VARIATIONAL-DIFFERENCE SCHEME FOR KIRCHHOFF TWO--DIMENSIONAL NONLINEAR DYNAMICAL EQUATION

Author: Zurab Vashakidze
Keywords: Kirchhoff nonlinear equation, Finite difference method, Variational method
Annotation:

In the present work, we consider the classical nonlinear Kirchhoff string equation and study its two-dimensional generalization. Our goal is to find an approximate solution to the initial-boundary value problem for this equation. To do so, we apply a three-layer symmetrical semi-discrete scheme with respect to time variable, in which the gradient value of a nonlinear term is taken at the middle point. This detail is essential, because the inversion of the linear operator is sufficient for computation of approximate solutions for each time step. The variation method is applied for spatial variables. Sine function and differences of the Legendre polynomials were used as coordinate functions. This choice of Legendre polynomials is also important for numerical realization. This way we obtain a system whose structure does not essentially differ from the corresponding difference equations system, allowing us to use the methods developed for solving difference equations system. Linear variation problem for one-dimensional Kirchhoff equation (for spatial dimension) is considered, error of approximate solution is estimated, and convergence order considering the number of spatial functions is found. General operator equation is considered for symmetric operators. We prove that the matrix corresponding to its variation system is positively defined, when coordinate functions satisfy some natural conditions that will be specified in this work. Numerical realization program with corresponding interface was created based on the offered algorithm, and numerical computations were carried out for model problems both for one-dimensional and two-dimensional cases. Based on the obtained theoretical results and numerical computations, the practical conclusions about the stability and convergence of the offered method were drawn.

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ვარიაციულ-სხვაობიანი სქემა კირხოფის ორგანზომილებიანი არაწრფივი დინამიური განტოლებისათვის [ka]