Hoff's Model on a Geometric Graph. Simulations

This article studies numerically the solutions to the Showalter-Sidorov (Cauchy) initial value problem and inverse problems for the generalized Hoff model. Basing on the phase space method and a modified Galerkin method, we develop numerical algorithms to solve initial-boundary value problems and inverse problems for this model and implement them as a software bundle in the symbolic computation package Maple 15.0. Hoff's model describes the dynamics of H-beam construction. Hoff's equation, set up on each edge of a graph, describes the buckling of the H-beam.
The inverse problem consists in finding the unknown coefficients using additional measurements, which account for the change of the rate in buckling dynamics at the initial and terminal points of the beam at the initial moment. This investigation rests on the results of the theory of semi-linear Sobolev-type equations, as the initial-boundary value problem for the corresponding system of partial differential equations reduces to the abstract Showalter-Sidorov (Cauchy) problem for the Sobolev-type equation. In each example we calculate the eigenvalues and eigenfunctions of the Sturm-Liouville operator on the graph and find the solution in the form of the Galerkin sum of a few first eigenfunctions. Software enables us to graph the numerical solution and visualize the phase space of the equations of the specified problems. The results may be useful for specialists in the field of mathematical physics and mathematical modelling.

Keywords

Sobolev-type equation; Hoff's model.

References

1. Bayazitova A.A. [The Showalter - Sidorov Problem for the Hoff Model on a Geometric Graph]. The Bulletin of Irkutsk State University. Series 'Mathematics', 2011, vol. 4, no. 1, pp. 2-8. (in Russian)
2. Sviridyuk G.A., Bayazitova A.A. On Direct and Inverse Problems for the Hoff Equations on Graph. Vestnik Samarskogo Gosudarstvennogo Tekhnicheskogo Universiteta. Seriya Fiz.-Mat. Nauki, 2009, no. 1 (18), pp. 6-17. (in Russian)
3. Sviridyuk G.A., Fedorov V.E. Linear Sobolev Type Equations and Degenerate Semigroups of Operators. Utrecht, Boston, K'oln, Tokyo, VSP, 2003. DOI: 10.1515/9783110915501
4. Sviridyuk G.A., Zagrebina S.A. [The Showalter - Sidorov Problem as a Phenomena of the Sobolev-type Equations]. The Bulletin of Irkutsk State University. Series 'Mathematics', 2010, vol. 3, no. 1, pp. 104-125. (in Russian)
5. Kostin, V.A. Towards the Solomyak - Yosida Theorem on Analytic Semigroups. St. Petersburg Mathematical Journal, 2000, vol. 11, no. 1, pp. 91-106.
6. Sviridyuk G.A., Brychev S.V. Numerical Solution of Systems of Equations of Leont'ev Type. Russian Mathematics (Izvestiya VUZ. Matematika), 2003, vol. 47, no. 8, pp. 44-50.
7. Sviridyuk G.A. Phase Spaces of Sobolev Type Semilinear Equations with a Relatively Strongly Sectorial Operator. St. Petersburg Mathematical Journal, 1995, vol. 6, no. 5, pp. 1109-1126.