Method Of Successive Approximations Research Articles

IRS-Assisted Secure UAV Communication System for Multiuser With Hardware Impairments

Perfect hardware conditions are often difficult to meet in practical networks due to the inevitable hardware impairments (HIs). The secure multiuser communication of an intelligent reflecting surface (IRS)-aided unmanned aerial vehicle (UAV) system with HIs at the transceiver and the IRS is studied in this article. Aiming for maximizing the average sum secrecy rate (ASSR) of the UAV system, we jointly optimize the transmit precoding (TPC) matrices, IRS phase shift matrices, and UAV trajectory. To deal with the nonconvexity, we break down this problem into three subproblems and solve them using an alternating optimization (AO) algorithm. Specifically, the semidefinite programming method is used to solve the TPC and phase shift matrices, while the successive convex approximation method is used to obtain the UAV trajectory. In addition, a low complexity mathod on the basis of the Riemannian Manifold Gradient is proposed to solve the IRS phase shift matrix. Simulation results verify the effectiveness of the algorithm. Our robust algorithm can improve ASSR by 22% compared to that without IRS. Furthermore, the nonrobust design without considering HIs will cause a loss of secrecy performance.

Further investigation on fixed point theorems via C-class functions in extended b-metric spaces

The purpose of this research project is to develop new theories, discuss, and extend some recent common fixed point results established when the underlying ambient space is an extended b-metric space and the contraction condition involves a new class of ψ-φ-C-contraction type mappings where ψ is the altering distance function and φ is the ultra-altering distance function. The unique fixed point theorems for such mappings in the setting of ψ-φ-complete metric spaces are proven. We also prove the fixed point theorem in partially ordered metric spaces. Moreover, some examples supporting the main results are given. Our results extend and generalize corresponding results in the literature. The start of the development of the theory of fixed points is tied to the end of the 19th century. The method of successive approximations is used in order to prove the solution's existence and uniqueness at the beginning of differential and integral equations. This branch of nonlinear analysis has been developed through various classes of spaces, such as metric spaces, topological spaces, probabilistic metric spaces, fuzzy metric spaces, and others. In developing the theory of fixed points, achievements are applied in various sciences, such as optimization, economics, and approximation theory. A very important step in the development of fixed point theory was taken by A.H. Ansari through the introduction of a C-class function. Using C-class functions, we generalize some known fixed point results, and Kamran et al. introduced a new intuitive concept of distance measure to extend the notion of b-metric space by further weakening the triangle inequality.

Stokes problem with the Coulomb stick–slip boundary conditions in 3D: Formulations, approximation, algorithms, and experiments

The paper deals with the approximation and numerical realization of the Stokes system in 3D with Coulomb’s slip boundary conditions. The weak velocity–pressure formulation leads to an implicit inequality type problem which is discretized by the P1+bubble/P1 elements. To regularize the discrete non-smooth slip term and to release the discrete impermeability condition the duality approach is used. For numerical realization of the resulting saddle-point problem two strategies are proposed, namely (i) its fixed-point formulation solved by the method of successive approximations (ii) the direct numerical solution of the saddle-point problem. The semi-smooth Newton method is used to solve non-smooth equations appearing in both these approaches.

Iterative sequential approximate solutions method to Hyers–Ulam stability of time-varying delayed fractional-order neural networks

There is a key issue in the field of stability analysis for fractional-order neural networks (FONNs) concerning whether the exact solution of the FONNs exists under the condition that a series of successive approximate solutions of the models can be obtained. If so, can the state differences between the approximate solutions and the exact solution be estimated? Hyers–Ulam stability, as one of the powerful tools to solve this problem, has attracted the attention of scholars, and various related research achievements have been reported. In view of this, this paper aims to investigate the Hyers–Ulam stability of FONNs with time-varying delays. Using the successive approximation method and the modified retarded Henry–Gronwall integral inequality, several conditions guaranteeing the existence and uniqueness of the exact solution of FONNs with time-varying delays are derived. Moreover, the corresponding results on the Hyers–Ulam–Rassias stability of the addressed models are given. Finally, a numerical example is designed to illustrate the efficiency and rationality of the theoretical results.

