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Numerical solution for non-stationary linearized Hoff equation defined on geometrical graph. (English) Zbl 07489417

Summary: The non-stationary linearized Hoff equation is considered in the article. For this equation, a solution is obtained both on the domain and on the geometric graph. For the five-edged graph, the Sturm-Liouville problem is solved to obtain a numerical solution of the non-stationary linearized Hoff equation on the graph. A numerical method for solving this equation on a graph is described. The graphics for obtained numerical solution are constructed at different instants of time for given values of the equation parameters and functions. The article besides the introduction and the bibliography contains four parts. The first part contains information on abstract non-stationary Sobolev type equations, and solutions for the non-stationary linearized Hoff equation on the domain are constructed. In the second one we consider the Sturm-Liouville problem on a graph and construct necessary spaces and operators on graphs. In the third one we study the solvability of the non-stationary linearized Hoff equation on the five-edged graph, and finally, in the last part we describe the numerical solution of the equation on the graph and the graphics of these solutions at different instants of time.

MSC:

65L99 Numerical methods for ordinary differential equations
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References:

[1] N. A. Hoff, “Greep Buckling”, Journal of the Aeronautical Sciences, 7:1 (1965), 1-20
[2] G. A. Sviridyuk, V. E. Fedorov, Lineinye uravneniya sobolevskogo tipa, Chelyab. gos. un-t, Chelyabinsk, 2003, 179 pp. · Zbl 1093.34001
[3] A. L. Shestakov, G. A. Sviridyuk, E. V. Zakharova, “Dinamicheskie izmereniya kak zadacha optimalnogo upravleniya”, Obozrenie prikladnoi i promyshlennoi matematiki, 16:4 (2009), 732-733
[4] A. V. Keller, E. I. Nazarova, “Svoistvo regulyarizuemosti i chislennoe reshenie zadachi dinamicheskogo izmereniya”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 2010, no. 5, 32-38 · Zbl 1234.34008
[5] A. L. Shestakov, “Modalnyi sintez izmeritelnogo preobrazovatelya”, Izvestiya RAN. Teoriya i sistemy upravleniya, 1995, no. 4, 67-75
[6] A. L. Shestakov, A. V. Keller, G. A. Sviridyuk, “The Theory of Optimal Measurements”, J. Comp. Eng. Math., 1:1 (2014), 3-16 · Zbl 1343.49005
[7] A. L. Shestakov, G. A. Sviridyuk, Yu. V. Khudyakov, “Dinamicheskie izmereniya v prostranstvakh «shumov»”, Vestnik YuUrGU. Seriya: Kompyuternye tekhnologii, upravlenie, radioelektronika, 13:2 (2013), 4-11
[8] G. A. Sviridyuk, V. O. Kazak, “The Phase Space of an Initial-Boundary Value Problem for the Hoff Equation”, Math. Notes, 71:2 (2002), 262-266 · Zbl 1130.37401 · doi:10.1023/A:1013919500605
[9] G. A. Sviridyuk, N. A. Manakova, “An Optimal Control Problem for the Hoff Equation”, J. Appl. Industr. Math., 1:2 (2007), 247-253 · doi:10.1134/S1990478907020147
[10] G. A. Sviridyuk, V. V. Shemetova, “Hoff Equations on Graphs”, Differ. Equ., 42:1 (2006), 139-145 · Zbl 1126.37047 · doi:10.1134/S0012266106010125
[11] G. A. Sviridyuk, S. A. Zagrebina, P. O. Pivovarova, “Ustoichivost uravnenii Khoffa na grafe”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 1 (20) (2010), 6-15 · Zbl 1449.35059 · doi:10.14498/vsgtu735
