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In fifth-generation (5G)-based vehicular networks, vehicles share information about the road's condition to make driving safer and alleviate traffic congestion. Given the public nature of the medium, addressing issues of confidentiality and security is of paramount importance. Existing systems are vulnerable to a distributed denial-of-service (DDoS) attack from an attacker or high traffic in a dense area because of the time-consuming and complex operations based on bilinear pair and elliptic curve cryptographies used. In order to protect vehicle-to-vehicle communication in 5G-based vehicular networks from DDoS attacks, a strategy based on the Chebyshev polynomial is proposed in this study. The semi-group and chaotic features of the Chebyshev polynomial are utilised in our proposal. Our solutions also meet all applicable standards for security and privacy, such as node authentication, message integrity, preserving identity privacy, traceability, unlinkability, and resistance to attacks. Finally, our innovation decreases the computational burden to produce a message signature by 66.67% and the burden to verify a signature by 33.33%. However, the size of the message-signature tuple is affected by the communication expenses of our work by 5.00%. [ABSTRACT FROM AUTHOR]

Surrogate-based aerodynamic robust design optimization uses a surrogate model to calculate the robustness indices, which strongly relies on the overall accuracy of the model, as well as the efficient exploration of the design space. For most surrogate modeling approaches, significant inaccuracies are often observed at the outlier region of the design space, where very few samples are spotted. A novel method using Chebyshev transformation is applied to re-allocate the orthogonal Latin hypercube sample set to alleviate the corner errors and eventually improve the overall accuracy. An inner Kriging model is developed using the sampling method, and robustness indices are calculated based on the subspaces adjacent to the sampling points. Subsequently, an outer robust model is constructed with the robustness indices as the target. Ultimately, a combination of the inner and outer models is utilized with the genetic algorithm to accomplish multi-objective robust optimization. Theoretical tests are undertaken for classic test functions, showing the advantage of the proposed approach. Based on this method, aerodynamic robust design optimizations are carried out on the RAE 2822 airfoil, for which the lift coefficient and drag coefficient are optimized for a given range of geometrical parameters. An increase of 1.94% lift coefficient and a reduction of 2.53% drag coefficient are achieved compared to the baseline design without sacrificing the robust performances. [ABSTRACT FROM AUTHOR]

Despite several applications of the fractional advection–diffusion equation (fADE) in studying sediment transport in an open channel flow, its application is limited to apprehending the non-local movement of sediment particles in an ice-covered channel with a steady, uniform flow field. An unsteady fADE is considered where the space term is non-local with a non-integer order and the mathematical model with Caputo fractional derivative is able to estimate the variation of sediment concentration along a vertical as well as with time in the ice-covered channel. An eddy viscosity expression is used, which includes the variation in roughness between the channel bed and ice cover surface. The Chebyshev collocation method and the Euler backward method are used to solve the fADE with the initial and boundary conditions and the convergence of the methods is established. The temporal variation of concentration shows that for a zero initial condition, the concentration profile first increases and then becomes stable after a certain time; for a non-zero initial concentration, the profile decreases with an increase in time and eventually a steady state is achieved. The effect of the order of the fractional derivative on the vertical variation of concentration at different times for zero and non-zero initial concentrations is studied and it is found that the order of the fractional derivative has a greater impact at smaller times. The impact of several parameters on concentration profiles is studied at different times and the validation of the model is done by comparing it with experimental studies under restricted conditions. [ABSTRACT FROM AUTHOR]

