Zurab Vashakidze
abstract:
In the present work, the classical nonlinear Kirchhoff string equation is
considered. A three-layer symmetrical semi-discrete scheme with respect to the
temporal variable is applied for finding an approximate solution to the
initial-boundary value problem for this equation, in which the value of the
gradient of a non-linear term is taken at the middle point. This approach is
essential because the inversion of the linear operator is sufficient for
computations of approximate solutions for each temporal step. The variation
method is applied to the spatial variable. Differences of the Legendre
polynomials are used as coordinate functions. This choice of Legendre
polynomials is also important for numerical realization. This way makes it
possible to get a system whose structure does not essentially differ from the
corresponding system of difference equations allowing us to use the methods
developed for solving a system of difference equations. An application of the
suggested variational-difference scheme for the numerical treatment of the
stated nonlinear problem gives us an opportunity to solve the system of linear
equations instead of a nonlinear one. It is proved that a matrix of the system
of Galerkin's linear equations is positively defined and the stability of the
factorization method is established.
The program of the numerical implementation with the corresponding interface is
created based on the suggested algorithm, and numerical computations are carried
out for the model problems.
Mathematics Subject Classification: 65F05, 65F50, 65M06, 65M60, 65N12, 65N22, 65Q30
Key words and phrases: Non-linear Kirchhoff string equation, Cauchy problem, three-layer semi-discrete scheme, Galerkin method, Cholesky decomposition