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In this paper, we obtain initial coefficient bounds for functions belong to a subclass of bi-univalent functions by using the Salagean differential operator and Chebyshev polynomials and also we find Fekete-Szego inequalities for functions in this class. [ABSTRACT FROM AUTHOR]

In this paper, the authors introduce the newly constructed subclass of bi-univalent functions defined by the Jackson (p,q)-derivative operator of Salagean type associated with Chebyshev polynomial. The initial coefficient bounds and Fekete-Szegö inequalities for the function class are obtained. Moreover, certain special cases are also pointed out. The results present in this paper generalizes and improves the results due to Altinkaya and Yalcin. [ABSTRACT FROM AUTHOR]

Inertial navigation computation is to acquire the attitude, velocity, and position information of a moving body by integrating inertial measurements from gyroscopes and accelerometers. Over half a century has witnessed great efforts in coping with the motion noncommutativity errors to accurately compute the navigation information as far as possible, so as not to compromise the quality measurements of inertial sensors. Highly dynamic applications and the forthcoming cold-atom precision inertial navigation systems demand for even more accurate inertial navigation computation. The paper gives birth to an inertial navigation algorithm to fulfill that demand, named the iNavFIter, which is based on a brand-new framework of functional iterative integration and Chebyshev polynomials. Remarkably, the proposed iNavFIter reduces the noncommutativity errors to almost machine precision, namely, the coning/sculling/scrolling errors that have perplexed the navigation community for long. Numerical results are provided to demonstrate its accuracy superiority over the state-of-the-art inertial navigation algorithms at affordable computation cost. [ABSTRACT FROM AUTHOR]

In mathematics, one always tries to get new structures from given ones. This also applies to the realm of graphs, where one can generate many new graphs from a given set of graphs. In this paper we define some classes of pyramid graphs and we derive simple formulas of the complexity, number of spanning trees, of these graphs, using linear algebra, Chebyshev polynomials and matrix analysis techniques. [ABSTRACT FROM AUTHOR]

Copyright of Journal of South China University of Technology (Natural Science Edition) is the property of South China University of Technology 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.)

• A simple LVIM is proposed for solving nonlinear differential equations. • The LVIM shows significant computational superiority over highly optimized FDM. • A highly efficient framework of parallel computational is involved in the LVIM. • Potential of LVIM in high performance computing of nonlinear systems is revealed. A very simple and efficient local variational iteration method (LVIM), or variational iteration method with local property, for solving problems of nonlinear science is proposed in this paper. The analytical iteration formula of this method is derived first using a general form of first order nonlinear differential equations, followed by straightforward discretization using Chebyshev polynomials and collocation method. The resulting numerical algorithm is very concise and easy to use, only involving highly sparse matrix operations of addition and multiplication, and no inversion of the Jacobian in nonlinear problems. Apart from the simple yet efficient iteration formula, another extraordinary feature of LVIM is that in each local domain, all the collocation nodes participate in the calculation simultaneously, thus each local domain can be regarded as one "node" in calculation through GPU acceleration and parallel processing. For illustration, the proposed algorithm of LVIM is applied to various nonlinear problems including Blasius equations in fluid mechanics, buckled bar equations in solid mechanics, the Chandrasekhar equation in astrophysics, the low-Earth-orbit equation in orbital mechanics, etc. Using the built-in highly optimized ode45 function of MATLAB as a comparison, it is found that the LVIM is not only very accurate, but also much faster by an order of magnitude than ode45 in all the numerical examples, especially when the nonlinear terms are very complicated and difficult to evaluate. [ABSTRACT FROM AUTHOR]

Let F (x)   =   ∑ n = 1 ∞ τ s 1 , s 2 ,   ...   , s k (n) x n be the generating function for the number τ s 1 , s 2 ,   ...   , s k (n) of spanning trees in the circulant graph Cn(s1, s2, ..., sk). We show that F(x) is a rational function with integer coefficients satisfying the property F(x) = F(1/x). A similar result is also true for the circulant graphs C2n(s1, s2, ..., sk, n) of odd valency. We illustrate the obtained results by a series of examples. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Jilin University (Science Edition) / Jilin Daxue Xuebao (Lixue Ban) is the property of Zhongguo Xue shu qi Kan (Guang Pan Ban) Dian zi Za zhi She 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.)