Novel approximate and analytical solutions to the fifth-order Duffing equation (FODE) are reported. These solutions are expressed in terms of the Jacobi elliptic and trigonometric functions. As a possible realization in the nature of the FODE, we study some nonlinear differential equations (NLDEs) that describe many physical problems. The numerical solution to the FODE is compared with the theoretical results. Moreover, the proposed method shows a new vision, which could be of great interest in the solution of the family of higher-order nonlinear Schrödinger equation such as the cubic-quintic NLSE (CQNLSE) which is used for interpreting the high Langmuir fields energy in magnetoplasmas. This method extends to find the solution of the derivative NLSE (DNLSE), which is used to describe the weakly modulated Alfvén waves propagation in magnetoplasmas. Furthermore, the equation of motion for the general pendulum could be solved using the techniques under consideration. All mentioned equations could be reduced to the FODE via a suitable transformation. Finally, this study is very interesting in describing several natural phenomena and solving many physical NLDEs that were difficult to solve them before. Moreover, the analytical bright soliton solution to the higher-order NLSE with incorporating cubic-quintic nonlinearity is obtained. [ABSTRACT FROM AUTHOR]

On the class of typically real odd polynomials of degree 2 N - 1 F (z) = z + ∑ j = 2 N a j z 2 j - 1 we consider two problems: 1) stretching the central unit disc under the above polynomial mappings and 2) estimating the coefficient a 2. It is shown that | F (z) | ≤ 1 2 csc 2 π 2 N + 2 , - 1 + 4 sin 2 π 2 N + 4 ≤ a 2 ≤ - 1 + 4 cos 2 π N + 2 for odd N , and - 1 + 4 (ν N) 2 ≤ a 2 ≤ - 1 + 4 cos 2 π N + 2 for even N , where ν N is a minimal positive root of the equation U N + 1 ′ (x) = 0 with U N + 1 ′ (x) being the derivative of the Chebyshev polynomial of the second kind of the corresponding order. The above boundaries are sharp, the corresponding estremizers are unique and the coefficients are determined. [ABSTRACT FROM AUTHOR]

The medical cyber-physical system utilizes various Internet of Medical Things (IoMT) devices that are connected to the network for real-time management and medication of patient. The resource constraint IoMT devices require energy-efficient lightweight security schemes to protect medical data. To the best of our knowledge, majority of the existing studies concentrate either on security or on energy efficiency issues. The main contribution of this research is to develop an energy-efficient lightweight authentication method without sacrificing the security level. The authentication is done using multiple factors, namely password, physical unclonable function (PUF), Chebyshev polynomial, smartcard and fuzzy extractor. Chebyshev polynomial is used for non-identification of the private key by any attacker, whereas PUF protects the smart card from cloning by generating unique challenge–response pairs. Apart from authentication, the proposed scheme includes Blockchain-enabled distributed trustable ecosystem among independent participants where miner is selected using lightweight fuzzy system. The proposed scheme carries out the formal security analysis using real or random model which is perseverance against different external attacks and the security verification has been done using AVISPA (Automate Validation of Internet Security Protocols and Applications) tool. As a major finding, the simulation result using NS-3 simulator confirms that the proposed study outperforms the existing studies in terms of packet loss rate, throughput, end-to-end delay, computational cost and communication cost. [ABSTRACT FROM AUTHOR]

Based on the Garlerkin method, the Galerkin finite block method (GFBM) is proposed to deal with two‐dimensional (2D) linear partial differential equations (PDEs) with variable coefficients in this paper. The mapping technique is utilized to transform a block in physical domain into normalized square. Physical variables are approximated with double layer Chebyshev polynomials for 2D problem. A set of linear algebraic equation is formulated with the Chebyshev polynomials from PDE and boundary conditions in weak form. Continuous conditions at interfacial surfaces between two blocks are introduced in either weak form or strong form. It is demonstrated that the GFBM is suitable to deal with complicated problems with high accuracy including discontinuous boundary values problem and concentrated heat sources in the domain. Several numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this study, we investigate and analyze an approximation of the Chebyshev polynomials for pantograph‐type fractional‐order differential equations. First, we construct the operational matrices of pantograph and Caputo fractional derivatives using Chebyshev interpolation. Then, the obtained matrices are utilized to approximate the fractional derivative. We also provide a detailed convergence analysis in terms of the weighted square norm. Finally, we describe and discuss the results of three numerical experiments conducted to confirm the applicability and accuracy of the computational scheme. [ABSTRACT FROM AUTHOR]

