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In this paper, we develop a mapped Jacobi spectral Galerkin method for solving the multi-term Fredholm integral equations (MFIEs) with two-sided weakly singularities. We introduce a new family of mapped Jacobi functions (MJFs) and establish the corresponding spectral approximation results on these MJFs in weighted Sobolev spaces involving the mapped Jacobi weight function. These MJFs serve as the basis functions in our algorithm design and are tailored to the two-sided end-points singularities of the solution by using suitable mapping. Moreover, we derive the error estimates of the proposed method for MFIEs. Finally, the numerical examples are provided to demonstrate the accuracy and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this work, the Caputo-type Hadamard fractional derivative is utilized to introduce a coupled system of time fractional Klein–Gordon-Schrödinger equations. The classical and shifted Jacobi polynomials are simultaneously applied to make a numerical technique for this system. To this aim, two operational matrices for the Caputo-type Hadamard fractional derivatives of the shifted Jacobi polynomials are gained. In the developed strategy, by considering a hybrid approximation of the problem's solution via the expressed polynomials and applying the obtained matrices, solving the original fractional system turns into solving an associated algebraic system of equations. Two test problems are examined to investigate the high accuracy of the developed procedure. [ABSTRACT FROM AUTHOR]

Frequency-hopping (FH) communication adversarial research is a key area in modern electronic countermeasures. To address the challenge posed by interfering parties that use deep neural networks (DNNs) to classify and identify multiple intercepted FH signals—enabling targeted interference and degrading communication performance—this paper presents a batch feature point targetless adversarial sample generation method based on the Jacobi saliency map (BPNT-JSMA). This method builds on the traditional JSMA to generate feature saliency maps, selects the top 8% of salient feature points in batches for perturbation, and increases the perturbation limit to restrict the extreme values of single-point perturbations. Experimental results in a white-box environment show that, compared with the traditional JSMA method, BPNT-JSMA not only maintains a high attack success rate but also enhances attack efficiency and improves the stealthiness of the adversarial samples. [ABSTRACT FROM AUTHOR]

We define and study refined Gopakumar-Vafa invariants of contractible curves in complex algebraic threefolds, alongside the cohomological Donaldson-Thomas theory of finite-dimensional Jacobi algebras. These Gopakumar-Vafa invariants can be constructed one of two ways: as cohomological BPS invariants of contraction algebras controlling the deformation theory of these curves, as defined by Donovan and Wemyss, or by feeding the moduli spaces that Sheldon Katz used to define genus zero Gopakumar-Vafa invariants into the machinery developed by Joyce et al. The conjecture that the two definitions give isomorphic results is a special case of a kind of categorified version of the strong rationality conjecture due to Pandharipande and Thomas, that we discuss and propose a means of proving. We prove the positivity of the cohomological/refined BPS invariants of all finite-dimensional Jacobi algebras. This result supports this strengthening of the strong rationality conjecture, as well as the conjecture of Brown and Wemyss stating that all finite-dimensional Jacobi algebras for appropriate symmetric quivers are isomorphic to contraction algebras. [ABSTRACT FROM AUTHOR]

In this paper, the direct similarity reduction method is used to reduce the modified variable coefficient Zakharove–Kusnetsov equation (vcmZK), which represents the propagation of quantum dust acoustic waves (QDAWs) in an inhomogeneous magnetized dusty plasma field, into a nonlinear ordinary differential equation (NODE). Moreover, the Jacobi elliptic method is used to solve the NODE, where many novel double periodic, kink, and solitary wave solutions are obtained. Additionally, we have chosen to plot periodic, kink, and soliton QDAWs solutions according to some choices for the variable coefficients. According to the a chieved results, we can conclude that the choices of the functional forms of the variable coefficients can control the QDAWs propagation. Finally, the new discovered results may have many important applications in astrophysics, atmospheric science, nanotechnology, and industrial plasma physics. [ABSTRACT FROM AUTHOR]

Moving scientific computation from high-performance computing (HPC) and cloud computing (CC) environments to devices on the edge, i.e., physically near instruments of interest, has received tremendous interest in recent years. Such edge computing environments can operate on data in situ, offering enticing benefits over data aggregation to HPC and CC facilities that include avoiding costs of transmission, increased data privacy, and real-time data analysis. Because of the inherent unreliability of edge computing environments, new fault-tolerant approaches must be developed before the benefits of edge computing can be realized. Motivated by algorithm-based fault tolerance, a variant of the asynchronous Jacobi (ASJ) method is developed that achieves resilience to data corruption by rejecting solution approximations from neighbor devices according to a bound derived from convergence theory. Numerical results on a two-dimensional Poisson problem show that the new rejection criterion, along with a novel approximation to the shortest path length on which the criterion depends, restores convergence for the ASJ variant in the presence of certain types data corruption. Numerical results are obtained for when the singular values in the analytic bound are approximated. Additional linear systems are also explored, one with a more dense sparsity pattern and one that includes advection. All results indicate that successful resilience to data corruption depends on whether the bound tightens fast enough to reject corrupted data before the iteration evolution deviates significantly from that predicted by the convergence theory defining the bound. This observation generalizes to future work on algorithm-based fault tolerance for other asynchronous algorithms, including upcoming approaches that leverage Krylov subspaces. [ABSTRACT FROM AUTHOR]