Joint optimization of altitude and resource allocation for aerial cooperative short packet communication system

Unmanned aerial vehicle (UAV) is considered as an important component of 6G Internet of Things (IoT) because of its deployment flexibility. In this paper, we consider a aerial cooperative short packet communication system model in urban environments. The number of transmitted bits is maximized by jointly optimizing the UAV’s altitude, transmit power and blocklength allocation with constraints on system reliability, latency and fairness. The successive convex approximation (SCA) method is adopted to solve the difficulty that the optimization problem is non-convex. Specifically, we firstly decompose the optimization problem into three subproblems to be solved separately, and then obtain the solution of the original problem by the proposed optimization algorithm. Finally, the performance of the proposed algorithm is verified by simulation results. In addition, we investigate the impact of different communication environments and the number of IoT devices on the system performance.

Computation bits enhancement for IRS-assisted multi-UAV wireless powered mobile edge computing systems

Mobile edge computing (MEC) and wireless energy transfer are two important paradigms to improve the computation bits and provide more cost-effective energy for low-power internet of things (IoT) devices. However, the performance of energy transfer and task offloading is significantly affected by propagation loss, and hence the links between IoT devices and transmitters are frequently blocked. To address this issue, in this paper, an intelligent reflecting surfaces (IRS) system is utilized to enhance the energy harvesting and task offloading performances in a multi-unmanned aerial vehicle (UAV)-assisted MEC system. Also, a power beacon (PB) is deployed in the coverage area to provide suitable and stable energy for UAVs via a laser charging system. The sum computation bits of all IoT devices are maximized by jointly optimizing transmit power, time allocation, the phase shift in the energy transfer stage, the phase shift in the task offloading stage, CPU frequency of IoT devices, and UAVs' trajectory. An alternate search method (ASM) is proposed to tackle the non-convexity of the formulated problem in which the closed-form expressions of transmit power, time allocation, the phase shift in the task offloading stage, and CPU frequency are derived. Also, the semidefinite relaxation (SDR) approach and successive convex approximation (SCA) methods are exploited to obtain the phase shifts in the energy transfer stage and UAVs' trajectory via CVX optimizer, respectively. The numerical evaluations confirm the performance gain of the proposed IRS-assisted multi-UAV MEC system compared to existing benchmark designs.

Semantic Communication-Based Dynamic Resource Allocation in D2D Vehicular Networks

The semantic communication mechanism enables wireless devices in vehicular networks to communicate more effectively with the semantic meaning. However, in high-dynamic vehicular networks, the transmission of semantic information faces challenges in terms of reliability and stability. To address these challenges, a long-term robust resource allocation scheme is proposed under the Device-to-Device (D2D) vehicular (D2D-V) networks, where multiple performance indicators (user satisfaction, queue stability, and communication delay) are considered. Due to the sophisticated probabilistic form with consideration of channel fluctuations, the Bernstein approximation is introduced to acquire the deterministic constraint more efficiently. The robust resource allocation problem is proposed and separated into two independent subproblems by the Lyapunov optimization method, which includes semantic access control in the application layer and power control in the physical layer. After that, the successive convex approximation method and Karush-Kuhn-Tucher conditions are adopted to solve the subproblems, thereby proposing a robust resource allocation algorithm. The simulations reveal the trade-off relationship between user satisfaction, queue stability, and communication delay, which is on the premise of meeting the user SINR requirement. Moreover, the simulations also prove the necessity of considering channel uncertainty in high-speed mobile vehicular communication scenarios.