This study explores efficient methods for computing eigenvalues and function values associated with Chebyshev-type prolate spheroidal wave functions (CPSWFs). Applying the expansion of the factor e i c x y and the inherent properties of Chebyshev polynomials, we present an exact and stable numerical approximation for the exact eigenvalues of the integral operator to CPSWFs. Additionally, we illustrate the efficiency of employing fast Fourier transform and barycentric interpolation techniques for computing CPSWF values and related quantities, which are essential for various numerical applications based on these functions. The analysis is supported by numerical examples, providing validation for the accuracy and reliability of our proposed approach. [ABSTRACT FROM AUTHOR]

This study advances dynamic force reconstruction in the time domain through the precise integration method (PIM), particularly effective for matrix exponential numerical solutions. Traditional PIM approaches, typically reliant on linear or step function assumptions within integration time steps, face limitations in handling large integration time step loads. Addressing this, our research introduces an innovative method employing velocity and displacement sensors in simple and complex structures, significantly enhancing load fitting accuracy, especially for nonlinear loads. We propose a refined version of PIM: the Chebyshev polynomial integration method (CPIM) and the adaptive precise integration method (APIM) based on wavelets. These methods are designed to fit force functions of varying orders across different time steps, thereby bolstering signal reconstruction robustness. The efficacy of these approaches is validated given adequate measurement response data. Furthermore, recognizing the growing emphasis on uncertainty in engineering, our work incorporates the collocation method for uncertainty quantification in load identification. To counteract the erratic results from polynomial fitting, particle swarm optimization (PSO) thresholding is employed to eliminate high-frequency noise, thus enhancing the Signal-to-Noise Ratio and minimizing signal distortion. Notably, we demonstrate that the Shannon wavelet surpasses the Chebyshev polynomial in fitting ability due to its tighter branch, rendering APIM more suitable for engineering applications compared to CPIM. APIM adeptly minimizes errors related to acquisition frequency by adaptively adjusting the number of discretization points and wavelet convergence levels to fulfill convergence criteria and matrix norm requisites. This paper proposes a multi-source uncertainty-oriented dynamic Force Reconstruction Framework, integrating adaptive fitting precise integration with PSO. Our numerical examples illustrate that APIM, as discussed herein, offers broader applications and is more time efficient than CPIM. [ABSTRACT FROM AUTHOR]

Various non-Fourier heat conduction models have been proposed to overcome the paradox of the infinite propagation speed of heat signals in Fourier's law. The three-phase-lag (TPL) heat model is a non-Fourier model based on the linearized theory of coupled thermoelasticity that combines heat flux, temperature gradient, and thermal displacement gradient. The present paper proposes a novel method with meshfree and spectral nature to solve the two-dimensional three-phase lag (TPL) bioheat model for tissue heating during hyperthermia. In the space domain, radial basis functions (RBFs) and, in the time direction, shifted Chebyshev polynomials are employed to solve the model. The use of Chebyshev polynomials in the time direction enables one to get the solution in the whole time domain simultaneously with fewer nodes. Further, there is no need for background meshes in space due to the meshless nature of RBFs, and the proposed method is equally applicable to irregular domains. The proposed method is validated with the existing semi-analytical solution of the TPL model in one spatial dimension and found to be in good agreement. The temperature profile and the impact of various parameters, such as phase lag times, blood perfusion rate, and heat source parameters on heat transfer in tissue, have also been discussed. [ABSTRACT FROM AUTHOR]

We present simple congruences modulo p of the Jones polynomial for the torus link T (p , λ) , where p is a prime number and λ is a nonzero integer. These congruences are expressed in terms of Chebyshev polynomials and quantum numbers. The approach makes elementary use of the Kauffman bracket and does not require any representation theory. [ABSTRACT FROM AUTHOR]