This paper presents a real‐time optimization method for nonlinear model predictive control (NMPC) of systems governed by partial differential equations (PDEs). The NMPC problem to be solved is formulated by discretizing the PDE system in space and time by using the finite difference method. The proposed method is called the double‐layer Jacobi method, which exploits both the spatial and temporal sparsities of the PDE‐constrained NMPC problem. In the upper layer, the NMPC problem is solved by ignoring the temporal couplings of either the state or costate (Lagrange multiplier corresponding to the state equation) equations so that the spatial sparsity is preserved. The lower‐layer Jacobi method is a linear solver dedicated to PDE‐constrained NMPC problems by exploiting the spatial sparsity. Convergence analysis indicates that the convergence of the proposed method is related to the prediction horizon. Results of a numerical experiment of controlling a heat transfer process show that the proposed method can be two orders of magnitude faster than the conventional Newton's method exploiting the banded structure of NMPC problems. [ABSTRACT FROM AUTHOR]

This article presents a new approach of shifted Jacobi spectral Galerkin methods to solve weakly singular Fredholm integral equations with non-smooth solutions. We have incorporated the singular part of the kernel into a single Jacobi weight function, by dividing the integration into two parts and using a simple variable transformation. Taking advantage of orthogonal projection operator and weighted inner product with respect to that same Jacobi weight function, we are able to obtain improved convergence rate for iterated shifted Jacobi spectral Galerkin method (SJSGM) and iterated shifted Jacobi spectral multi-Galerkin method (SJSMGM) in both weighted and infinity norms. Further, we obtain improved superconvergence rate for iterated SJSGM and iterated SJSMGM, by improving the regularity of exact solution, using smoothing transformation. Increasing the value of the smoothing parameter we can improve the regularity of the exact solution upto the desired degree. Numerical results with a comparative study of pre and post smoothing transformation are given to illustrate the theoretical results and efficiency of our proposed methods. [ABSTRACT FROM AUTHOR]

This paper is concerned with the numerical approximation of initial‐boundary‐value problems of a three‐parameter family of Bona–Smith systems, derived as a model for the propagation of surface waves under a physical Boussinesq regime. The work proposed here is focused on the corresponding problem with Dirichlet boundary conditions and its approximation in space with spectral methods based on Jacobi polynomials, which are defined from the orthogonality with respect to some weighted L2$L^{2}$ inner product. Well‐posedness of the problem on the corresponding weighted Sobolev spaces is first analyzed and existence and uniqueness of solution, locally in time, are proved. Then, the spectral Galerkin semidiscrete scheme and some detailed comments on its implementation are introduced. The existence of numerical solution and error estimates on those weighted Sobolev spaces are established. Finally, the choice of the time integrator to complete the full discretization takes care of different stability issues that may be relevant when approximating the semidiscrete system. Some numerical experiments illustrate the results. [ABSTRACT FROM AUTHOR]

Weakly singular Volterra integral equations of the second kind typically have nonsmooth solutions near the initial point of the interval of integration, which seriously affects the accuracy of spectral methods. We present Jacobi spectral-collocation method to solve two-dimensional weakly singular Volterra-Hammerstein integral equations based on smoothing transformation and implicitly linear method. The solution of the smoothed equation is much smoother than the original one after smoothing transformation and the spectral method can be used. For the nonlinear Hammerstein term, the implicitly linear method is applied to simplify the calculation and improve the accuracy. The weakly singular integral term is discretized by Jacobi Gauss quadrature formula which can absorb the weakly singular kernel function into the quadrature weight function and eliminate the influence of the weakly singular kernel on the method. Convergence analysis in the L ∞ -norm is carried out and the exponential convergence rate is obtained. Finally, we demonstrate the efficiency of the proposed method by numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, a reliable Jacobi Galerkin method is developed and analyzed to solve a particular class of cordial Volterra integral equations. Existence and uniqueness theorems approve that these kinds of equations possess smooth solutions versus smooth data, so representing their Galerkin solutions based on suitable polynomial basis produces spectrally accurate approximations. Moreover, in order to control condition number growth, we designed a reliable approach that calculates the approximate solutions recursively without solving any high-conditioning systems. Indeed, this approach deals with solving equations in a long integration domain with even high oscillatory solutions. The convergence analysis of the presented approach is established, and the familiar spectral accuracy is justified in L ∞ -norm. Numerical results are presented to demonstrate the effectiveness of the proposed method. Finally, we provide an application of this method to approximate solution of a two-dimensional case. [ABSTRACT FROM AUTHOR]

