Method Of Successive Approximations Research Articles

Optimisation of energy efficiency of ambient backscatter communication and reconfigurable intelligent surfaces in non‐orthogonal multiple access downlink

AbstractThe authors study the energy efficiency (EE) of ambient backscatter communication (AmBC) device‐assisted and reconfigurable intelligent surfaces (RIS)‐assisted non‐orthogonal multiple access (NOMA) downlinks. The authors establish two optimisation problems based on the two collaborative devices (AmBC devices, RIS) with the objective of maximising the EE of the system, taking into account the requirements of power limitation and rate limitation, etc. and also obtain the solutions of two problems by optimising the relevant performance metrics based on the alternating optimisation algorithm. For the backscatter device (BD)‐aided downlink NOMA network, the problem is first decoupled into three subproblems, where the power allocation optimisation subproblem is solved by using the quadratic transformation method and the subgradient algorithm. The maximum EE is obtained by iterating according to the Dinkelbach's algorithm. For the RIS‐aided downlink NOMA network, the power allocation problem is solved by the same method as above and the phase optimisation problem is solved by the successive convex approximation method. Numerical results show that the proposed algorithm can achieve convergence after several iterations, and the EE of systems with BD‐assisted and RIS‐assisted have different levels of sensitivity to different influencing factors.

Modeling and stability analysis of the transmission dynamics of Monkeypox with control intervention

Monkeypox is a deadly disease from the Orthopox family. The paper explores a nonlinear trend in the dynamics of Monkeypox propagation. In this model, we include vaccination, treatment, and level of awareness as the main parameters of the propagation of Monkeypox. We also include the quarantined infected individuals with severe complications in the Monkeypox dynamics. We show the boundedness and positivity of the model and also demonstrate the stability of the model using the Banach fixed point theory and the Picard successive approximation method to ensure that it is meaningful mathematically and epidemiologically. We obtain the unique solution of the model under appropriate conditions. This work finds three equilibrium points: Monkeypox free, Rodents free, and endemic. We show the local stability of monkeypox free equilibria and rodent-free endemic equilibria through reproduction number. Furthermore, Dulac’s function technique demonstrates the global stability of endemic equilibrium. We show the transformation from endemic equilibria to Monkeypox-free equilibria via rodent-free. We also obtain the condition of transcritical bifurcation. We analyze the system’s dynamic behavior to develop efficient infection control strategies. We find that if we increase awareness, vaccination, treatment, and quarantine rates, we effectively control the transmission. The model is connected with a continuous model using an ordinary differential equation. We examine the complex dynamics of Monkeypox infection under diverse system input factors through numerical simulation of the proposed model with variable input parameters in reducing Monkeypox. We present the theory of Monkeypox disease control in humankind as an application in real-world problems. The work can be helpful in the vector-borne disease control system.

On the issue of contactless alternators on movable objects

BACKGROUND: This article discusses automotive inductor generators of domestic and foreign production with different power and design. A comparative analysis of inductor generators according to the main electromagnetic parameters and design designs is carried out. Based on the results of the comparative analysis, conclusions are drawn about the competitiveness of domestic inductor generators and their superiority over foreign analogues. AIMS: research and comparative analysis of tractor inductor generators in order to determine the optimal design, as well as the main characteristics and parameters. METHODS: The calculation of the magnetic circuit and the idling characteristic is based on the method of successive approximations using iterations. A well-known technique using the Blondel diagram is used to calculate the current-speed characteristic. RESULTS: A review and comparative analysis of automotive inductor generators of domestic and foreign production, with various magnetic systems and constructive equipment, is carried out. The advantages and disadvantages of the magnetic circuit of inductor generators of both domestic and foreign production are determined, which are reflected in comparative tables and current-speed characteristics. To carry out the calculation, the optimal design of the inductor generator with the most effective technical and economic indicators was chosen. Conclusions: The calculation of a three-phase single-pole, single-pack inductor generator 11.3701 with a classical toothed zone and a two-half-period rectifier is carried out. The calculation of the magnetic circuit and the idling characteristics was carried out by the method of successive approximations using iterations. The calculation of the current-velocity characteristic was performed using a well-known technique using the Blondel diagram.

