Search Results (26)

Aiming at the cooperative path-planning problem of multiple autonomous underwater vehicles in underwater three-dimensional terrain and dynamic ocean current environments, a hybrid algorithm based on the Improved Multi-Objective Particle Swarm Optimization (IMOPSO) and Dynamic Window (DWA) is proposed. The traditional particle swarm optimization algorithm is prone to falling into local optimization in high-dimensional and complex marine environments. It is difficult to meet multiple constraint conditions, the particle distribution is uneven, and the adaptability to dynamic environments is poor. In response to these problems, a hybrid initialization method based on Chebyshev chaotic mapping, pre-iterative elimination, and boundary particle injection (CPB) is proposed, and the particle swarm optimization algorithm is improved by combining dynamic parameter adjustment and a hybrid perturbation mechanism. On this basis, the Dynamic Window Method (DWA) is introduced as the local path optimization module to achieve real-time avoidance of dynamic obstacles and rolling path correction, thereby constructing a globally and locally coupled hybrid path-planning framework. Finally, cubic spline interpolation is used to smooth the planned path. Considering factors such as path length, smoothness, deflection Angle, and ocean current kinetic energy loss, the dynamic penalty function is adopted to optimize the multi-AUV cooperative collision avoidance and terrain constraints. The simulation results show that the proposed algorithm can effectively plan the dynamic safe path planning of multiple AUVs. By comparing it with other algorithms, the efficiency and security of the proposed algorithm are verified, meeting the navigation requirements in the current environment. Experiments show that the IMOPSO–DWA hybrid algorithm reduces the path length by 15.5%, the threat penalty by 8.3%, and the total fitness by 3.2% compared with the traditional PSO algorithm. Full article

Massive multiple input–multiple output (M-MIMO) is a promising and pivotal technology in contemporary wireless communication systems that can effectively enhance link reliability and data throughput, especially in uplink scenarios. Even so, the receiving end requires more computational complexity to reconstitute the signal. This problem has emerged in fourth-generation (4G) MIMO system; with the dramatic increase in demand for devices and data in beyond-5G (B5G) systems, this issue will become yet more obvious. To take into account both complexity and signal-revested capability at the receiver, this study uses the matrix iteration method to avoid the staggering amount of operations produced by the inverse matrix. Then, we propose a highly efficient multi-user detector (MUD) named hybrid SOR-based Chebyshev acceleration (CHSOR) for the uplink of M-MIMO orthogonal frequency-division multiplexing (OFDM) and universal filtered multi-carrier (UFMC) waveforms, which can be promoted to B5G developments. The proposed CHSOR scheme includes two stages: the first consists of successive over-relaxation (SOR) and modified successive over-relaxation (MSOR), combining the advantages of low complexity of both and generating a better initial transmission symbol, iteration matrix, and parameters for the next stage; sequentially, the second stage adopts the low-cost iterative Chebyshev acceleration method for performance refinement to obtain a lower bit error rate (BER). Under constrained evaluation settings, Section (Simulation Results and Discussion) presents the results of simulations performed in MATLAB version R2022a. Results show that the proposed detector can achieve a 91.624% improvement in BER performance compared with Chebyshev successive over-relaxation (CSOR). This is very near to the performance of the minimum mean square error (MMSE) detector and is achieved in only a few iterations. In summary, our proposed CHSOR scheme demonstrates fast convergence compared to previous works and as such possesses excellent BER and complexity performance, making it a competitive solution for uplink M-MIMO B5G systems. Full article

This paper focuses on a transmission problem describing the scattering of a TE-wave on a slab having an absolutely conducting wall at the bottom and covered with graphene at the top, accounting for the optical nonlinearity of graphene. This problem is reduced to a nonlinear hypersingular boundary integral equation defined on $\mathbb{R}$. To find an approximate solution to this equation, we develop a novel mathematical approach that combines the collocation method using Chebyshev series to represent a solution (it allows to calculate hypersingular integrals analytically) with an iterative scheme (it allows to account for the nonlinearity of graphene). Using this approach, we numerically simulate the scattering of TE-wave at 3 THz by a ten-micron graphene-coated slab filled with silica. It is shown that by tuning the chemical potential of graphene, one can modulate both the phase and amplitude of the reflected wave. The presented simulation results also demonstrate the effect of the nonlinearity of graphene on the reflected wave. Full article

