APPLICATION OF FUNCTIONAL RELATIONS TO MODELING BY MEANS OF DIFFERENTIAL EQUATIONS

Modeling by means of differential equations is considered in the paper. Their solutions are constructed on the base of functional relations connecting values of a solution of the equation in different points (infinite or finite set of values). For examples, even, odd and periodical solutions, Vallée-Poussin’s assertion, Lagrange interpolation polynomial, Hermite interpolation polynomial, spline-functions for ordinary differential equations, Asgeirsson’s identity and its generalizations for partial differential equations of hyperbolic type, “mean value” for partial differential equations of elliptic type are considered. Also, if an equation is close to one of considered types then an assertion is to be fulfilled approximately. Some estimations are found for such examples. An application of such relations to investigate some problems of interpolation and extrapolation is demonstrated.

A High-Order Weakly L-Stable Time Integration Scheme with an Application to Burgers’ Equation

In this paper, we propose a 7th order weakly L-stable time integration scheme. In the process of derivation of the scheme, we use explicit backward Taylor’s polynomial approximation of sixth-order and Hermite interpolation polynomial approximation of fifth order. We apply this formula in the vector form in order to solve Burger’s equation, which is a simplified form of Navier-Stokes equation. The literature survey reveals that several methods fail to capture the solutions in the presence of inconsistency and for small values of viscosity, e.g., 10−3, whereas the present scheme produces highly accurate results. To check the effectiveness of the scheme, we examine it over six test problems and generate several tables and figures. All of the calculations are executed with the help of Mathematica 11.3. The stability and convergence of the scheme are also discussed.

Full Hermite interpolation of the reliability of a hammock network

Although the hammock networks were introduced more than sixty years ago, there is no general formula of the associated reliability polynomial. Using the full Hermite interpolation polynomial, we propose an approximation for the reliability polynomial of a hammock network of arbitrary size. In the second part of the paper, we provide combinatorial formulas for the first two non-zero coefficients of the reliability polynomial.

ON THE REPRESENTATION AERODROME SURFACEROUTING NETWORK EDGE BY SMOOTH CURVES

The actual problem of ways to represent aerodrome surface route network is considered. Based on the analysis of various options, an approach is proposed for representing route network sections as smooth curves, which are described by parametric vector functions. Each of the vector function components is represented by a two-point Hermite interpolation polynomial, which uses derivatives up to some order inclusive. Within this approach, the optimization problem related to the coefficients selection of these polynomials based on minimizing the distance between the broken line and smooth curve is solved. The problem is reduced to solving a system of linear equations by the derivatives values at the ends of the route network section. The corresponding finite formulas for approximating broken lines by smooth curves are proposed. Based on the formulas obtained, algorithm and program for approximating route network sections using information about taxi lines, which are stored in aerodrome mapping database (AMDB), were developed. The program also allows you to calculate statistical indicators, what allow to get a quantitative approximation estimate. Numerical experiments based on the Sheremetyevo aerodrome dataset have shown the promise of this approach to presenting aerodrome surface route network, which can significantly (2 – 4 times) reduce the amount of data and increase the realism of the aerodrome model.

Minimum time path planning of robotic manipulator in drilling/spot welding tasks

In this paper, a minimum time path planning strategy is proposed for multi points manufacturing problems in drilling/spot welding tasks. By optimizing the travelling schedule of the set points and the detailed transfer path between points, the minimum time manufacturing task is realized under fully utilizing the dynamic performance of robotic manipulator. According to the start-stop movement in drilling/spot welding task, the path planning problem can be converted into a traveling salesman problem (TSP) and a series of point to point minimum time transfer path planning problems. Cubic Hermite interpolation polynomial is used to parameterize the transfer path and then the path parameters are optimized to obtain minimum point to point transfer time. A new TSP with minimum time index is constructed by using point-point transfer time as the TSP parameter. The classical genetic algorithm (GA) is applied to obtain the optimal travelling schedule. Several minimum time drilling tasks of a 3-DOF robotic manipulator are used as examples to demonstrate the effectiveness of the proposed approach. Highlights In this paper, an optimization strategy is proposed for solving minimum time manufacturing path planning in multi points manufacturing tasks. According to the start-stop movement in drilling/spot welding task, the path planning problem is converted into a traveling salesman problem (TSP) and a series of point to point minimum time transfer path planning problems. Cubic Hermite interpolation polynomial is used to parameterize the transfer path and then the path parameters are optimized to obtain minimum point to point transfer time. A new TSP with minimum time index is constructed and then solved by using a classical genetic algorithm (GA). Numerical test is executed to demonstrate the effectiveness of the proposed approach.

Calculating Methods for the Irregular Hermite Interpolation Polynomial

Based on the theory of the regular Hermite interpolation polynomial, several calculating methods including basic function, multiple difference quotients, etc., have been proposed to solve the complex irregular Hermite interpolation polynomial.