Robust Downlink Transmission Design in IRS-Assisted Cognitive Satellite and Terrestrial Networks

Cognitive satellite and terrestrial network (CSTN) is considered as a promising technology to provide ubiquitous connectivity for various users within wide-coverage. This paper proposes a robust downlink transmission scheme for multiple intelligent reflecting surfaces (IRSs) assisted CSTN. Here, the satellite network adopts multigroup multicast transmission scheme to serve many earth stations, while the terrestrial network exploits space division multiple access and multi-IRS-enhanced non-orthogonal multiple access technology to communicate with many terrestrial users. By assuming that these two networks share the same frequency band having only the angular information based imperfect channel state information of each user, we formulate an optimization problem to minimize the total transmit power subject to the constraints of quality-of-service requirement for each user, per-antenna transmit power budgets of satellite and BS, and unit-modulus requirement for each reflecting element. To tackle this mathematically intractable problem, we then employ angular discretization together with the successive convex approximation method to obtain the active beamforming (BF) vectors of satellite and BS, the passive BF vector of IRS, and the power allocation coefficients. Moreover, we propose a generalized zero forcing BF and alternative optimization to obtain the suboptimal solutions of the optimization problem with low computational complexity. Finally, simulation results are given to demonstrate the effectiveness and superiority of the proposed two schemes over the benchmarks.

Gain Without Pain: Recycling Reflected Energy From Wireless-Powered RIS-Aided Communications

In this paper, we investigate and analyze energy recycling for a reconfigurable intelligent surface (RIS)-aided wireless-powered communication network. As opposed to the existing works where the energy harvested by Internet of things (IoT) devices only comes from the power station, IoT devices are also allowed to recycle energy from other IoT devices. In particular, we propose group switching-and user switching-based protocols with time-division multiple access to evaluate the impact of energy recycling on the system performance. Two different optimization problems are respectively formulated for maximizing the sum throughput by jointly optimizing the energy beamforming vectors, the transmit power, the transmission time, the receive beamforming vectors, the grouping factors, and the phase-shift matrices, where the constraints of the minimum throughput, the harvested energy, the maximum transmit power, the phase shift, the grouping, and the time allocation are taken into account. In light of the intractability of the above problems, we respectively develop two alternating optimization-based iterative algorithms by combining the successive convex approximation method and the penalty-based method to obtain corresponding sub-optimal solutions. Simulation results verify that the energy recycling-based mechanism can assist in enhancing the performance of IoT devices in terms of energy harvesting and information transmission. Besides, we also verify that the group switching-based algorithm can obtain more sum throughput of IoT devices, and the user switching-based algorithm can harvest more energy.

Two-Phases Flying Process Based Cruise Time Minimization for Collecting Data in a UAV-Enabled Untraveled Wireless IoT Network

This work studies a practical scenario where the unmanned aerial vehicle (UAV) is dispatched from the data center to collect the data from the Internet of Things (IoT) devices and return to the data center. These IoT devices with different amounts of data and are sparsely located in an untraveled area. In this scenario, we consider minimizing the UAV's cruise time by jointly optimizing the clustering strategy of the IoT devices, the association strategy between the UAV and the IoT devices, the UAV's trajectory, and the bandwidth allocation strategy. Traditionally, the minimal cruise time is obtained by visiting all IoT devices one by one and this will lead to huge cost. To address this problem, affinity propagation (AP) algorithm is proposed to cluster all these IoT devices first of all. Then, to avoid visiting all devices one by one and realize parallel computation of data collection time, the entire UAV's flying process is divided into two phases, namely, the inter-cluster phase and the intra-cluster phase. For the inter-cluster phase, the minimal cruise time is formulated to minimize total distance among clusters using the traveling salesman problem with a neighborhood (TSPN)-based algorithm. Subsequently, for the intra-cluster phase, we formulate the collecting time minimization problem in each cluster as a mix-integer non-convex optimization problem. To address these issues, we design a two-loop algorithm involving the bisection search, the altering optimization (AO), and the successive convex approximation (SCA) methods. Simulation results validate the effectiveness of our proposed scheme.

On the Existence of the Exact Solution of Quaternion-Valued Neural Networks Based on a Sequence of Approximate Solutions.