The development of laminations and mineral composition significantly determine the quality of shale oil reservoirs. The quantitative characterization of lamination development indicators and accurate calculation of mineral composition are key issues in logging evaluation. The Shahejie Formation continental shale oil reservoir in the Nanpu Sag, Bohai Bay Basin, was taken as a case study. Based on electrical imaging logging data, a high-pass filter was designed using the Chebyshev optimal approximation method to extract high-frequency information from the microelectrode curves of the electrical images. A high-resolution quantitative characterization method for millimeter-scale laminated structures of laminae was established, which improved the resolution by 2 to 3 times compared to the static and dynamic image resolutions of electrical imaging. By constructing lamination indices to characterize the sedimentary structural features of reservoirs, we effectively enhanced the fine recognition capability of electrical imaging logging data for sedimentary structures. Utilizing stratigraphic elemental well-log data, we employed an elemental–mineral component conversion model and optimized iterative techniques for accurate mineral composition calculation. We constructed a lithofacies classification scheme based on well-log data using the “rock types + sedimentary structures “approach, combined with research findings on lithofacies identification from well logs, and we identified 12 lithofacies types in the continental shale oil reservoirs of the Nanpu Sag, achieving fine-grained lithofacies logging identification across the entire area. The detailed lithofacies logging classification results were consistent with fine core descriptions. Full article

High-order methods are particularly crucial for achieving highly accurate solutions or satisfying high-order optimality conditions. However, most existing high-order methods require solving complex high-order Taylor polynomial models, which pose significant computational challenges. In this paper, we propose a Chebyshev–Halley method with gradient regularization, which retains the convergence advantages of high-order methods while effectively addressing computational challenges in polynomial model solving. The proposed method incorporates a quadratic regularization term with an adaptive parameter proportional to a certain power of the gradient norm, thereby ensuring a closed-form solution at each iteration. In theory, the method achieves a global convergence rate of $O\left(k^{-3}\right)$ or even $O\left(k^{-5}\right)$, attaining the optimal rate of third-order methods without requiring additional acceleration techniques. Moreover, it maintains local superlinear convergence for strongly convex functions. Numerical experiments demonstrate that the proposed method compares favorably with similar methods in terms of efficiency and applicability. Full article

Chebyshev-type methods have replaced the Chebyshev method in practice for solving nonlinear equations in abstract spaces. These methods are of the same R-order of three. However, they are easier to deal with, since the computationally expensive second derivative of the operator involved does not appear on these methods. However, the invertibility of the first derivative is still required at each step of the iteration. In this article, the inverse is replaced by a finite sum of linear operators. The convergence of the new Hybrid Chebyshev-Type Method (HCTM) is established under relaxed generalized continuity assumptions on the derivative and majorizing sequences. The iterates of the new methods converge to the original ones, but they are easier to find. Moreover, the numerical examples demonstrate that the new iterates converge essentially as fast to the solution. The methodology of this article can be used on other methods with inverses along the same lines due to its generality. Full article

In this paper, we have established a numerical method for a class of time-fractional and Riesz space distributed-order advection–diffusion equation with time-delay. Firstly, we transform the Riesz space distributed-order derivative term of the diffusion equation into multi-term fractional derivatives by using the Gauss quadrature formula. Secondly, we discretize time by using second-order finite differences, discretize space by using second kind Chebyshev polynomials, and convert the multi-term fractional equation to a system of algebraic equations. Finally, we solve the algebraic equations by the iterative method, and prove the stability and convergence. Moreover, relevant examples are shown to verify the validity of our algorithm. Full article

As information systems become more widespread, data security becomes increasingly important. While traditional encryption methods provide effective protection against unauthorized access, they often struggle with multimedia data like images and videos. This necessitates specialized image encryption approaches. With the rise of mobile and Internet of Things (IoT) devices, lightweight image encryption algorithms are crucial for resource-constrained environments. These algorithms have applications in various domains, including medical imaging and surveillance systems. However, the biggest challenge of lightweight algorithms is balancing strong security with limited hardware resources. This work introduces a novel nonlinear matrix permutation approach applicable to both confusion and diffusion phases in lightweight image encryption. The proposed method utilizes three different chaotic maps in harmony, namely a 2D Zaslavsky map, 1D Chebyshev map, and 1D logistic map, to generate number sequences for permutation and diffusion. Evaluation using various metrics confirms the method’s efficiency and its potential as a robust encryption framework. The proposed scheme was tested with 14 color images in the SIPI dataset. This approach achieves high performance by processing each image in just one iteration. The developed scheme offers a significant advantage over its alternatives, with an average NPCR of 99.6122, UACI of 33.4690, and information entropy of 7.9993 for 14 test images, with an average correlation value as low as 0.0006 and a vast key space of $2^{800}$. The evaluation results demonstrated that the proposed approach is a viable and effective alternative for lightweight image encryption. Full article

This paper deals with the convergence and dynamics of Chebyshev’s method for simple and multiple zeros of analytic functions. We establish a local convergence theorem that provides error estimates and exact domains of initial approximations to guarantee the Q-cubic convergence of the method right from the first iteration. Applications to some real-world problems as well as theoretical and numerical comparison with the famous Halley’s method are also provided. Full article

