Algorithmic solutions to the problem of assessing the information impact on the electorate during election campaigns

In the article, to solve the problem of assessing the information impact on the electorate during election campaigns, algorithmic solutions, including a mathematical model, a numerical scheme and algorithmic implementations, are formed. This assessment is reduced to determining the instantaneous values of the number of voters who prefer a candidate (party), taking into account: the positive or negative stochastic nature of the impact of mass media; interpersonal interaction; two-step assimilation of information; the presence of a variety of mass media, social groups and a list of candidates. The mathematical model is based on a generalized model of information confrontation in a structured society and, with the introduction of stochastic components in the intensity of agitation, it is reduced to solving the FokkerPlanckKolmogorov equation. For its study in the formulation of the Galerkin method, a numerical scheme is proposed and the order of its convergence is determined. In relation to the basic procedures of the numerical scheme, the features of the algorithmic implementation are clarified.

Investigation of dynamics of elastic element of vibration device

ва подхода к решению аэрогидродинамической части задачи, основанные на методах теории функций комплексного переменного и методе Фурье. В результате применения каждого подхода решение исходной задачи сведено к исследованию дифференциального уравнения с частными производными для деформации элемента, позволяющего изучать его динамику. На основе метода Галеркина произведены численные эксперименты для конкретных примеров механической системы, подтверждающие идентичность решений, найденных для каждого дифференциального уравнения с частными производными. The dynamics of an elastic element of a vibration device, simulated by a channel, inside which a stream of a liquid flows, is investigated. Two approaches to solving the aerohydrodynamic part of the problem, based on the methods of the theory of functions of a complex variable and the Fourier method, are given. As a result of applying each approach, the solution to the original problem is reduced to the study of a partial differential equation for the deformation of an element, which makes it possible to study its dynamics. Based on the Galerkin method, the numerical experiments were carried out for specific examples of mechanical system, confirming the identity of the solutions found for each partial differential equation.

Computing Natural Frequencies and Mode Shapes of an Axially Moving Non-Uniform Beam

The partial differential equation of motion of an axially moving beam with spatially varying geometric, mass and material properties has been derived. Using the theory of linear time-varying systems and numerical optimization, a general algorithm has been developed to compute complex eigenvalues/natural frequencies, mode shapes, and the critical speed for stability. Numerical results from the new method are presented for beams with spatially varying rectangular cross sections with sinusoidal variation in thickness and sine-squared variation in width. They are also compared to those from the Galerkin method. It has been found that critical speed of the beam can be significantly reduced by non-uniformity in a beam's cross section.

An Alternative Algorithm for Simulating Flash Flood

Most of the approaches in numerical modeling techniques are based on the Eulerian coordinate system. This approach faces difficulty in simulating flash flood front propagation. This paper shows an alternative method that implements a numerical modeling technique based on the Lagrangian coordinate system to simulate the water of debris flow. As for the interaction with the riverbed, the simulation uses an Eulerian coordinate system. The method uses the conservative and momentum equations of water and sediment mixture in the Lagrangian form. Source terms represent deposition and erosion. The riverbed in the Eulerian coordinate system interacts with the flow of the mixture. At every step, the algorithm evaluates the relative position of moving nodes of the flow part to the fixed nodes of the riverbed. Computation of advancing velocity and depth uses the riverbed elevation, slope data, and the bed elevation change computation uses the erosion or deposition data of the flow on the moving nodes. Spatial discretization is implementing the Galerkin method. Furthermore, temporal discretization is implementing the forward difference scheme. Test runs show that the algorithm can simulate downward, upward, and reflected backward 1-D flow cases. Two-D model tests and comparisons with SIMLAR software show that the algorithm works in simulating debris flow.

Galerkin Method for Nonlinear Higher-Order Boundary Value Problems Based on Chebyshev Polynomials

In the current paper, we develop an algorithm to approximate the analytic solution for the nonlinear boundary value problems in higher-order based on the Galerkin method. Chebyshev polynomials are introduced as bases of the solution. Meanwhile, some theorems are deducted to simplify the nonlinear algebraic set resulted from applying the Galerkin method, while Newton’s method is used to solve the resulting nonlinear system. Numerous examples are presented to prove the usefulness and effectiveness of this algorithm in comparison with some other methods.