In many practical applications, it is difficult or impossible to obtain the exact solution of the mathematical model due to the limitations of solving methods and the complexity of the neural network itself. A natural problem is given as follows: does the exact solution of quaternion-valued neural networks (QVNNs) exist when successively improved approximate solutions can be obtained? Fortunately, the Hyers-Ulam stability happens to be one of the important means to deal with this problem. In this article, the issue of Hyers-Ulam stability of QVNNs with time-varying delays is addressed. First, inspired by the Hyers-Ulam stability of general functional equations, the concept of the Hyers-Ulam stability of QVNNs is proposed along with the QVNNs model. Then, by utilizing the successive approximation method, both delay-dependent and delay-independent Hyers-Ulam stability criteria are obtained to ensure the Hyers-Ulam stability of the QVNNs considered. Finally, a simulation example is given to verify the effectiveness of the derived results.

Analysis of conditions for selecting the number of teeth of planetary gearboxes

The issues of selecting the number of teeth for 2k-h type A mechanisms based on the study of the known conditions for their assembly used in the development of planetary gearboxes of various shipboard both deck and main power ship installations are highlighted in the paper. Considering the design practice, when selecting the number of teeth, it is important to maintain a given gear ratio, ensure the alignment of the mechanism gears, withstand the condition of the neighborhood of satellites, ensure the assembly condition and the absence of interference of internal gearing. These conditions are expressed in terms of the number of teeth of gears and impose restrictions on the parameters of 2k-h mechanisms. Checking the interference condition of internal gearing at the stage of selecting the number of teeth in existing methods is usually not done, but it is performed at the stage of checking the quality of gearing according to geometric conditions. With this approach, the selection of tooth numbers is considered as a problem solved by the method of successive approximations: selection of tooth numbers — calculation of geometric dimensions — checking the quality of gearing — selection of a new combination of tooth numbers, etc. The condition of interference of internal gearing obtained in the work is represented by a closed formula, which eliminates the use of the iteration method, which requires a large amount of computational work and allows it to be tested at an early stage of product development. The neighborhood condition is expressed by an equation in which the design parameter is associated with the maximum possible number of satellites and a guaranteed gap between their initial circles. It makes it possible to theoretically explain well-known constructions that do not agree with the theoretical views existing at that time. The alignment condition allows for both an even and an odd difference in the number of teeth of the central wheels. Thus, the results obtained make it possible to solve the problems of creating a mechanism with the maximum possible number of satellites for reducing the size of a planetary mechanism as a unit completing a marine gearbox of any complexity, to expand the set of possible combinations of the teeth numbers that provide a given design and require less computational work.

Solving the Power Flow Problem in Bipolar DC Asymmetric Distribution Networks Using Broyden's Method.

This research addresses the power flow analysis in bipolar asymmetric direct current (DC) networks by applying Broyden's numerical method. This general successive approximations method allows for a simple Newton-based recursive formula to reach the roots of multiple nonlinear equations. The main advantage of Broyden's approach is its simple but efficient structure which can be applied to real complex nonlinear equations.The power flow problem in bipolar DC networks is still challenging, as multiple operating options must be considered, e.g., the possibility of having a solidly grounded or floating neutral wire. The main goal of this research is to contribute with a generalization of Broyden's method for the power flow solution in bipolar DC networks, with the main advantage that, under well-defined conditions, this is a numerical method equivalent to the matricial backward/forward power flow, which is equivalent to the successive approximations power flow method. Numerical results in the 21-, 33-, and 85-bus grids while considering two connections for the neutral wire (i.e., solidly grounded at any node or floating) show the effectiveness of Broyden's method in the power flow solution for bipolar asymmetric DC networks. All numerical simulations were carried out in the MATLAB programming environment.