One of the goals of sixth-generation mobile networks (6G) is to achieve a larger network coverage area. Satellite networks enable global coverage and aerial nodes such as Unmanned Aerial Vehicle (UAV) can serve as a supplement to ground networks in remote environments. Therefore, 6G networks are gradually evolving towards Space-Air-Ground integrated networks. The combination of UAV networks and satellite networks is a research hotspot in the field of Space-Air integrated networks. However, the combination of UAV networks and satellite networks currently faces many challenges in terms of security. The characteristics of large propagation delay and unstable communication links in satellite networks make them vulnerable to various attacks, including eavesdropping, tampering, and impersonation. Meanwhile, existing research on UAV networks mainly focuses on UAV-Ground networking authentication mechanisms, which are not suitable for resource-constrained nodes in the Space-Air integration scenario. Therefore, based on elliptic curve public key cryptography and Chebyshev polynomial, we propose a secure networking authentication scheme for satellite nodes and UAV nodes in the Space-Air integration scenario. The security analysis indicates that our scheme possesses security attributes such as mutual authentication, key agreement, identity anonymity, unlinkability, perfect forward-backward security, and resistance against various protocol attacks, among other security properties. Performance analysis also indicates certain advantages of our scheme over existing schemes in terms of signaling, bandwidth, and computational overhead. [ABSTRACT FROM AUTHOR]

In a recent work (J. Sci. Comput. 85 (2020), no. 3), the author generalized the Chebyshev-Davidson method appeared in standard eigenvalue problems to symmetric generalized eigenvalue problems. The theoretical derivation indicates that the Chebyshev-Davidson method for symmetric generalized eigenvalue problems only admits local convergence; thus, in this paper, we adopt a flexible strategy to improve the global convergence and to save number of iteration steps. Moreover, the deflation technique used for computing several eigenpairs in the proposed Chebyshev-Davidson method cannot be implemented in parallel; therefore, we construct a flexible block Chebyshev-Davidson method for computing several eigenpairs of symmetric generalized eigenvalue problems. The block implementation is important in scientific computing since it allows parallelism and efficient use of local memory. Numerical experiments are carried out to show great superiority and robustness over some state-of-the-art iteration methods. [ABSTRACT FROM AUTHOR]

This paper introduces an innovative method for numerically integrating fourth-order initial value problems by utilizing Chebyshev polynomials as the fundamental basis function. The block integrator base on Chebyshev polynomial demonstrates significant improvements in accuracy and stability, rendering it a valuable tool across various scientific and engineering fields. By leveraging the characteristics of Chebyshev polynomials, this approach accurately estimates solutions for fourth-order differential equations without reducing it to a system of first order Ordinary Differential Equations while at the same time effectively managing error accumulation within a block integration framework and thereby enhancing its accuracy over extended intervals. Through rigorous numerical experiments, the effectiveness and reliability of the new integrator are demonstrated and compared with existing methods. The new method is consistent, zero stable and convergent. The method also shows an appreciable error constants. The new method performed better in terms of accuracy than the existing methods in the literature in both linear and nonlinear problems.

Spectral relationships explain many physical phenomena, especially in quantum physics and astrophysics. Therefore, in this paper, we first attempt to derive spectral relationships in position and time for an integral operator with a singular kernel. Second, using these relations to solve a mixed integral equation (MIE) of the second kind in the space $ {L}_{2}\left[-\mathrm{1, 1}\right]\times C\left[0, T\right], T < 1. $ The way to do this is to derive a general principal theorem of the spectral relations from the term of the Volterra-Fredholm integral equation (V-FIE), with the help of the Chebyshev polynomials (CPs), and then use the results in the general MIE to discuss its solution. More than that, some special and important cases will be devised that help explain many phenomena in the basic sciences in general. Here, the FI term is considered in position, in $ {L}_{2}\left[-\mathrm{1, 1}\right], $ and its kernel takes a logarithmic form multiplied by a general continuous function. While the VI term in time, in $ C\left[0, T\right], T < 1, $ and its kernels are smooth functions. Many numerical results are considered, and the estimated error is also established using Maple 2022.