In this study, the travelling wave solutions of the general Drinfel'd–Sokolov–Wilson (DSW)-system, introduced as a model of water waves, are obtained and wave dynamics are investigated. Jacobi elliptic functions are valuable mathematical tools that can be used to various aspects of physics engineering. The method used is effective in generating periodic solutions. It has been observed that the periodic solutions obtained by using Jacobi elliptic function expansions containing different Jacobi elliptic functions may be different, and some new periodic solutions can be obtained. To examine the behaviour of the solutions obtained for appropriate values of the parameters, three-dimensional simulations were conducted using Maple TM . Additionally, 2D simulations are presented to physicists for easy observation of wave motion. The usage of these simulations aids in better understanding and predicting the behaviour of waves, tidal forces and other phenomena, ultimately contributing to the development of safer and more efficient structures and systems. The stability property of the obtained solutions was tested to demonstrate their reliability. The novelty of this study is that the rich solution group obtained with Jacobi elliptic functions is different from the solutions in the literature and thus provides new physical interpretations. The findings are expected to contribute to the development of new analytical and numerical tools for solving other nonlinear systems in the future, containing a large number of travelling wave solutions crucial for explaining certain scientific phenomena in fluid media. [ABSTRACT FROM AUTHOR]

This study undertakes a comprehensive examination of the classical Lonngren wave equation, a fundamental computational model used for simulating electrical signals in semiconductor materials, with specific emphasis on the tunnel diode. The primary objective of this study is to attain novel and more comprehensive solutions beyond those documented in existing literature. To achieve this goal, we have employed well-established mathematical methods, specifically analysis via Lie symmetry, coupled with other specialized techniques such as the power series method and Jacobi elliptic expansion technique. Notably, this marks the inaugural application of these methodologies to the classical Lonngren wave equation, signifying a pioneering endeavor in the exploration of this equation using these analytical tools. These methodologies yield solutions characterized by elliptic functions. The results are visually presented through 3D, 2D, and density plots, effectively illustrating the characteristics of these solutions. The visual representations reveal a range of patterns, including periodic and singular periodic solutions. Furthermore, the paper applies the multiplier method and leverages the conservation theorem introduced by Ibragimov to derive conserved vectors. These conserved vectors play a pivotal role in the examination of physical quantities, such as energy and momentum conservation, thereby enhancing our understanding of the underlying physics within the system. [ABSTRACT FROM AUTHOR]

The Landau–Ginzburg–Higgs equation is one of the significant evolution equation in physical phenomena. In this work, the exact solutions of this equation are gained by applying an analytical method depends on twelve Jacobi elliptic functions. This equation is turned into an ordinary differential equation by the proposed method. When solving the Landau–Ginzburg–Higgs equation, an auxiliary ordinary differential equation is considered. Some theorems and corollaries utilized in the solutions of this auxiliary equation are given. Using these solutions, the elliptic and elementary solutions of the Landau–Ginzburg–Higgs equation are obtained and illustrated by tables. Many solutions are given in the form of the complex, rational, hyperbolic, and trigonometric functions. The soliton solutions and the complex valued solutions are also found by proposed method. These solutions include the largest set of solutions in the literature. Some of them are shown graphically by 2-dimensional and 3-dimensional with the help of Mathematica software. The obtained solutions are beneficial for the farther development of a concerned model. The presented method does not need initial and boundary conditions, perturbation, or linearization. Besides, this method is easy, efficient, and reliable for solutions of many partial differential equations. [ABSTRACT FROM AUTHOR]

The two-sided Jacobi method for singular value decomposition (SVD) has the advantage of obtaining singular vectors quickly and accurately. In previous research, fast and accurate implementations of the two-sided Jacobi method have been achieved for real matrices. In this study, we implemented SVD for complex matrices using the two-sided Jacobi method. In SVD, given rectangular matrices can be converted into upper-triangular matrices by conducting QR decomposition as a preprocessing step. Then, the upper-triangular matrices are decomposed. In the case where the given matrices are complex, the upper-triangular matrices are complex. Two implementations are proposed: the first requires SVD for complex matrices, whereas the second requires SVD for only real matrices. [ABSTRACT FROM AUTHOR]

In this paper, we employ Tikhonov regularization method with the projection methods using Jacobi polynomial bases to the first kind of Fredholm integral equations to find the approximate solution. We discuss the convergence analysis and obtain the convergence rates in L w α , β 2 norm under a priori parameter choice strategy. We also consider the Engl-type discrepancy principle as a posteriori parameter strategy for finding the regularization parameter and also evaluate the convergence rate which is of optimal order. Finally, we provide the numerical experiments to justify the theoretical results. [ABSTRACT FROM AUTHOR]

