Hoff's Model on a Geometric Graph. Simulations

Abstract

This article studies numerically the solutions to the Showalter Sidorov (Cauchy) initial value problem and inverse problems for the generalized Ho model. Basing on the phase space method and a modi ed 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. Ho 's model describes the dynamics of H-beam construction. Ho 's equation, set up on each edge of a graph, describes the buckling of the H-beam. The inverse problem consists in nding the unknown coe cients 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 di erential 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 nd the solution in the form of the Galerkin sum of a few rst eigenfunctions. Software enables us to graph the numerical solution and visualize the phase space of the equations of the speci ed problems. The results may be useful for specialists in the eld of mathematical physics and mathematical modelling.