BOUSSINESQ-L"OVE MATHEMATICAL MODEL ON A GEOMETRICAL GRAPH

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Published: Jul 6, 2015
Keywords:
Sobolev type model geometrical graph Fourier method Sturm - Liouville problem.

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A. A. Zamyshlyaeva
South Ural State University
A. V. Lut
South Ural State University

Abstract

Of concern is the Boussinesq - L"ove mathematical model describing longitudinal vibrations in the elements of construction that can be represented in the form of a finite connected oriented graph. Differential equations on graphs is a relatively new piece of mathematical knowledge. The article deals with Sobolev type equations on graphs. The Fourier method is used for solution of the problem.  The article besides introduction, conclusion and list of references contains four paragraphs. The first paragraph describes the properties of eigenvalues and eigenfunctions of the problem on a graph. The second section is devoted to specific examples of solutions of the Sturm - Liouville problem. In the third paragraph, by applying the Fourier method, the solutions of the Boussinesq - L"ove problem are found. In the last paragraph, the possibility of usage of the Fourier method for Boussinesq - L"ove problem for some finite connected oriented graphs is justified.

Article Details

Issue
Vol. 2 No. 2 (2015): JOURNAL OF COMPUTATIONAL AND ENGINEERING MATHEMATICS
Section
Survey Articles

References

L"ove A.E.H. A Tretise on the Mathematical Theory of Elasticity. Cambridge, Univ. Press, 1927.

Zamyshlyaeva A.A. Phase Spaces of Some Class of Linear Second Order Sobolev Type Equations. Vychislitel'nyye Tehnologii - Computational Technologies, 2003, vol. 8, no. 4, pp. 45-54. (in Russian)

Zamyshlyaeva A.A., Yuzeeva A.V. The Initial-finish Value Problem for the Boussinesq - L"ove Equation. Bulletin of the South Ural State University. Series "Mathematical Modelling, Programming & Computer Software", 2010, vol. 5, no. 16(192), pp. 23-31. (in Russian)

Zamyshlyaeva A.A., Tsyplenkova O.N. The Optimal Control over Solutions of the Initial-finish Value Problem for the Boussinesque - L"ove Equation. Bulletin of the South Ural State University. Series "Mathematical Modelling, Programming & Computer Software", 2012, vol. 11, no. 5(264), pp. 13-24. (in Russian)

Zamyshlyaeva A.A., Tsyplenkova O.N. Optimal Control of Solutions of the Showalter - Sidorov - Dirichlet Problem for the Boussinesq - L"ove Equation. Differential Equations, 2013, vol. 49, no. 11, pp. 1356-1365.

Zamyshlyaeva A.A., Bychkov E.V. The Phase Space of the Modified Boussinesq Equation. Bulletin of the South Ural State University. Series "Mathematical Modelling, Programming & Computer Software", 2012, vol. 12, no. 18(277), pp. 13-19. (in Russian)

Zamyshlyaeva A.A. On Algorithm of Numerical Modelling of the Boussinesq - L"ove Waves. Bulletin of the South Ural State University. Series "Computer Technologies, Automatic Control & Radioelectronics", 2013, vol. 13, no. 4, pp. 24-29. (in Russian)

Pokorny Y.V., Penkin O.M., Pryadiev V.L., Borovskih A.V., Lazarev K.P., Shabrov S.A. Differential Equations on Geometric Graphs. Moscow, Fizmatlit Publ., 2005. (in Russian)

Sviridyuk G.A., Shemetova V.V. The Phase Space of One Nonclssical Model. Izv. Vyssh. Uchebn. Zaved. Mat., 2005, no. 11, pp. 47-52. (in Russian)

Sviridyuk G.A., Bayazitova A.A. On Direct and Inverse Problems for the Hoff Equations on Graph. Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki, 2009, vol. 18, no. 1, pp. 6-17. (in Russian)

Bayazitova A.A. The Sturm - Liouville Problem on Geometric Graph. Bulletin of the South Ural State University. Series "Mathematical Modelling, Programming & Computer Software", 2010, vol. 5, no. 16(192), pp. 4-10. (in Russian)