The use of optical fiber for communication has grown at a rapid pace due to the demands of the modern information era. In this paper, the modified extended direct algebraic method is implemented to explore optical solitons and other exact wave solutions for the coupled system of perturbed highly dispersive complex Ginzburg–Landau equation with polynomial nonlinearity law which describe the transmission of solitons in birefringent fibers. The obtained solutions include bright solitons, dark solitons, singular solitons and combo dark-singular solitons. Additionally, Jacobi elliptic function solutions, exponential solutions, and singular periodic solutions are also offered. To ensure the existence of the obtained soliton solutions, we set some constraints on the parameters. In addition, some selected solutions are visually shown to demonstrate the physical properties of the exact solutions. [ABSTRACT FROM AUTHOR]

The Schrödinger–Hirota equation are studied in light transmission, optical fiber communication, and nonlinear effects in optics. The auxiliary equation method is not only suitable for solving specific types of nonlinear partial differential equations, but also has strong applicability to all kinds of different types of equations. It helps us to deduce the exact solution of the equation faster and analyzes the dynamic behavior of the system further. The extended fourth Jacobi elliptic equation is used in this paper to seek different types of exact solutions, which include bright soliton solutions, kink solutions, periodic wave solutions, and singular traveling wave solutions via selecting appropriate parameters.The characteristics of some solutions are graphically presented using two- and three-dimensional graphs such as the real part, the imaginary part, and their modulus via providing suitable values to arbitrary parameters. Compared to other methods, the method is more direct and easier for calculations. Kindly check and confirm the edit made in the title. The edit made in the title is correct. The word "extended" in the title can be deleted if necessary. [ABSTRACT FROM AUTHOR]

In this paper, we investigate pseudo-Riemannian manifolds those eigenvalues of the Weyl conformal Jacobi operators are constant on the unit sphere bundles. Using a result of [4], we give an explicit construction of conformally Osserman manifold which is not locally conformally flat. [ABSTRACT FROM AUTHOR]

In this paper, we study fractional reaction–diffusion equation using Jacobi spectral method. Reaction–diffusion equation is used as model for spatial effects in ecology. In this method, the reaction–diffusion equation is changed to the systems of equations. Convergence analysis for the proposed spectral method is established from theoretical as well as numerical point of view. Some numerical examples are solved using proposed numerical method showing the feasibility of the method. The efficiency of the spectral method is shown by listing CPU time of computation. [ABSTRACT FROM AUTHOR]

The major aim of this article is to obtain the numerical solution of a fractional mathematical model with a nonsingular kernel for thrombin receptor activation in calcium signals using two numerical schemes based on the collocation techniques. We present the computational solution of the considered fractional model using the Laguerre collocation method (LCM) and Jacobi collocation method (JCM). An operational matrix of the fractional order derivative in the Caputo sense is needed for the recommended approach. The computational scheme converts fractional differential equations (FDEs) into an algebraic set of equations using the collocation method. The technique is used more quickly and successfully than in other existing schemes. A comparison between LCM and JCM is also presented in the form of figures. We obtained very good results with a great agreement between both the schemes. Additionally, an error analysis of the suggested procedures is provided. [ABSTRACT FROM AUTHOR]

The goal of this paper is to find exact solutions to the improved modified Korteweg-de Vries (mKdV) equation with a conformable derivative using the Jacobi elliptic function expansion method. The improved mKdV equation is a prominent mathematical model in the realm of nonlinear partial differential equations, with widespread applicability in diverse scientific and engineering domains. This study leverages the well-known effectiveness of the Jacobi elliptic function expansion method in solving nonlinear differential equations, specifically focusing on the intricacies of the improved mKdV problem. The investigation reveals innovative and explicit solutions, providing insight into the dynamics of the related physical processes. This paper provides a comprehensive examination of these solutions, emphasizing their distinct features and depictions using Jacobi elliptic functions. These findings are especially advantageous for specialists in the fields of nonlinear science and mathematical physics, providing significant insights into the behavior and development of nonlinear waves in various physical situations. This work also contributes to our knowledge of the improved mKdV equation and shows that the Jacobi elliptic function expansion method is a useful tool for solving complex nonlinear situations. The study is enhanced with graphical illustrations of various solutions, which further enhance its analytical complexity. [ABSTRACT FROM AUTHOR]

This study addresses the integrable Akbota equation which is a Heisenberg ferromagnet-type equation and has a significant study of the curve and surface geometry. The conservation laws of the model are computed using the multipliers approach. The Jacobi elliptic function solutions for the model are derived by utilizing the new extended auxiliary equation method. The Jacobi elliptic functions solutions are degenerated to dark, bright, singular and singular periodic solitons in their limit conditions. Further, the modulation instability analysis of the equation is studied. Physical interpretations of the obtained results are represented graphically. To the best of our knowledge, the obtained results in this article are novel and have not been reported before. [ABSTRACT FROM AUTHOR]

