Features of mathematical modelling in the analysis of console-type structures

The article deals with the mathematical model of console-type structural elements. The dynamic load is presented as quasi-static one. The differential equation of bending of an object is nonlinear and has movable singular points in which the solution has discontinuity. From a physical point of view, the object will break (collapse) in this place. The application of the majorant method to the solution of the problem allows, in contrast to the classical approach, establishing the boundaries of the solution area and to construct an analytical approximate solution to the problem with a given accuracy. As a result, it’s possible to calculate the displacement at any point of the cantilever structure and estimate the stress-strain state of the object.

ON THE NEWTON–KANTOROVICH THEOREM

The Newton–Kantorovich theorem enjoys a special status, as it is both a fundamental result in Numerical Analysis, e.g., for providing an iterative method for computing the zeros of polynomials or of systems of nonlinear equations, and a fundamental result in Nonlinear Functional Analysis, e.g., for establishing that a nonlinear equation in an infinite-dimensional function space has a solution. Yet its detailed proof in full generality is not easy to locate in the literature. The purpose of this article, which is partly expository in nature, is to carefully revisit this theorem, by means of a two-tier approach. First, we give a detailed, and essentially self-contained, account of the classical proof of this theorem, which essentially relies on careful estimates based on the integral form of the mean value theorem for functions of class [Formula: see text] with values in a Banach space, and on the so-called majorant method. Our treatment also includes a careful discussion of the often overlooked uniqueness issue. An example of a nonlinear two-point boundary value problem is also given that illustrates the power of this theorem for establishing an existence theorem when other methods of nonlinear functional analysis cannot be used. Second, we give a new version of this theorem, the assumptions of which involve only one constant instead of three constants in its classical version and the proof of which is substantially simpler as it altogether avoids the majorant method. For these reasons, this new version, which captures all the basic features of the classical version could be considered as a good alternative to the classical Newton–Kantorovich theorem.

Nonlinear Interaction of Submerged Cylinder With Free Surface

The fully nonlinear problem on the unsteady water waves generated by submerged moving cylinder is considered. Using the analytic majorant method we prove local in time unique solvability of this problem. For the case when the dimensionless cylinder radius is small, the solution estimate obtained predicts rigorously dipole-like structure for the lowest order far field flow. The strength of dipole concentrated at the cylinder axis depends on the instantaneous wave form and fluid velocity at the free surface. A special case of the lifting accelerated cylinder starting from the rest is studied analytically in more detail.

Nonlinear Interaction of Submerged Cylinder With Free Surface

The fully nonlinear problem on the unsteady water waves generated by submerged moving cylinder is considered. Using the analytic majorant method we prove local in time unique solvability of this problem. For the case when the dimensionless cylinder radius is small, the solution estimate obtained predicts rigorously dipole-like structure for the lowest order far field flow. The strength of dipole concentrated at the cylinder axis depends on the instantaneous wave form and fluid velocity at the free surface. Special case of the lifting accelerated cylinder starting from the rest is studied analytically in more detail.