Based on Chebyshev spectral collocation and numerical techniques for handling highly oscillatory integrals, we propose a numerical method for a class of highly oscillatory Volterra integral equations frequently encountered in engineering applications. Specifically, we interpolate the unknown function at Chebyshev points, and substitute these points into the integral equation, resulting in a system of linear equations. The highly oscillatory integrals are treated using either the numerical steepest descent method or the Filon-Clenshaw-Curtis method. Additionally, we derive an error estimation formula for this method using error analysis techniques and validate the convergence and effectiveness of the proposed approach through numerical examples.

Purpose: According to the interaction between multiple cracks, a method for calculating the stress intensity factors of multiple closed collinear cracks in an infinite plate is proposed. In addition, the effects of the crack angle, crack length, crack spacing, lateral pressure coefficient and crack surface friction coefficient on the stress intensity factors are analyzed. Method: The integral equation of the stress intensity factors is derived by the superposition principle, and the approximate solution of the integral equation is derived by the Chebyshev polynomial. The results obtained are compared with the finite element solutions of the same problem and the solutions of other scholars to verify the correctness of the equation. Results: The results show that the intensity factor solution obtained by the proposed method is in good agreement with the finite element solution and other scholars' solutions. A larger crack length, smaller crack spacing, smaller friction coefficient and smaller lateral pressure coefficient can produce larger stress intensity factors, and the stress intensity factors are maximized when the crack angle is 45°. Conclusions: It is proved that the method based on the initial solution and superposition principle can be effectively applied to the calculation of the stress intensity factors of multiple collinear cracks. [ABSTRACT FROM AUTHOR]

Spectral relationships explain many physical phenomena, especially in quantum physics and astrophysics. Therefore, in this paper, we first attempt to derive spectral relationships in position and time for an integral operator with a singular kernel. Second, using these relations to solve a mixed integral equation (MIE) of the second kind in the space L2[-1,1] x C[0, T], T < 1. The way to do this is to derive a general principal theorem of the spectral relations from the term of the Volterra-Fredholm integral equation (V-FIE), with the help of the Chebyshev polynomials (CPs), and then use the results in the general MIE to discuss its solution. More than that, some special and important cases will be devised that help explain many phenomena in the basic sciences in general. Here, the FI term is considered in position, in L2[-1,1], and its kernel takes a logarithmic form multiplied by a general continuous function. While the VI term in time, in C[0, T], T < 1, and its kernels are smooth functions. Many numerical results are considered, and the estimated error is also established using Maple 2022. [ABSTRACT FROM AUTHOR]

This study focuses on the numerical solution of the space–time variable-order fractional derivative mobile–immobile advection–dispersion equation. These equations are preferred for dynamical system modeling, which includes determining the solute transport mechanism. We take into account the Caputo class of fractional derivatives. A Chebyshev collocation method in both the space and time directions is utilized for the approximate solution of the equation. The convergence analysis of the method is shown by using the Chebyshev-weighted Sobolev space. Finally, some examples are provided to illustrate the accuracy and efficiency of the present approach. Comparisons between the results obtained by our method and those obtained by the existing methods are presented to demonstrate the superiority of the proposed methodology. The spectral or exponential convergence of the suggested approach is one of its essential characteristics, which is verified through numerical results and provides additional support for the effectiveness of the suggested technique. [ABSTRACT FROM AUTHOR]

We propose a multiscale spatio graph neural network (MSGCN) for 3D point cloud. The core of MSGCN is a multiscale spatio graph(MSG) that explicitly models the relations at various spatial scales. Different from many previous hierarchical structures, the MSG is built in a data adaptive fashion. MSG supports multiscale analysis of point clouds in the scale space and can obtain the dimensional features of point cloud data at different scales. Because traditional convolutional neural networks are not applicable to graph data with irregular vertex neighborhoods, this paper presents an sef-adaptive graph convolution kernel that uses the Chebyshev polynomial to fit an irregular convolution filter based on the theory of optimal approximation. In experiments conducted on four widely used public datasets, The results show that the proposed model outperforms most state-of-the-art methods. [ABSTRACT FROM AUTHOR]