Secure Active Intelligent Reflecting Surface Communication against Colluding Eavesdroppers

An active intelligent reflecting surface (IRS)-assisted, secure, multiple-input–single-output communication method is proposed in this paper. In this proposed scheme, a practical and unfavorable propagation environment is considered by assuming that multiple colluding eavesdroppers (Eves) coexist. In this case, we jointly optimize the beamformers of the base station (BS) and the active IRS for the formulated sum secrecy rate (SSR) maximization problem. Because the formulated problem is not convex, we apply the alternating optimization method to optimize the beamformers for maximizing the SSR. Specifically, we use the semi-definite relaxation method to solve the sub-problem of the beamforming vector of the BS, and we use the successive convex approximation method to solve the sub-problem of the power amplification matrix of the active IRS. Based on the solutions obtained using these stated methods, numerical results show that deploying an active IRS is superior compared to the cases of a passive IRS and a non-IRS for improving the physical layer security of wireless communication with multiple colluding Eves under different settings, such as the numbers of users, Eves, reflecting elements, and BS antennas as well as the maximum transmit power budget at the BS.

Local solutions for nonhomogeneous Navier–Stokes equations with large flux

The local existence of solutions to nonhomogeneous Navier–Stokes equations in cylindrical domains with arbitrary large flux is demonstrated. The existence is proved by the method of successive approximations. To show the existence with the lowest possible regularity the special Besov spaces called the Sobolev–Slobodetskii spaces are used. The inflow and outflow are prescribed on the parts of the boundary which are perpendicular to the ‐axis. Since the inflow and outflow are positive the crucial point of this paper is to verify that ‐coordinate of velocity is also positive. Finally, we conclude the local existence such that the velocity belongs to , the gradient of pressure to and the density to , where .

Low-Resolution Optimization for an Unmanned Aerial Vehicle Communication Network under a Passive Reconfigurable Intelligent Surface and Active Reconfigurable Intelligent Surface

This paper investigates the optimization of an unmanned aerial vehicle (UAV) network serving multiple downlink users equipped with single antennas. The network is enhanced by the deployment of either a passive reconfigurable intelligent surface (RIS) or an active RIS. The objective is to jointly design the UAV’s trajectory and the low-bit, quantized, RIS-programmable coefficients to maximize the minimum user rate in a multi-user scenario. To address this optimization challenge, an alternating optimization framework is employed, leveraging the successive convex approximation (SCA) method. Specifically, for the UAV trajectory design, the original non-convex optimization problem is reformulated into an equivalent convex problem through the introduction of slack variables and appropriate approximations. On the other hand, for the RIS-programmable coefficient design, an efficient algorithm is developed using a penalty-based approximation approach. To solve the problems with the proposed optimization, high-performance optimization tools such as CVX are utilized, despite their associated high time complexity. To mitigate this complexity, a low-complexity algorithm is specifically tailored for the optimization of passive RIS-programmable reflecting elements. This algorithm relies solely on closed-form expressions to generate improved feasible points, thereby reducing the computational burden while maintaining reasonable performance. Extensive simulations are created to validate the performance of the proposed algorithms. The results demonstrate that the active RIS-based approach outperforms the passive RIS-based approach. Additionally, for the passive RIS-based algorithms, the low-complexity variant achieves a reduced time complexity with a moderate loss in performance.

Method of successive approximation in mathematical modelling of geotechnical mechanics problems

The purpose of this work was to develop a mathematical model by the results of mathematical modeling of the problem of loading a circular cross-sectional roadway under the action of hydrostatic pressure. Instead of finite element method, it proposed to solve this problem by method of successive approximation, which assumes finding a solution in vicinity of one point in the domain of definition in a multiplicative form where the product functions are one-dimensional functions of parameters. Such representation advantageously differs from classical representation of approximation results by visualization and convenience and by possibility to investigate function sensitivity to the parameter variation. The obtained model provides errors of 5-7%. The results of qualitative comparison of volumetric functions of the tangential stress intensity obtained by the successive approximation method with the representation of numerical results of solving the chosen classical problem presented. Mathematical models of the tangential stresses intensity function in vicinity of a roadway in analytical form are obtained. It is proposed to use this method for representing mathematical models in analytical form and determining function sensitivity to the parameter variation in the problems of geotechnical mechanics. The effectiveness of the method increases when analyzing results of multiparametric problems.

