Parallel Solving Method for the Variable Coefficient Nonlinear Equation

In view of the inaccuracy of traditional methods for solving nonlinear equations with variable coefficients in parallel, a new method for solving nonlinear equations with variable coefficients is proposed. Using the generalized symmetry group, the variable coefficient of the equation is taken as a new variable which is the same as the state of the original actual physical field. Some relations between variable coefficient equations and their solutions are found. This paper analyzes the meaning of linear differential equation and nonlinear differential equation, the difference between linear differential equation and nonlinear differential equation and their role in physics, and the necessity of solving nonlinear differential equation. By solving the nonlinear equation with variable coefficients, it can be seen that the general methods to solve the nonlinear equation include scattering inversion, Backlund transform and traveling wave solution. Based on the existing methods for solving nonlinear equations with variable coefficients, the homogeneous balance method is combined with the improved truncated expansion method, truncated expansion method and function reduction method, and the Hopf Cole transform and trial function are combined respectively. The above three methods are used to solve nonlinear equations with variable coefficients. Based on KdV Painleve principle, a parallel method for solving nonlinear equations with variable coefficients is proposed. Finally, it is proved that the method is accurate and effective for the parallel solution of nonlinear equations with variable coefficients.

Non-stationary oscillations of an instantly loaded oscillator under conditions of nonlinear resistance

The motion of an oscillator instantaneously loaded with a constant force under conditions of nonlinear external resistance, the components of which are quadratic viscous resistance, dry and positional friction, are considered. Using the first integral of the equation of motion and the Lambert function, compact formulas for calculating the ranges of oscillations are derived. In order to simplify the search for the values of the Lambert function, asymptotic formulas are given that, with an error of less than one percent, express this special function in terms of elementary functions. It is shown that as a result of the action of the resistance force, including dry friction, the oscillation process has a finite number of cycles and is limited in time, since the oscillator enters the stagnation region, which is located in the vicinity of the static deviation of the oscillator caused by the applied external force. The system dynamic factor is less than two. Examples of calculations that illustrate the possibilities of the stated theory are considered. In addition to analytical research, numerical computer integration of the differential equation of motion was carried out. The complete convergence of the results obtained using the derived formulas and numerical integration is established, which confirms that using analytical solutions it is possible to determine the extreme displacements of the oscillator without numerical integration of the nonlinear differential equation. To simplify the calculations, the literature is also recommended, where tables of the Lambert function are printed, allowing you to find its value for interpolating tabular data. Under conditions of nonlinear external resistance, the components of which are quadratic viscous resistance, dry and positional friction, the process of oscillations of an instantly loaded oscillator has a limited number of cycles. The dependences obtained in this work using the Lambert function make it possible to determine the range of oscillations without numerical integration of the nonlinear differential equation of motion both for an oscillator with quadratic viscous resistance and dry friction, and for an oscillator with quadratic resistance and positional and dry friction. Keywords: nonlinear oscillator, instantaneous loading, quadratic viscous resistance, Lambert function, oscillation amplitude.

Design of a beam of variable cross-section on the elastic base by the quasi-analytical method considering boundary conditions

Purpose. Improvement of the quasi-analytical method of nonlinear differential equation solution and its approbation with reference to beams of variable cross-section on the elastic base with two base factors. Research methods. Boundary conditions in the form of required number of correspondently transformed equations are added to the system of the linear algebraic equations which results from substitution of approximating function with constant factors (for example – power function) in the nonlinear differential equation and fixation of a set of variable values. The total number of the equations have to correspond to quantity of constant factors if the further solution will be carried out by an analytical method. Results. Deflection diagram of a trapezoid concrete beam with rectangular cross-section of variable height on the elastic base with two base factors has been calculated during approbation. Average solution error was equal to 0.06%. Distributions of the bending moments and normal stresses along the beam have been researched. Scientific novelty. The authors did not meet in literature such method of nonlinear differential equation solution. Practical value. The quasi-analytical method with realised consideration of boundary conditions that has been offered can be used for solution of differential equations of any order with various types of nonlinearity, including calculations of beams of variable cross-section on the elastic base.

Interaction of electromagnetic wave and metamaterial with inductive type chiral inclusions

The principle of calculation of a plate from a metamaterial with inductive type chiral inclusions is submitted. It is shown that distribution of an electromagnetic wave to such substance can be investigated with the help of introduction of a chiral parameter and on the basis of a detailed method of calculation. By comparison of two methods the dependence of chiral parameter from frequency of electromagnetic radiation falling on a plate is found. With the help of a detailed method the nonlinear differential equation for potential on the chiral plate is found. It is shown that this equation has solutions as traveling solitary and standing waves but not traveling sine waves. The analysis of the received solutions of the nonlinear equation is carried out. Transition from the multiwave solution to the solution as standing waves is graphically shown at reduction ofdistance between the chiral elements.