Legendre-Galerkin method for the nonlinear Volterra-Fredholm integro-differential equations

The study of solving nonlinear integro-differential equations in Volterra-Fredholm type presents in this paper. The proposed method tends to use Legendre polynomials as a basis in the Galekin method to obtain the numerical solution. We use the Newton method to get the numerical solution of the nonlinear equations resulted from applying the Galerkin method. The comparison of the present study with the existing results in the literature shows an excellent agreement. Numerical examples explain the convergence, applicability, and efficiency of algorithm.

Stability Analysis of a Diffusive Three-Species Ecological System with Time Delays

In this study, the dynamics of a diffusive Lotka–Volterra three-species system with delays were explored. By employing the Galerkin Method, which generates semi-analytical solutions, a partial differential equation system was approximated through mathematical modeling with delay differential equations. Steady-state curves and Hopf bifurcation maps were created and discussed in detail. The effects of the growth rate of prey and the mortality rate of the predator and top predator on the system’s stability were demonstrated. Increase in the growth rate of prey destabilised the system, whilst increase in the mortality rate of predator and top predator stabilised it. The increase in the growth rate of prey likely allowed the occurrence of chaotic solutions in the system. Additionally, the effects of hunting and maturation delays of the species were examined. Small delay responses stabilised the system, whilst great delays destabilised it. Moreover, the effects of the diffusion coefficients of the species were investigated. Alteration of the diffusion coefficients rendered the system permanent or extinct.

Technique to Solve Linear Fractional Differential Equations Using B-Polynomials Bases

A multidimensional, modified, fractional-order B-polys technique was implemented for finding solutions of linear fractional-order partial differential equations. To calculate the results of the linear Fractional Partial Differential Equations (FPDE), the sum of the product of fractional B-polys and the coefficients was employed. Moreover, minimization of error in the coefficients was found by employing the Galerkin method. Before the Galerkin method was applied, the linear FPDE was transformed into an operational matrix equation that was inverted to provide the values of the unknown coefficients in the approximate solution. A valid multidimensional solution was determined when an appropriate number of basis sets and fractional-order of B-polys were chosen. In addition, initial conditions were applied to the operational matrix to seek proper solutions in multidimensions. The technique was applied to four examples of linear FPDEs and the agreements between exact and approximate solutions were found to be excellent. The current technique can be expanded to find multidimensional fractional partial differential equations in other areas, such as physics and engineering fields.

Controlling Resonator Nonlinearities and Modes through Geometry Optimization

Controlling the nonlinearities of MEMS resonators is critical for their successful implementation in a wide range of sensing, signal conditioning, and filtering applications. Here, we utilize a passive technique based on geometry optimization to control the nonlinearities and the dynamical response of MEMS resonators. Also, we explored active technique i.e., tuning the axial stress of the resonator. To achieve this, we propose a new hybrid shape combining a straight and initially curved microbeam. The Galerkin method is employed to solve the beam equation and study the effect of the different design parameters on the ratios of the frequencies and the nonlinearities of the structure. We show by adequately selecting the parameters of the structure; we can realize systems with strong quadratic or cubic effective nonlinearities. Also, we investigate the resonator shape effect on symmetry breaking and study different linear coupling phenomena: crossing, veering, and mode hybridization. We demonstrate the possibility of tuning the frequencies of the different modes of vibrations to achieve commensurate ratios necessary for activating internal resonance. The proposed method is simple in principle, easy to fabricate, and offers a wide range of controllability on the sensor nonlinearities and response.

An Analytical Buckling Load Formula for a Cylindrical Shell Subjected to Local Axial Compression

This paper proposes an analytical buckling load formula for a cylindrical shell subjected to local axial compression for the first time, which is achieved by a carefully constructed load description and perturbation procedure. Local axial load is described by introducing an arctangent function firstly. Then, the analytical solutions of local buckling load coefficients and buckling modes for a locally compressed shell are derived after solving governing differential equations by the perturbation method. For validation, using the presented analytical buckling modes, the Galerkin method is applied to obtain numerical result, which is an infinite order determinant about local buckling load coefficient. Comparative calculation results show that local buckling load coefficients by the analytical formula are in perfect agreement with numerical ones by the Galerkin method and known results in literature. Therefore, the validity and accuracy of the presented formula are verified. Engineering application of the analytical formula is also discussed to evaluate local buckling loads of thin-walled cylindrical shell structures such as silos, pressure vessels, large storage tanks and so on.