Successive upper approximation methods for generalized fractional programs

The majorization approximation procedure consists in replacing the resolution of a nonlinear optimization problem by solving a sequence of simpler ones, whose objective and constraint functions upper estimate those of the original problem. For generalized fractional programming, i.e., constrained minimization programs whose objective functions are maximums of finite ratios of functions, we propose an adapted scheme that simultaneously upper approximates parametric functions formed by the objective and constraint functions. For directionally convex functions, that is, functions whose directional derivatives are convex with respect to directions, we will establish that every cluster point of the generated sequence satisfies Karush-Kuhn-Tucker type conditions expressed in terms of directional derivatives. The proposed procedure unifies several existing methods and gives rise to new ones. Numerical problems are solved to test the efficiency of our methods, and comparisons with different approaches are given.

Finding and implementing the numerical solution of an optimal control problem for oscillations in a coupled objects system

Finding and implementing the numerical solution of an optimal control problem for oscillations in a coupled objects system

Собственные колебания упругой прямоугольной CFCF-пластинки при одноосном растяжении-сжатии ее плоскости

The influence of a uniaxial distributed load in the plane of a thin rectangular plate on the value of the first natural frequency of its oscillations has been studied. We considered a plate, two opposite faces of which are clamped, and the other two are free (CFCF-plate, C-clamped edge, F-free edge). The load is applied to the clamped edges. The main differential equation for the coordinate component of the deflection function and all the boundary conditions of the problem are fulfilled exactly using two hyperbolic-trigonometric series in two coordinates and an additional function depending on one variable x. The problem was reduced to an infinite system of linear algebraic equations with respect to one sequence of uncertain coefficients, containing as parameters the magnitude of the load and the frequency of oscillations. For a number of load values, the natural frequencies of oscillations were found by enumerating frequency values in combination with the method of successive approximations when solving a reduced system of linear algebraic equations. To ensure acceptable accuracy of calculations, the number of terms in the series (the size of the reduced system), the number of iterations and the number of significant digits in the mantissa when calculating non-trivial coefficients of the system were changed. A square plate was considered as an example. Based on the calculation results, a graph was constructed of the dependence of the first natural frequency of oscillations on the magnitude of tension-compression forces, which is a curve close to a parabola. Under Eulerian compressive load, the oscillations stop. The shapes of natural vibrations changed slightly and were similar to the shape of the curved surface of a plate under the action of a uniform transverse load. The purpose of this study is to create an effective algorithm for calculating the natural frequencies of oscillations of a CFCF-plate when the uniaxial tensile-compression load of its clamped faces changes.

Pilot and data power optimization problems in multiple Aerial Relay Stations cell‐free communication systems

SummaryA cell‐free (CF) technology and unmanned aerial vehicles (UAVs) acting as aerial relay stations (ARSs) are gradually viewed as viable technologies for usage in sixth‐generation (6G) wireless networks owing to their outstanding advantages, such as large coverage, uniform service quality, huge connections, flexible deployment in all geographical areas, and high line‐of‐sight (LoS) probability. As a result, the combination of the CF model and UAVs is suitable for densely populated urban areas, shopping malls, festivals, and locations with complex geography. Furthermore, the integration of the CF model and UAV communication can improve system quality and is considered an entirely new research direction as there is no comprehensive investigation of this combined model. In this study, we investigate the quality of the multi‐ARS CF communication system, where ARSs are equipped with multiple antennas and stochastic distribution within a specific region to simultaneously serve numerous ground users. We establish a closed‐form formulation for the uplink and downlink throughput of individual users by using the matched filtering method and conjugate beamforming technology, respectively. Moreover, we propose a novel method to optimize the pilot and data transmission power coefficients for improving channel estimation and increasing throughput per user, thereby ensuring a high quality of service. The power optimization method is executed via the successive convex approximation (SCA) method, second‐order cone program (SOCP), and linear program. We evaluate system performance according to the cumulative distribution function (CDF) of user throughput based on various parameters, such as the number of users, the number of ARSs, and the length of pilot sequences. The analytical findings also reveal that the system with the proposed power optimization method is always superior to the system without the proposed method in both uplink and downlink.