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This study explores the complex dynamics of the predator–prey interactions, with a specific emphasis on the influence of the Allee effect on the predator population. We examined the fundamental mathematical characteristics of the model under consideration, such as the positivity of the system and the boundedness of the solutions. We investigated the equilibrium points and analyzed their stability using the Jacobi and Lyapunov methods. A comprehensive examination was carried out on the geometric properties of the dynamical system to compute the five invariants of the KCC theory. In particular, the deviation curvature tensor and its eigenvalues are investigated to demonstrate the behavior of the system stability. We have also obtained the necessary and sufficient conditions for the given set of parameters of the system in order to have the Jacobi stability (instability) near the equilibrium point. To visualize the dynamical behavior of the predator–prey model with the Allee effect in the predator density, numerical simulations were conducted. The investigation encompasses an examination of the system's behavior from both geometric and numerical standpoints, with the objective of attaining a thorough comprehension using few examples. [ABSTRACT FROM AUTHOR]

As an improvement on the traditional model-based Yamaguchi four-component decomposition method, in recent years, to fully utilize the polarization information in the coherency matrix, four-component target decomposition methods Y4R and S4R have been proposed, which are based on the rotation of the coherency matrix and the expansion of the volume model, respectively. At the same time, there is also an improved G4U method proposed based on Y4R and S4R. Although these methods have achieved certain decomposition results, there are still problems with overestimation of volume scattering and insufficient utilization of polarization information. In this paper, a unitary transformation extension to the four-component target decomposition method of PolSAR based on the properties of the Jacobi method is proposed. By analyzing the terms in the basic scattering models, such as volume scattering, in the existing four-component decomposition methods, it is clear that the reason for the existence of the residual matrix in the existing decomposition methods is that the off-diagonal term T 13 and the real part of T 23 of the coherency matrix T do not participate in the four-component decomposition. On this basis, a matrix transformation method is proposed to decouple terms T 13 and Re T 23 , and the residual matrix decomposed based on this method is derived. The performance of the proposed method was validated and evaluated using two datasets. The experimental results indicate that, compared with model-based methods such as Y4R, S4R and G4U, the proposed method can enhance the contribution of double-bounce scattering and odd-bounce scattering power in urban areas in both sets of data. The computational time of the proposed method is equivalent to Y4R, S4R, etc. [ABSTRACT FROM AUTHOR]

In this study, moving contact problem for a rigid punch and a functionally graded orthotropic layer bonded to an isotropic homogeneous layer is solved, semi-analytically. The cylindrical punch moves steadily with a constant subsonic velocity over the upper layer and transmits concentrated normal and tangential forces. Using Fourier transform and Galilean transformation, the contact problem is converted to a Cauchy-type singular integral equation of the second kind, in which the contact stress and contact width are the unknowns. The numerical solution of the singular integral equation is obtained by using Gauss–Jacobi integration formulas. Numerical results for the contact stress, in-plane stress, and the contact width are given. [ABSTRACT FROM AUTHOR]

The solution of the Poisson equation raised from large-scale problems requires iterative techniques. Multilevel and Krylov subspace methods have been proven to be the most efficient among various iterative methods. Recently, the scheduled relaxation Jacobi technique has been developed, which can be used as a stand-alone iterative solver as well as a preconditioner to the Krylov subspace methods. The relaxed Jacobi (RJ) method has also been implemented as a smoother in the multigrid method, with no-post smoothening. Here, the focus is to test the performance of relaxed Jacobi when applied to the large scale Poisson problems. Multigrid methods with both post-smoothening and pre-smoothening operations have been conducted in this paper. The methodology is applied to a rectangular domain using structured Cartesian grids. Also, multigrid methods with relaxed Jacobi smoothers are used as preconditioners to Krylov subspace solvers, and their performances are compared with the red-black Gauss-Seidel smoothers. Similar to the results in the literature, RJ smoothers to multigrid techniques provide better results than Gauss-Seidel smoothers when no-post smoothening operations are carried out. However, the use of post-smoothening operations changed the entire dynamic of the RJ smoothers. [ABSTRACT FROM AUTHOR]

We consider the Alternating Direction Implicit (ADI) method to compute the numerical solution of a continuous Sylvester equation A X + X B = C , based on the recently developed inexact ADI iteration, and we propose classical acceleration techniques to enhance its convergence rate. An extrapolated variant (EADI) and a block successive overrelaxation variant (block SOR-ADI) of the ADI iterative method are described. These relaxation approaches are similar to what is used in Gauss-Seidel and Jacobi methods for linear systems, and, to our knowledge, novel, especially the block SOR-ADI scheme. Convergence properties of these two relaxed variants are analyzed when the matrix A is positive definite and the matrix B is positive semi-definite (not necessarily Hermitian matrices), or conversely. Our numerical experiments suggest that these new schemes are computationally attractive. The convergence rate of the ADI method is usually increased, particularly with the block SOR-ADI variant. A comparison with the well-known Hermitian and skew-Hermitian splitting (HSS) method emphasizes the efficiency of the proposed methods. [ABSTRACT FROM AUTHOR]