In this paper, a collocation method based on shifted second‐order Chebyshev polynomials is implemented to obtain the approximate solution of the stochastic Itô–Volterra integral equation of Abel type with weakly singular kernel. In this method, operational matrices are used to convert the stochastic Itô–Volterra integral equation to algebraic equations that are linear. The algorithm of the proposed numerical scheme has been presented in this paper. Also, the error bound and convergence of the proposed method are well established. Consequently, two illustrative examples are provided to demonstrate the efficiency, plausibility, reliability, and consistency of the current methodology. [ABSTRACT FROM AUTHOR]

Copyright of Machine Tool & Hydraulics is the property of Guangzhou Mechanical Engineering Research Institute (GMERI) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The inherent non-smoothness of the vibroimpact system leads to complex behaviors and a strong sensitivity to parameter changes. Unfortunately, uncertainties and errors in system parameters are inevitable in mechanical engineering. Therefore, investigations of dynamical behaviors for vibroimpact systems with stochastic parameters are highly essential. The present study aims to analyze the dynamical characteristics of the three-degree-of-freedom vibroimpact system with an uncertain parameter by means of the Chebyshev polynomial approximation method. Specifically, the vibroimpact system model considered is one with unilateral constraint. Firstly, the three-degree-of-freedom vibroimpact system with an uncertain parameter is transformed into an equivalent deterministic form using the Chebyshev orthogonal approximation. Then, the ensemble means responses of the stochastic vibroimpact system are derived. Numerical simulations are performed to verify the effectiveness of the approximation method. Furthermore, the periodic and chaos motions under different system parameters are investigated, and the bifurcations of the vibroimpact system are analyzed with the Poincaré map. The results demonstrate that both the restitution coefficient and the random factor can induce the appearance of the periodic bifurcation. It is worth noting that the bifurcations fundamentally differ between the stochastic and deterministic systems. The former has a bifurcation interval, while the latter occurs at a critical point. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a subclass L (q , b , α , t) by means of Chebyshev polynomials and q - derivative operator expansions of univalent functions of complex order in the open unit disk. Later, we find initial coefficients | a 2 | and | a 3 | and sharp bounds of Fekete–Szegő functional | a 3 - ξ a 2 2 | for functions in this subclass. [ABSTRACT FROM AUTHOR]

This study explores efficient methods for computing eigenvalues and function values associated with Chebyshev-type prolate spheroidal wave functions (CPSWFs). Applying the expansion of the factor eicxy and the inherent properties of Chebyshev polynomials, we present an exact and stable numerical approximation for the exact eigenvalues of the integral operator to CPSWFs. Additionally, we illustrate the efficiency of employing fast Fourier transform and barycentric interpolation techniques for computing CPSWF values and related quantities, which are essential for various numerical applications based on these functions. The analysis is supported by numerical examples, providing validation for the accuracy and reliability of our proposed approach.

We prove that the walk matrix of the Dynkin graph D n has the Smith normal form diag [ 1 , 1 , ... , 1 ︸ n 2 − 1 , 2 , 2 , ... , 2 ︸ n 2 − 1 , 0 , 0 ] when n ≡ 0 (mod 4). This gives an affirmative answer to a question in (Wang et al., 2022 [22]). [ABSTRACT FROM AUTHOR]

The exact decrease rate of the best approximations of non-integer numbers by polynomials with integer coefficients of growing degrees is found on a disk in the complex plane, a cube in ℝd, and a ball in ℝd. The sup-norm is used in the first two cases, and the norm in Lp, 1 ≤ p < ∞, is applied in the third one. Detailed comments are given (two remarks at the end of the paper). [ABSTRACT FROM AUTHOR]

