Geodesic
Numerical solution for non-stationary linearized Hoff equation defined on geometrical graph
Journal of computational and engineering mathematics, Tome 5 (2018) no. 3, pp. 61-74.

Voir la notice de l'article provenant de la source Math-Net.Ru

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.
Keywords: Sobolev type equation, relatively bounded operator, Sturm – Liouville problem, Laplace operator on graph. 
@article{JCEM_2018_5_3_a5,
     author = {M. A. Sagadeeva and A. V. Generalov},
     title = {Numerical solution for non-stationary linearized {Hoff} equation defined on geometrical graph},
     journal = {Journal of computational and engineering mathematics},
     pages = {61--74},
     publisher = {mathdoc},
     volume = {5},
     number = {3},
     year = {2018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JCEM_2018_5_3_a5/}
}
TY  - JOUR
AU  - M. A. Sagadeeva
AU  - A. V. Generalov
TI  - Numerical solution for non-stationary linearized Hoff equation defined on geometrical graph
JO  - Journal of computational and engineering mathematics
PY  - 2018
SP  - 61
EP  - 74
VL  - 5
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/JCEM_2018_5_3_a5/
LA  - en
ID  - JCEM_2018_5_3_a5
ER  - 
%0 Journal Article
%A M. A. Sagadeeva
%A A. V. Generalov
%T Numerical solution for non-stationary linearized Hoff equation defined on geometrical graph
%J Journal of computational and engineering mathematics
%D 2018
%P 61-74
%V 5
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/JCEM_2018_5_3_a5/
%G en
%F JCEM_2018_5_3_a5
M. A. Sagadeeva; A. V. Generalov. Numerical solution for non-stationary linearized Hoff equation defined on geometrical graph. Journal of computational and engineering mathematics, Tome 5 (2018) no. 3, pp. 61-74. http://geodesic.mathdoc.fr/item/JCEM_2018_5_3_a5/

[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. | MR

[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

[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

[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 | DOI | DOI | MR | Zbl

[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 | MR

[10] G. A. Sviridyuk, V. V. Shemetova, “Hoff Equations on Graphs”, Differ. Equ., 42:1 (2006), 139–145 | DOI | MR | Zbl

[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 | DOI

[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

[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 | MR | Zbl

[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 | DOI | DOI | MR | Zbl

[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 | DOI | Zbl

[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 | DOI | MR | Zbl

[18] M. A. Sagadeeva, “Mathematical Bases of Optimal Measurements Theory in Nonstationary Case”, J. Comp. Eng. Math., 3:3 (2016), 19–32 | DOI | MR

[19] G. A. Sviridyuk, “Uravneniya sobolevskogo tipa na grafakh”, Neklassicheskie uravneniya matematicheskoi fiziki, Novosibirsk, 2002, 221–225 | Zbl

[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 \flqq Mathematical Modelling, Programming Computer Software\frqq. Issue 1, 2008, no. 15 (115), 23–26 | Zbl

[21] A. A. Zamyshlyaeva, “On a Sobolev Type Equation Defined on the Graph”, Bulletin of the South Ural State University. Series \flqq Mathematical Modelling, Programming Computer Software\frqq. Issue 2, 2008, no. 27 (127), 45–49 | Zbl

[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 | MR

[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

[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 | DOI | MR

[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 | DOI | Zbl