Evaluating the lifetime distribution parameters and reliability of products using successive approximation method

AbstractThe Weibull distribution is an extensively used statistical model for analyzing the reliability of mechanical and electrical components. Due to the complexity of the nonlinear equations and the scarcity of failure data, the common method may not provide satisfactory results of the reliability. In this case, a new approach for Weibull parameter estimation and reliability analysis, based on successive approximation schemes, is presented. The shape and scale parameters are estimated by maximizing the likelihood functions, and the location parameter is obtained by constructing an approximate correction model between it and the failure data. In order to show the performance of the proposed method, an extensive Monte‐Carlo simulation study is conducted. Simulation results show that the proposed method provides better estimates and efficient confidence intervals for Weibull parameters. In addition, the proposed method works well in presence of small sample sizes. Finally, two real examples are analyzed to illustrate the application of the proposed method.

Joint optimization for computation offloading and 3C resource allocations over wireless-powered and NOMA-enabled multi-access MEC

Multi-access mobile edge computing (MEC) enables mobile users (MUs) to offload tasks to proximal multiple MEC servers for fast task processing. Since MUs generally have stringent delay requirements and limited energy and MEC servers have finite communication, computation, and caching (3C) resources, the joint co-design and optimization over computation offloading and 3C resource allocation for wireless-powered multi-access MEC systems have been highly demanded, while having not been well studied yet. We consider a wireless-powered multi-access MEC, where multiple energy harvesting (EH) based MUs and multiple base stations (BSs) each equipped with an MEC server coexist. Each MU first harvests energy from nearby MEC servers, and then offloads its sub-tasks to multiple MEC servers concurrently based on non-orthogonal multiple access (NOMA), so as to reduce data offloading delay. We first formulate a delay minimization problem, by jointly optimizing MUs’ computation offloading, MEC servers’ computation and caching resources allocation, and system communication resources allocation. Then we propose an alternating direction multiplier method (ADMM) based distributed scheme to decompose the formulated optimization problem into several sub-problems, and use the block coordinate descending (BCD) method and the successive convex approximation (SCA) method to transform all sub-problems each corresponding to one MEC server to convex subproblems. Finally, we validate and evaluate our proposed scheme through numerical analyses, which show that our proposed distributed scheme can greatly reduce the min–max delay of all MUs by allowing each MU’s concurrent data offloading to multiple MEC servers using NOMA and by jointly optimizing computation offloading and 3C resource allocations. Also, the delay imposed by our proposed scheme is much smaller than that imposed by the Interior Point method, which shows the effectiveness of our proposed scheme.

Power Losses Reduction in Three-Phase Unbalanced Distribution Networks Using a Convex Optimization Model in the Complex Domain

This article presents a solution methodology to minimize power losses in three-phase unbalanced distribution networks. This approach involved an efficient complex-domain model that is categorized under mixed-integer convex optimization. The methodology employed consisted of efficient load rotation at each constant power node via a three-phase rotation matrix that allows defining each load connection to minimize the expected power imbalance at the terminals of the substation, as well as the total grid power losses, and improve voltage profile performance at each system phase. The load imbalance, expressed as a percentage, can be defined as a function of the active, reactive, or apparent power. In addition, considering the complex-domain representation of three-phase electrical networks under steady-state conditions, a mixed-integer convex model was formulated to reduce the power imbalances. With the purpose of determining the initial and final power losses of these distribution systems, the successive approximations method was employed to address the three-phase power flow problem. As a result, numerical validations in the IEEE 25-bus system and a 35-node three-phase feeder showed that the final active power losses vary depending on the objective function analyzed. Therefore, for the test feeders studied, it is necessary to evaluate each objective function, with the aim of finding the one that yields the best numerical results. Power losses reductions of about 3.8056 % and 6.8652 % were obtained for both test feeders via the proposed optimization methodology. All numerical validations were performed in the Julia programming environment, using the JuMP optimization tool and the HiGHS solver.

About one approach for solving bounary-value problems of differential equation of pantograph type

Pantograph equations have been studied for a long time. In 1940, K. Mahler introduced functional differential equations of this type into number theory. In 1971, J. Ockendon used a functional differential equation with a transformed argument, , to describe the dynamics of the pantograph of an electric locomotive. Subsequently, Pantograph-type equations were studied in the works of many authors. This article considers a boundary value problem for a differential equation of the pantograph type. To solve the posed boundary value problem, the parameterization method proposed by Professor D. Dzhumabaev is used. To do this, we denote the value of the function at the initial point of the segment under consideration by a parameter and perform a change of variable Then the solvability of the original boundary value problem is reduced to studying the solvability of the resulting Cauchy problem for the original equation and to a linear algebraic equation to determine the introduced parameter. Next, using the method of successive approximations, we find solutions to the Cauchy problem for the Pantograph type equation. The convergence of the resulting sequence and the convergence of its solution to the solution of the Cauchy problem of Pantograph type are proved. By requiring continuity of the free term, we establish its unique solvability. Substituting the resulting solution into a linear algebraic equation to determine the entered parameter, we calculate the values of the entered parameter using the original data. Substituting the resulting expressions into we find the solution to the original problem . And assuming the unique solvability of a linear algebraic equation, we establish the solvability of the boundary value problem for an equation of Pantograph type.