2D Jackiw–Rebbi and trivial localized states in square interfaced binary waveguide lattices

We investigate optical analogs of two-dimensional (2D) Jackiw–Rebbi (JR) states and 2D trivial localized states in square interfaced binary waveguide lattices in the linear and nonlinear regimes of Kerr type. The 2D JR states can be formed in the square interfaced binary lattice during propagation of optical beams and turn out to be quite robust, especially in the linear and defocusing nonlinearity regime where established 2D JR state profiles are well conserved during propagation. The established profiles of nonlinear 2D JR states formed during propagation agree well with exact solutions obtained by using the Newton–Raphson method. The 2D nonlinear trivial localized states found by the successive approximation method are much broader than 2D JR states with the same peak amplitudes. The 2D JR states can also be well trapped within the central interface of the lattices even in the presence of an initial tilt of the input beam which can easily displace 2D trivial localized states in square interfaced binary waveguide lattices, 2D Dirac solitons in perfectly periodic binary waveguide lattices and other solitonic structures in the continuous media.

Generalized Function Projective Synchronization of Two Different Chaotic Systems with Uncertain Parameters

This study proposes a new approach to realize generalized function projective synchronization (GFPS) between two different chaotic systems with uncertain parameters. The GFPS condition is derived by converting the differential equations describing the synchronization error systems into a series of Volterra integral equations on the basis of the Laplace transform method and convolution theorem, which are solved by applying the successive approximation method in the theory of integral equations. Compared with the results obtained by constructing Lyapunov functions or calculating the conditional Lyapunov exponents, the uncertain parameters and the scaling function factors considered in this paper have fewer restrictions, and the parameter update laws designed for the estimation of the uncertain parameters are simpler and easier to realize physically.

Plane Strain State of Composite Material Reinforced by Two Rows of Crystalline Fibres in Dynamic Elastic-Plastic Formulation

A generalized approach was developed for solving contact problems in a dynamic elastic-plastic formulation. For the design of composite and reinforced materials, a technique for solving dynamic contact problems in more adequate an elastic-plastic mathematical formulation is used. To consider the physical nonlinearity of the deformation process, the method of successive approximations is used, which makes it possible to reduce the nonlinear problem to a solution of the sequences of linear problems. In contrast to the traditional plane strain, when one normal stress is equal to a certain constant value, for a more accurate description of the deformation of the sample, taking into ac-count the possible increase in longitudinal elongation, we present this normal stress as a function that depends on the parameters that describe the bending of a prismatic body that is in a plain strain state. The problem of a plane strain state of a beam made from the composite reinforced one-layer material is being solved. The reinforced or armed composite material consists of two materials: the main material of glass and two rows of the reinforcing crystalline fourteen fibres of basalt. Glass has high strength and is not affected by the processes of aging of the material, corrosion, and creep. In addition, this material is cheap and widely available. The reinforced composite beam is rigidly linked to an absolutely solid base and on which an absolutely solid impactor acts from above in the centre on a different size of the area of initial contact.

About the existence and uniqueness of solutions for some second-order nonlinear BVPs

The significance of our work is to solve some second-order nonlinear boundary value problems. To do this, we take into account the equivalence of the problems considered with certain integral equations, we will obtain a fixed-point-type result for these integral equations. This result provides us the existence and uniqueness of solutions for the second-order nonlinear boundary value problems considered. As a novelty, we will use for this fixed-point-type result a family of third order iterative processes to approximate the solution, instead of the usually considered method of Successive Approximations of linear convergence.

Duality Method for Solving 3D Contact Problems with Friction

The article studies a 3D contact problem with Coulomb friction for an elastic body resting on a rigid support. The solution of the quasi-variational formulation of the problem is defined as a fixed point of some mapping that associates the given force of the normal reaction of the support with the value of the normal stress in the contact zone. The fixed point is sought by the method of successive approximations, the convergence of which is proved using modified Lagrange functionals. The results of the numerical solution using finite element modeling and the proximal gradient method are presented.

Convergence Improvement Technique in the Method of Successive Approximations in Boundary-Value Problems of Thermoviscoplasticity

Convergence Improvement Technique in the Method of Successive Approximations in Boundary-Value Problems of Thermoviscoplasticity

Stability Analysis of First Order Integro-Differential Equations With the Successive Approximation Method

Stability Analysis of First Order Integro-Differential Equations With the Successive Approximation Method