In this paper, a new method to compute lower and upper bounds for Salem numbers with a given trace and a given degree is given. With this method, it is proven that the smallest trace of Salem numbers of degree 22 is -1. Further, new lower bounds for degree of Salem numbers with minimal trace -5 and -6 are given. All Salem numbers of trace -2 and degree 24, 26 are given. This includes 7 additional Salem numbers of degree 26 beyond what was previously known. The auxiliary functions related to Chebyshev polynomials, which are adapted to Salem number, are used in this work. [ABSTRACT FROM AUTHOR]

• A formula for characterizing the excitation condition of the nth-order Bragg resonance is derived. • The influence of various bed parameters on wave reflection, including Bragg resonant reflection, is investigated. • Zero reflections are closely related to the zero points of the Chebyshev polynomial of the second kind. • When there is no slope, there are N − 1 zero reflections between two Bragg peaks, otherwise all zero reflections disappear. • Higher-order Bragg resonance peaks beat faster, and the nth-order Bragg resonance peak has n local maxima. For water waves propagating over a finite periodic array of congruent trapezoidal bars on a sloping seabed or trenches dug on a sloping seabed, an analytical solution in terms of Taylor series to the modified mild-slope equation (MMSE) is constructed. Based on the analytical solution, influences of various topographic parameters (e.g., the slope of seabed, bar/trench number, bar height or trench depth, and bar/trench width) on the peak value, peak phase and zero reflection of Bragg resonances are investigated. Especially, starting from the original definition of Bragg resonances, a modified Bragg's law for characterizing the excitation condition of the n th order Bragg resonance is derived, which shows that the excitation of Bragg resonances depends entirely on the average phase velocity of the water wave within the bar/trench field. When the influence of the bar/trench field on the average phase velocity is quite small, the modified Bragg's law degenerates into the traditional Bragg's law borrowed from X-ray crystallography, which also reflects the essential difference between the water wave as a mechanical wave and the non-mechanical X-ray wave. In addition, for N bars/trenches on a flat seabed, there are at least N − 1 zero reflections between any two adjacent Bragg resonance peaks. When the bars are very low or the trenches are very shallow, the phase of the N − 1 zero reflections in those zero reflections is exactly the position of the N − 1 zero points of the Chebyshev polynomial of the second kind, U N − 1 (cos k 0 d) , where k 0 is the wavenumber on the flat seabed and d is the bar/trench spacing. However, once a sloping seabed is added, all these N − 1 zero reflections disappear possibly due to the destruction of the symmetry of the background bar/trench field. Finally, regardless of the number of bars/trenches, when the bar/trench width varies, the peak value of the n th order Bragg resonance oscillates periodically, leading to n local maxima. A comparative study shows that these characteristics are consistent with the band structure shown in the Bloch periodic gap map (PGM). [ABSTRACT FROM AUTHOR]

A system of hypersingular integral equations gets formed while solving curved crack or multi-crack problems in fracture mechanics. However, the issue is that its analytical solution is only available for some particular cases. For this, we develop a residual-based Galerkin's numerical method using Chebyshev polynomials of the second kind with its complete analysis. We prove the system's well-posedness and the theoretical and numerical convergence of the approximation scheme. We derive the convergence rate, validate error bounds via test examples, and compare them with the Legendre polynomial-based numerical method in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the interfacial behavior of a thin one-dimensional (1D) hexagonal quasicrystal (QC) film bonded on an elastic substrate subjected to a mismatch strain due to thermal variation. The contact interface is assumed to be non-slipping, with both perfectly bonded and debonded boundary conditions. The Fourier transform technique is adopted to establish the integral equations in terms of interfacial shear stress, which are solved as a linear algebraic system by approximating the unknown phonon interfacial shear stress via the series expansion of the Chebyshev polynomials. The expressions are explicitly obtained for the phonon interfacial shear stress, internal normal stress, and stress intensity factors (SIFs). Finally, based on numerical calculations, we briefly discuss the effects of the material mismatch, the geometry of the QC film, and the debonded length and location on stresses and SIFs. [ABSTRACT FROM AUTHOR]