This research focuses on calculating the translational vibration modes of a three-level frame building, in order to determine its periods and way of vibrating, the evaluation was carried out using interactive methods of Stodola and Holzer, the numerical method of Jacobi, and the results obtained were verified using functions of the eigenvals vector and eigenvecs matrix of the Mathcad software. In addition, a verification was carried out with modeling in the Etabs software, following the properties established by the Technical Standard E.030 for Seismic-Resistant Design of the Reglamento Nacional de Edificaciones (RNE). Throughout the investigation, accurate results were obtained through manual calculations as well as verifications with the use of Mathcaf and ETABS software. In each instance, it was determined that these methods represent the most appropriate way to obtain approximate values with respect to the verification carried out with its period and way of vibrating. These reliable results underline the effectiveness of the tools and methods used, highlighting the deep understanding of the dynamic characteristics of the analyzed building. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the "Walk on Equations" (WE) Monte Carlo algorithm, a novel approach for solving linear algebraic systems. This algorithm shares similarities with the recently developed WE MC method by Ivan Dimov, Sylvain Maire, and Jean Michel Sellier. This method is particularly effective for large matrices, both real- and complex-valued, and shows significant improvements over traditional methods. Our comprehensive comparison with the Gauss–Seidel method highlights the WE algorithm's superior performance, especially in reducing relative errors within fewer iterations. We also introduce a unique dominancy number, which plays a crucial role in the algorithm's efficiency. A pivotal outcome of our research is the convergence theorem we established for the WE algorithm, demonstrating its optimized performance through a balanced iteration matrix. Furthermore, we incorporated a sequential Monte Carlo method, enhancing the algorithm's efficacy. The most-notable application of our algorithm is in solving a large system derived from a finite-element approximation in constructive mechanics, specifically for a beam structure problem. Our findings reveal that the proposed WE Monte Carlo algorithm, especially when combined with sequential MC, converges significantly faster than well-known deterministic iterative methods such as the Jacobi method. This enhanced convergence is more pronounced in larger matrices. Additionally, our comparative analysis with the preconditioned conjugate gradient (PCG) method shows that the WE MC method can outperform traditional methods for certain matrices. The introduction of a new random variable as an unbiased estimator of the solution vector and the analysis of the relative stochastic error structure further illustrate the potential of our novel algorithm in computational mathematics. [ABSTRACT FROM AUTHOR]

In this article, we construct a Jacobi spectral collocation scheme to approximate the Caputo fractional derivative based on Jacobi–Gauss quadrature. The convergence analysis is provided in anisotropic Jacobi-weighted Sobolev spaces. Furthermore, the convergence rate is presented for solving Caputo fractional derivative with noisy data by invoking the mollification regularization method. Lastly, numerical examples illustrate the effectiveness and stability of the proposed method. [ABSTRACT FROM AUTHOR]

Hadamard type fractional calculus involves logarithmic function of an arbitrary exponent as its convolutional kernel, which causes challenge in numerical treatment. In this paper we present a spectral collocation method with mapped Jacobi log orthogonal functions (MJLOFs) as basis functions and obtain an efficient algorithm to solve Hadamard type fractional differential equations. We develop basic approximation theory for the MJLOFs and derive a recurrence relation to evaluate the collocation differentiation matrix for implementing the spectral collocation algorithm. We demonstrate the effectiveness of the new method for the nonlinear initial and boundary problems, i.e, the fractional Helmholtz equation, and the fractional Burgers equation. [ABSTRACT FROM AUTHOR]

Plane and axisymmetric problems of interfacial cracks in a power-law graded infinite medium are examined. It is shown that the models are governed by integral equations of Mellin's convolution type whose kernels are expressed through the hypergeometric function. Exact solutions are derived by the method of orthogonal Jacobi polynomials in a series form and by the Wiener–Hopf method by quadratures. Mode-I stress intensity factors and the associated weight functions for power-law graded materials in the plane and axisymmetric cases are introduced and evaluated. It is shown that, although the displacement jump and the normal traction component have power singularities at the crack tip different from 1/2, the strain energy variation is proportional to the crack length change as in the case of homogeneous materials. A Griffith-type criterion of crack propagation in power-law graded materials is proposed and results of numerical tests are reported. [ABSTRACT FROM AUTHOR]

The radiative transfer equation (RTE) is an integro-differential equation that describes the radiation energy absorbing, emitting, and scattering in both space and angle, which can be up to five-dimensional problems. It is difficult for a RTE solver to satisfy both accuracy and efficiency (less computational resources) for such high dimensional problems. In this paper, an iterative solver for one-dimensional cylindrical radiative transfer problems using the space-angle discontinuous Galerkin (DG) method is developed to achieve both accuracy and efficiency. The iterative solver is based on the angular decomposition (AD) scheme, which slices the spatial-angular domain into slabs and decouples the angular integration between slabs. Both Jacobi and successive over-relaxation (SOR) iterative schemes are investigated by numerical analysis and examples. The comparison of the two iterative schemes suggests that an appropriate relaxation factor for the SOR method may accelerate the convergence. Finally, the iterative scheme is more efficient than the direct solution of the system both in terms of memory usage and computational time, especially for finer meshes. [ABSTRACT FROM AUTHOR]