Phase unwrapping of high phase noise and steep phase gradient has always been a challenging problem in interferometric synthetic aperture radar (InSAR), in which case the least squares (LS) phase unwrapping method often suffers from significant unwrapping errors. Therefore, this paper proposes an improved LS phase unwrapping method based on the Chebyshev filter, which solves the problem of incomplete unwrapping and errors under high phase noise and steep phase gradient. Firstly, the steep gradient phase is transformed into multiple flat gradient phases using the Chebyshev filter. Then the flat gradient phases are unwrapped using the LS unwrapping method. Finally, the final unwrapped phase is obtained by iteratively adding the unwrapping results of the flat gradient phases. The simulation results show that the proposed method has the best accuracy and stability compared to LS, PCUA, and RPUA. In the real InSAR phase unwrapping experiment, the RMSE of the proposed method is reduced by 63.91%, 35.38%, and 54.39% compared to LS, PCUA, and RPUA. The phase unwrapping time is reduced by 62.86% and 11.64% compared to PCUA and RPUA. Full article

The reentry trajectory planning problem of hypersonic vehicles is generally a continuous and nonconvex optimization problem, and it constitutes a critical challenge within the field of aerospace engineering. In this paper, an improved sequential convexification algorithm is proposed to solve it and achieve online trajectory planning. In the proposed algorithm, the Chebyshev pseudo-spectral method with high-accuracy approximation performance is first employed to discretize the continuous dynamic equations. Subsequently, based on the multipliers and linearization methods, the original nonconvex trajectory planning problem is transformed into a series of relaxed convex subproblems in the form of an augmented Lagrange function. Then, the interior point method is utilized to iteratively solve the relaxed convex subproblem until the expected convergence precision is achieved. The convex-optimization-based and multipliers methods guarantee the promotion of fast convergence precision, making it suitable for online trajectory planning applications. Finally, numerical simulations are conducted to verify the performance of the proposed algorithm. The simulation results show that the algorithm possesses better convergence performance, and the solution time can reach the level of seconds, which is more than 97% less than nonlinear programming algorithms, such as the sequential quadratic programming algorithm. Full article

In this paper, a family of fifth-order Chebyshev–Halley-type iterative methods with one parameter is presented. The convergence order of the new iterative method is analyzed. By obtaining rational operators associated with iterative methods, the stability of the iterative method is studied by using fractal theory. In addition, some strange fixed points and critical points are obtained. By using the parameter space related to the critical points, some parameters with good stability are obtained. The dynamic plane corresponding to these parameters is plotted, visualizing the stability characteristics. Finally, the fractal diagrams of several iterative methods on different polynomials are compared. Both numerical results and fractal graphs show that the new iterative method has good convergence and stability when $\alpha =\frac{1}{2}$. Full article

In this work, an innovative technique is presented to solve Emden–Fowler-type singular boundary value problems (SBVPs) with derivative dependence. These types of problems have significant applications in applied mathematics and astrophysics. Initially, the differential equation is transformed into a Fredholm integral equation, which is then converted into a system of nonlinear equations using the collocation technique based on Chebyshev polynomials. Subsequently, an iterative numerical approach, such as Newton’s method, is employed on the system of nonlinear equations to obtain an approximate solution. Error analysis is included to assess the accuracy of the obtained solutions and provide insights into the reliability of the numerical results. Furthermore, we graphically compare the residual errors of the current method to the previously established method for various examples. Full article

The aim of this paper is to delve into the dynamic study of the well-known Chebyshev–Halley family of iterative methods for solving nonlinear equations. Our objectives are twofold: On the one hand, we are interested in characterizing the existence of extraneous attracting fixed points when the methods in the family are applied to polynomial equations. On the other hand, we are also interested in studying the free critical points of the methods in the family, as a previous step to determine the existence of attracting cycles. In both cases, we want to identify situations where the methods in the family have bad behavior from the root-finding point of view. Finally, and joining these two studies, we look for polynomials for which there are methods in the family where these two situations happen simultaneously. The rational map obtained by applying a method in the Chebyshev–Halley family to a polynomial has both super-attracting extraneous fixed points and super-attracting cycles different from the roots of the polynomial. Full article

In this paper, we present a new third-order family of iterative methods in order to compute the multiple roots of nonlinear equations when the multiplicity $(m\ge 1)$ is known in advance. There is a plethora of third-order point-to-point methods, available in the literature; but our methods are based on geometric derivation and converge to the required zero even though derivative becomes zero or close to zero in vicinity of the required zero. We use the exponential fitted curve and tangency conditions for the development of our schemes. Well-known Chebyshev, Halley, super-Halley and Chebyshev–Halley are the special members of our schemes for $m=1$. Complex dynamics techniques allows us to see the relation between the element of the family of iterative schemes and the wideness of the basins of attraction of the simple and multiple roots, on quadratic polynomials. Several applied problems are considered in order to demonstrate the performance of our methods and for comparison with the existing ones. Based on the numerical outcomes, we deduce that our methods illustrate better performance over the earlier methods even though in the case of multiple roots of high multiplicity. Full article