This article presents a method for modeling extended small celestial (ESC) objects, which are mainly cometary systems. Special attention was given to the possibility of analyzing their structure and physical properties in line with the modern theories of the Solar System’s formation suggesting a rather complicated evolutionary dynamics. Modeling and investigating the structure of different extended celestial objects advance our understanding of the general evolutionary processes that have taken place in the Solar System because all its objects are evolutionarily related. The isoline modeling (IM) method was tested on the real comet data and proved effective in assessing the activity of the processes that occur as ESC objects move in space. The IM method is particularly useful for studying long-period comets that, in many cases, cross the perihelion only once within a foreseeable period of human existence.

This article presents an implicit two-layer finite element scheme for solving the Kirchhoff equation, a nonlinear nonlocal equation of hyperbolic type with the Dirichlet integral. The discrete scheme was designed considering the solution of the problem and its derivative for the time variable. It ensures total energy conservation at a discrete level. The use of the Newton method was proven to be effective for solving the scheme on the time layer despite the nonlocality of the equation. The test problems with smooth solutions showed that the scheme can define both the solution of the problem and its time derivative with an error of O(h²+τ²) in the root-mean-square norm, where τ and h are the grid steps in time and space, respectively.

Necessary and sufficient conditions for the existence of a valid Dirichlet solution were obtained. A method was developed to find Dirichlet solutions of the functional differential equation with non-delayed linear argument deviation.

A cutting method was proposed for solving the convex programming problem. The method assumes that the constraint region of the problem is embedded into some polyhedral sets for constructing iteration points. It involves the construction of a sequence of approximations that belongs to the admissible set and is relaxed, as well as implies that the ε-solution of the initial problem is fixed after a finite number of steps. The method also allows to obtain mixed convergent algorithms by using, if desired, any known or new relaxation algorithms for constructing the main iteration points.

For the problem of eigenvibrations of the shallow shell with an attached oscillator, a new symmetric variational statement in the Hilbert space was proposed. It was established that there exist a sequence of positive eigenvalues of finite multiplicity with a limit point at infinity and the corresponding complete orthonormal system of eigenvectors. The problem was approximated by the mesh scheme of the finite element method with Hermite finite elements. Theoretical error estimates for the approximate solutions were proved. The theoretical findings were verified by the results of numerical experiments.

In this study, the development, design, and software implementation of the methods for solving the nonlinear diffraction problem were performed. The influence of nonlinear medium defined by the Kerr law on the propagation of a wave passing through an object was examined. The differential and integral formulations of the problem and the nonlinear integral equation were considered. The problem was solved for different bodies with the use of various computational grids. Convergence graphs of the iterative processes were generated. The obtained graphical results were presented. The explicit and implicit methods for solving the integral equation were compared.