The paper presents iterative methods for solving interval linear system of equations. We present a generalization of interval Jacobi method and interval Gauss-Seidel method by generalizing interval diagonal matrices to band interval matrices, and discuss the convergence analysis of the proposed methods. More specifically, we prove that both the proposed methods converge for any initial approximation if the coefficient interval matrix of the system is either an interval strictly diagonally dominant matrix, or interval M-matrix or interval H-matrix. Numerical experiment are carried out to assess the effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

A previous study by Mantas and Vecil (Int J High Perform Comput Appl 34(1): 81–102, 2019) describes an efficient and accurate solver for nanoscale DG MOSFETs through a deterministic Boltzmann-Schrödinger-Poisson model with seven electron–phonon scattering mechanisms on a hybrid parallel CPU/GPU platform. The transport computational phase, i.e. the time integration of the Boltzmann equations, was ported to the GPU using CUDA extensions, but the computation of the system's eigenstates, i.e. the solution of the Schrödinger-Poisson block, was parallelized only using OpenMP due to its complexity. This work fills the gap by describing a port to GPU for the solver of the Schrödinger-Poisson block. This new proposal implements on GPU a Scheduled Relaxation Jacobi method to solve the sparse linear systems which arise in the 2D Poisson equation. The 1D Schrödinger equation is solved on GPU by adapting a multi-section iteration and the Newton-Raphson algorithm to approximate the energy levels, and the Inverse Power Iterative Method is used to approximate the wave vectors. We want to stress that this solver for the Schrödinger-Poisson block can be thought as a module independent of the transport phase (Boltzmann) and can be used for solvers using different levels of description for the electrons; therefore, it is of particular interest because it can be adapted to other macroscopic, hence faster, solvers for confined devices exploited at industrial level. [ABSTRACT FROM AUTHOR]

In this paper, we present a new generalized model of partial integro‐differential equation of fractional order (PIDEFO). The new model of the PIDEFO is defined in terms of the B operator presented recently. The B operator takes form of the Riemann‐Liouville and Caputo derivatives in special case. The generalized model of the PIDEFO reduces to (1) linear space‐time fractional reaction‐diffusion equation (FRDE), (2) space‐time fractional diffusion equations (STFDE), and (3) time fractional telegraph equations (TFTE) in particular cases. The considered model is solved with collocation method using Jacobi polynomials. For theoretical accuracy, we prove that the Jacobi approximation converges to the exact solution uniformly. Two test examples are considered to verify the accuracy of the proposed method. Further, we present two applications of the proposed model and solve it using the presented method. The obtained numerical results are compared with the existing results. Finally, the numerical stability of the presented method is analyzed under the effect of the random noise in the force term. [ABSTRACT FROM AUTHOR]

Calibration is vital for the binocular visual system. The high-precision measurement volume of the traditional binocular visual system is very small; therefore, it is difficult to measure large objects accurately. Accurate calibration of the binocular visual system is also extremely difficult in a large-scale field-of-view (FOV). Several attempts have been made to calibrate the binocular visual system in a large-scale FOV without recognizing the importance of depth information. A new method based on multimodal constraints has been proposed to calibrate the binocular visual system in a large-scale FOV whose height-width-depth is 2*2*2 m 3 . The method fuses multiple constraints, i.e. the distance error constraint, the reprojection error constraint, the epipolar error constraint, the spatial points error constraint, and the regularization constraint. An objective function that integrates the above constraints has been derived. The construction method of the global Jacobi matrix, the adaptive weight adjustment method, and the Levenberg–Marquardt iterative algorithm are used to minimize the objective function. Comparison experiments have been carried out to validate the performance of our proposed method by measuring a 1.4 m long calibration gauge. The results show that the root mean square (RMS) of the residual error of our method is 0.0929 mm, Zhang's traditional method is 0.523 mm, and Xiao's photogrammetric method is 0.348 mm. The RMS of our method is 82% less than that of Zhang's method. Noise experiments show that the RMS of the residual error of our method is 45% less than Zhang's method with 0.1 level noise inference, which verifies the stability and reliability of our method. [ABSTRACT FROM AUTHOR]

In this paper, flat potentials (μ│x / a│N) are numerically investigated by pseudo-spectral method. The Schrodinger equation of this type of potential has become an eigen system using the pseudo-spectral method. The eigen system is then diagonalized by the Jacobi method. Energy eigen values for different Ns have been compared with similar articles. The limit behavior of this potential for the states N = 2 and N → ∞ is related to the harmonic oscillator and the particle in the box with length 2a, respectively. For each N, a function is proposed for energy eigen values in terms of the quantum number n. By using of data fitting, the correctness of the proposed equation is checked.