On models of longitudinal vibrations of inhomogeneous rods

The main provisions of the model of longitudinal vibrations of rods are presented, taking into account certain phenomena. To construct a model of longitudinal vibrations of rods from the system, the method of successive approximation was used. Models are based on three components: the basic law or principle of dynamics, a hypothesis or basic statement, and a construction algorithm method.

Existence Theorem for a Fractal Sturm-Liouville Problem

In this article, using a new calculus defined on fractal subsets of the set of real numbers, a Sturm--Lioville type problem is discussed, namely the fractal Sturm--Liouville problem. The existence and uniqueness theorem has been proved for such equations. In this context, the historical development of the subject is discussed in the introduction. In Section 2, the basic concepts of $F^{\alpha}$-calculus defined on fractal subsets of real numbers are given, i. e., $F^{\alpha}$-continuity, $F^{\alpha}$-derivative and fractal integral definitions are given and some theorems to be used in the article are given. In Section 3, the existence and uniqueness of the solutions for the fractal Sturm--Liouville problem are obtained by using the successive approximations method. Thus, the well-known existence and uniqueness problem for Sturm--Liouville equations in ordinary calculus is handled on the fractal calculus axis, and the existing results are generalized.

Fractional periodic boundary value and Cauchy problems with Hilfer–Prabhakar operator

We introduce a successive approximations method to study one fractional periodic boundary value problem of the Hilfer-Prabhakar type. The problem is associated to the corresponding Cauchy problem, whose solution depends on an unknown initial value. To find this value we numerically solve the so-called ’determining system’ of algebraic or transcendental equations. As a result, we determine an approximate solution of the studied problem, written in a closed form. Finally, we evaluate efficiency of our method on a nonlinear numerical example.

Numerical simulation of the dynamic problem of a generalized system with distributed parameters

Introduction. Questions of mathematical modeling of the dynamic problem in the form of a generalized system with one degree of freedom are discussed. Such systems include high-rise tower-type structures. Seismic stability of unique tower-type objects represents a relevant research problem.Aim. To determine the stress-strain state of a studied object under the action of external factors in the form of an earthquake accelerogram.Materials and methods. The methods of structural mechanics, dynamics of structures, and numerical simulation were used. The Lagrange equation was used as a basis for obtaining the motion equation of a generalized system with distributed parameters. The research methodology also included mathematical modeling of the considered systems, their numerical analysis, comparison of the obtained results with literature data.Results. A mathematical model was developed to investigate the stress-strain state of engineering structures under various external, including seismic, effects. The differential equation of the generalized system is solved directly using the method of successive approximations and the Duhamel integral at each time step. The developed algorithm was used to compile a software application in the FORTRAN language followed by obtaining the kinematic and static data of the investigated object. Using the example of a tower-type structure, free vibrations from the action of an instantaneous impulse were investigated. The results from a given earthquake accelerogram are presented.Conclusions. The results obtained on the free vibrations of the object under study agree well with those obtained by numerical simulation. The results obtained by numerical differentiation are effectively identical with those obtained by numerical integration, under the action of various effects. The validity of the results is confirmed by comparing the results obtained by the two methods. The developed software applications can be used for monitoring the state of unique tower-type objects.

Existence and Stability Results for Time-Dependent Impulsive Neutral Stochastic Partial Integrodifferential Equations with Rosenblatt Process and Poisson Jumps

Abstract In this paper, we discuss the existence, uniqueness and stability of mild solutions of time-dependent impulsive neutral stochastic partial integrodifferential equations with the Rosenblatt process and Poisson jumps. The existence of mild solutions for the equations is discussed by means of the semigroup theory and theory of the resolvent operator. Next, under some sufficient conditions, the results are obtained by using the method of successive approximation and Bihari’s inequality. Finally, an example is provided to illustrate our results.