[12] S. A. Zagrebina, “Mnogotochechnaya nachalno-konechnaya zadacha dlya lineinoi modeli Khoffa”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 2012, no. 11, 4-12 · Zbl 1355.35117
[13] N. A. Manakova, “An Optimal Control to Solutions of the Showalter - Sidorov Problem for the Hoff Model on the Geometrical Graph”, J. Comp. Eng. Math., 1:1 (2014), 26-33 · Zbl 1339.49004
[14] N. A. Manakova, A. G. Dylkov, “Optimal Control of the Solutions of the Initial-Finish Problem for the Linear Hoff Model”, Math. Notes, 94:2 (2013), 220-230 · Zbl 1510.49017 · doi:10.1134/S0001434613070225
[15] M. A. Sagadeeva, Investigation of Solutions’ Stability for Linear Sobolev Type Equations, Disertation of PhD (Math), Chelyabinsk, 2006, 120 pp.
[16] A. V. Keller, M. A. Sagadeeva, “The Optimal Measurement Problem for the Measurement Transducer Model with a Deterministic Multiplicative Effect and Inertia”, Vestnik YuUrGU. Ser. Mat. Model. Progr., 7:1 (2014), 134-138 · Zbl 1303.93093 · doi:10.14529/mmp140111
[17] M. A. Sagadeeva, G. A. Sviridyuk, “The Nonautonomous Linear Oskolkov Model on a Geometrical Graph: the Stability of Solutions and Optimal Control Problem”, Semigroups of Operators - Theory and Applications, Proc. Int. Conference (Bedlewo, Poland, 2013, October 6-10), Springer Proceedings in Mathematics and Statistics, 113, eds. J. Banasiak, A. Bobrowski, M. Lachowicz, Springer International Publishing, 2015, 257-271 · Zbl 1331.47091 · doi:978-3-319-12145-1_16
[18] M. A. Sagadeeva, “Mathematical Bases of Optimal Measurements Theory in Nonstationary Case”, J. Comp. Eng. Math., 3:3 (2016), 19-32 · Zbl 1429.49028 · doi:10.14529/jcem160303
[19] G. A. Sviridyuk, “Uravneniya sobolevskogo tipa na grafakh”, Neklassicheskie uravneniya matematicheskoi fiziki, Novosibirsk, 2002, 221-225 · Zbl 1015.35088
[20] S. A. Zagrebina, N. P. Solovyeva, “The Initial-Finish Problem for the Evolution of Sobolev-Type Equations on a Graph”, Bulletin of the South Ural State University. Series «Mathematical Modelling, Programming & Computer Software». Issue 1, 2008, no. 15 (115), 23-26 · Zbl 1231.35282
[21] A. A. Zamyshlyaeva, “On a Sobolev Type Equation Defined on the Graph”, Bulletin of the South Ural State University. Series «Mathematical Modelling, Programming & Computer Software». Issue 2, 2008, no. 27 (127), 45-49 · Zbl 1225.35248
[22] A. A. Zamyshlyaeva, A. V. Yuzeeva, “The Initial-Finish Value Problem for the Boussinesque-Löve Equation Defined on Graph”, IIGU Ser. Matematika, 3:2 (2010), 18-29 · Zbl 1269.35032
[23] A. G. Dylkov, “Numerical Solution of an Optimal Control Problem for One Linear Hoff Model Defined on Graph”, Vestnik YuUrGU. Ser. Mat. Model. Progr., 2012, no. 13, 128-132 · Zbl 1413.35420
[24] N. A. Manakova, K. V. Vasiuchkova, “Numerical Investigation for the Start Control and Final Observation Problem in Model of an I-beam Deformation”, J. Comp. Eng. Math., 4:2 (2017), 26-40 · Zbl 1457.74143 · doi:10.14529/jcem170203
[25] A. A. Zamyshlyaeva, A. V. Lut, “Numerical Investigation of the Boussinesq-Love Mathematical Models on Geometrical Graphs”, Vestnik YuUrGU. Ser. Mat. Model. Progr., 10:2 (2017), 137-143 · Zbl 1401.74291 · doi:10.14529/mmp170211
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