In this paper, the variable-order (VO) time fractional version of the 2D Wu-Zhang system is introduced. A numerical method based on the spline approximation of the VO fractional terms in this system and the Jacobi polynomials (JPs) approximation of the solution is proposed to solve this system. In the presented method, the VO fractional derivatives are approximated using the spline expansions at first. Then, by employing the expressed spline expansions as well as the θ -weighted method a recurrence relation is obtained. Next, the unknown functions are expanded in terms of the 2D JPs (with unknown coefficients) and substituted into the recurrence relation. Finally, by solving a system of linear algebraic equations, the unknown coefficients and subsequently an approximate solution of the problem is obtained. The accuracy of the established method is investigated by solving some numerical examples. [ABSTRACT FROM AUTHOR]

In order to enhance the anti-interference ability of the speed to external load and achieve the rapid response control for turboshaft engine of next-generation helicopters, an incremental nonlinear dynamic inverse (INDI) control method for turboshaft engine based on Jacobi matrix is proposed. First, the control law of turboshaft engine based on state variable model (SVM) is obtained according to the mathematical derivation of INDI. Then, based on adopting Jacobi matrix and sequential quadratic programming (SQP) algorithm, the SVM is online obtained, so as to express the steady and dynamic characteristics of turboshaft engines under different flight conditions precisely. Finally, several numerical simulations are conducted to validate the control effect of INDI control method. The results demonstrate that compared with cascade PID controller, INDI control method based on single SVM can significantly decrease the overshoot and sag of power turbine speed by more than 50% under constant flight conditions. Under variable flight conditions, INDI control method based on Jacobi matrix can effectively decrease the overshoot and sag of power turbine speed by more than 30% compared with INDI control method based on single SVM, which has more significant control effect and superior robustness, and realizes the fast response control of turboshaft engines. [ABSTRACT FROM AUTHOR]

The fundamental formula in an optical system is Rayleigh diffraction integral. In practice, we deal with Fresnel diffraction integral as approximate diffraction formula. Fresnel transforms and inverse Fresnel transforms have been formulated systematically and defined as a bounded additive, a unitary operator in Hilbert space. To investigate the band-limited effect in Fresnel transform plane, we seek the function that its total power in finite Fresnel transform plane is maximized, on condition that an input signal is zero outside the bounded region. This problem is a variational one with an accessory condition. This leads to the eigenvalue problems of Fredholm integral equation of the first kind. The kernel of the integral equation is of Hermitian conjugate and positive definite. Moreover, we prove that the eigenfunctions corresponding to distinct eigenvalues have dual orthogonal property. By discretizing the kernel and integral range to seek the approximate solutions, the eigenvalue problems of the integral equation can depend on a one of the Hermitian matrix in finite-dimensional complex value vector space. We use the Jacobi method to compute all eigenvalues and eigenvectors of the matrix. We consider the application of the eigenvectors to the problems of the approximating any functions. We show the validity and limitation of the eigenvectors in computer simulations. [ABSTRACT FROM AUTHOR]

The aim of this study is to investigate the dynamical behavior of (1 + 1) -dimensional Kudryashov's equation with generalized anti-cubic nonlinearity by extracting the exact closed form solitons and other traveling wave solutions. This nonlinear mathematical model explains the propagation of pulses though optical fibers. It is based on the well-known nonlinear Schrödinger equation with an arbitrary power nonlinearity. In this paper, two techniques are used to extract the optical solitons and other solitary wave solutions of Kudryashov's equation. One is the general projective Riccati equation method and the other is the Jacobi elliptic method. The proposed techniques retrieve the explicit solution expressions by calculating the coefficients of a truncated power series. The exact solutions are obtained in terms of hyperbolic, trigonometric and Jacobi elliptic functions. The constructed solitons include kink, combo dark-bright and dark solitons. Periodic traveling wave solutions are also reported. The derived solutions are shown as three-dimensional graphs for the appropriate choice of parametric parameters. The corresponding two dimensional contour graphs are included to understand dynamical behavior of obtained results. [ABSTRACT FROM AUTHOR]

In this paper, an efficient Jacobi spectral collocation method is developed for multi-dimensional weakly singular volume-constrained nonlocal models including both nonlocal diffusion (ND) models and peridynamic (PD) models. The model equation contains a weakly singular integral operator with the singularity located at the center of the integral domain, and the numerical approximation of it becomes an essential difficulty in solving nonlocal models. To approximate such singular nonlocal integrals in an accurate way, a novel nonlocal quadrature rule is constructed to accurately compute these integrals for the numerical solutions produced by spectral methods. Numerical experiments are given to show that spectral accuracy can be obtained by using the proposed Jacobi spectral collocation methods for several different nonlocal models. Besides, we numerically verify that the numerical solution of our Jacobi spectral method can converge to its correct local limit as the nonlocal interactions vanish. [ABSTRACT FROM AUTHOR]