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The crack tip stress intensity factor is an important criterion for judging crack extension and structural failure. Investigating the crack tip stress intensity factor for the interaction between circular hole and crack under tensile loading is of great significance for material fracture criterion and residual strength analysis. Based on the superposition principle and the initial solution of elastodynamics, the integral equation of the stress intensity factors of the crack at the edge of a uniaxially tensile circular hole plate was obtained by using the Westergaard stress function, and the approximate solution of the integral equation was obtained by using Chebyshev polynomials, and the approximate solution was modified by using the exponential function to obtain the modified solution of the stress intensity factors at the crack tip. The same problem was simulated using Abaqus and compared with the corrected solution results. The effects of crack size, circular hole radius, crack location angle, and crack inclination on the stress intensity factor at the crack tip were analyzed. The results show that the modified solution and the Abaqus simulation solution basically agree. The stress intensity factors increases with the increase of crack size and circular hole radius, and decreases with the increase of crack location angle and crack inclination angle. [ABSTRACT FROM AUTHOR]

Excitation identification for nonlinear structures is still a challenging problem due to the convergence and accuracy in this process. In this study, a load estimation method is proposed with orthogonal decomposition, the order for which can be fairly accurately determined by a regression. In this process, the force time history is represented by the orthogonal basis and the coefficients of the orthogonal decomposition are taken as unknowns and augmented to the state variable, which can be identified recursively in state space. A general energy-conserving method is selected to a step-by-step integration to guarantee the convergence of this integration. The proposed method is first validated by numerical simulation studies of a truss structure considering its geometric property. The identification results of the numerical studies demonstrate that the proposed excitation identification method and the orthogonal decomposition order determination method work well for nonlinear structures. The laboratory work of a 7-story frame is investigated to consider the geometric nonlinearity in impact force identification. The results of experimental studies show that uncertainties such as measurement noise and model error are included in the investigation of the accuracy and robustness of the proposed force identification method, while the time history of external forces could be identified with promising results. [ABSTRACT FROM AUTHOR]

The utilization of Internet of Things (IoT) devices in various smart city and industrial applications is growing rapidly. Within a trusted authority (TA), such as an industry or smart city, all IoT devices are closely monitored in a controlled infrastructure. However, in cases where an IoT device from one TA needs to communicate with another IoT device from a different TA, the trust establishment between these devices becomes extremely important. Obtaining a digital certificate from a certificate authority for each IoT device can be expensive. To solve this issue, a group authentication framework is proposed that can establish trust between group IoT devices owned by different entities. The Chebyshev polynomial has many important properties, semigroup is one of the most important. These properties make the Chebyshev polynomial a good candidate for the proposed group authentication mechanism. The secure exchange of information between trusted authorities is supported by Blockchain technology. The proposed framework was implemented and tested using Python and deployed on Blockchain using Ethereum's Goerli's testnet. The results show that the proposed framework can reasonably use Chebyshev polynomials with degrees up to four digits in length. The values of various parameters related to Blockchain are also discussed to understand the usability of the proposed framework. [ABSTRACT FROM AUTHOR]

The finite block method (FBM) with Chebyshev polynomials is developed in this paper for the functionally graded material (FGM) of two-dimensional fracture problem. The main idea is to divide domain into few blocks. By introducing the mapping technique, a block of quadratic type is transformed from Cartesian coordinate to normalised coordinate with eight seeds two dimensions. The differential matrices in physical domain are determined from that in the normalised transformed system. The accuracy of the FBM with Chebyshev polynomials is demonstrated with two examples and comparisons have been implemented with different numerical approaches. [ABSTRACT FROM AUTHOR]