeneous partial differential equation of infinite order with respect to space variables. J. Math. Sci. 2010, 167 (1), 1-15. doi: 10.1007/s10958-010-9898-9 (translation of Mat. Metody Fiz.-Mekh. Polya 2008, 51 (4), 17-26. (in Ukrainian)) · Zbl 1212.35050
[10] Kostin D.V. Application of Maslov’s formula to finding an asymptotic solution to an elastic deformation problem. Math. Notes 2008, 83 (1-2), 48-56. doi: 10.1134/S0001434608010069 (translation of Mat. Zametki 2008, 83 (1), 50-60. (in Russian)) · Zbl 1156.35097
[11] Nakhushev A.M. Nonlocal boundary problems with displacement and their relation to loaded equations. Differ. Equations 1985, 21 (1), 92-101. (in Russian)
[12] Ptashnyk B.Yo., Il’kiv V.S., Kmit’ I.Ya., Polishchuk V.M. Nonlocal boundary value problems for partial differential equations. Naukova Dumka, Kyiv, 2002. (in Ukrainian)
[13] Sabitov K.B. Fluctuations of a beam with clamped ends. J. Samara State Tech. Univ. Ser. Phys. Math. Sci. 2015, 19 (2), 311-324. doi: 10.14498/vsgtu1406 (in Russian) · Zbl 1413.35141
[14] Savka I.Y. A nonlocal boundary-value problem for partial differential equations with constant coefficients belonging to smooth curves. J. Math. Sci. 2011, 174 (2), 136-158. doi: 10.1007/s10958-011-0286-x (translation of Mat. Metody Fiz.-Mekh. Polya 2009, 52 (4), 18-33. (in Ukrainian))
[15] Symotyuk M.M., Savka I.Ya. Initial-nonlocal boundary value problems for factorized partial differential equations. Bull. Lviv Polytech. National Univ. Ser Phys. & Math. Sci. 2013, 768, 19-25. (in Ukrainian) · Zbl 1289.35061
[16] Symotyuk M.M., Savka I.Y. Metric estimates of small denominators in nonlocal boundary value problems. Bull. South Ural State Univ. Ser. Math. Mechanics. Phys. 2015, 7 (3), 48-53. (in Russian) · Zbl 1336.35255
[17] Vasylyshyn P.B., Savka I.Ya., Klyus I.S. Multipoint nonlocal problem for factorized equation with dependent coefficients in conditions. Carpathian Math. Publ. 2015, 7 (1), 22-27. doi: 10.15330/cmp.7.1.22-27 · Zbl 1326.35080
[18] Vlasii O.D., Goy T.P., Ptashnyk B.Yo. A problem with nonlocal conditions for weakly nonlinear equations with variable coefficients in the principal part of the operator. Mat. Metody Fiz.-Mekh. Polya 2004, 47 (4), 101-109. (in Ukrainian) · Zbl 1122.35316
[19] Vlasii O.D., Goy T.P., Savka I.Ya. Boundary value problem with nonlocal conditions of a second kind for the hyperbolic factorized operator. Uzhhorod Univ. Sci. Bull. Ser. Math. & Inform. 2014, 25 (1), 33-46. (in Ukrainian)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.