We propose a novel self-shrinking chaos based pseudo-random number output algorithm. The result of the analysis shows that the presented generator ensures a secure way for sending electronic information with critical applications in data encryption. [ABSTRACT FROM AUTHOR]

Numerical analysis is the area of mathematics that creates, analyzes, and implements algorithms for solving numerically the problems from real-world applications of algebra, geometry, and calculus, and they involve variables which vary continuously. Till now, numerous numerical methods have been introduced. Spectral method is one of those techniques used in applied mathematics and scientific computing to numerically solve certain differential equations, potentially involving the use of the Fast Fourier Transform (FFT). This study presents some of the fundamental ideas of spectral method. Orthogonal basis are used to establish computational algorithm. The accuracy and efficiency of proposed model are discussed. This manuscript estimates for the error between the exact and approximated discrete solutions. This paper shows that, grid points for polynomial spectral methods should lie approximately in a minimal energy configuration associated with inverse linear repulsion between points. The wave equation, linear and non-linear boundary value problems are solved using spectral method on the platform of MATLAB language. [ABSTRACT FROM AUTHOR]

In this paper, a high gain cascaded single stage distributed amplifier design method is proposed. This new design method is based on the Chebyshev polynomial approximation of the amplifier transducer gain. By this approximation an improvement of 12 dB in gain, is obtained, compared with the conventional cascaded single stage distributed amplifier with a simplified unilateral single field effect transistor model. This 12 dB are obtained by eliminating two resistors which consume the power; but, by doing this, a mismatch is provoked in the circuit, so a loss of gain is obtained at high frequencies. By the use of the proposed method a stable gain is obtained from dc to the cut-off frequency of the circuit. In addition, a more compact circuit is obtained, by eliminating four components, compared with the cascaded single stage distributed amplifier. [ABSTRACT FROM AUTHOR]

Let K be a quasidisk on the complex plane. We construct a sequence of monic polynomials p n = p n (· , K) with all their zeros on K such that ‖ p n ‖ K ≤ O (1) cap (K) n as n → ∞ . [ABSTRACT FROM AUTHOR]

RodFIter is a promising method of attitude reconstruction from inertial measurements based on the functional iterative integration of Rodrigues vector. The Rodrigues vector is used to encode the attitude in place of the popular rotation vector because it has a polynomial-like rate equation and could be cast into theoretically sound and exact integration. This paper further applies the approach of RodFIter to the unity-norm quaternion for attitude reconstruction, named QuatFIter, and shows that it is identical to the previous Picard-type quaternion method. The Chebyshev polynomial approximation and truncation techniques from the RodFIter are exploited to speed up its implementation. Numerical results demonstrate that the QuatFIter is comparable in accuracy to the RodFIter, although its convergence rate is relatively slower with respect to the number of iterations. Notably, the QuatFIter has about two times better computational efficiency, thanks to the linear quaternion kinematic equation. [ABSTRACT FROM AUTHOR]

Numerical inverse Laplace transform is employed in solving some fractional order differential equations that convert the linear fractional differential equations into the linear system of algebraic equations. The unknown function or the solution of fractional differential equations can be expressed in a series of orthogonal polynomials; Chebyshev polynomials of the second kind are so developed using odd cosine series. The efficacy of the technique is well tested on initial value problems of fractional order differential equations, fractional oscillation equation and system of a fractional algebraic equation. We have also calculated the L∞-errors of obtained results which shows that the proposed method produces satisfactory results. [ABSTRACT FROM AUTHOR]

• The definition of Implicit Chebyshev Moments (ICMs) was given. • The algorithm for determination of degrees of implicit Chebyshev polynomials was proposed. • The method of generating level sets from original shapes was proposed. • Shapes were efficiently represented by ICMs based on level sets and boundary points on shapes. • Geometric invariants were derived from ICMs. Chebyshev Moments(CMs) have been applied to representation and recognition of 2D object shapes in image processing and computer vision. However they still suffer from poor representation power and difficulty in computing invariants for shapes. In this work, we present Implicit Chebyshev Moments (ICMs) to overcome these issues. Firstly, we use Euclid distance transformation to generate a series of level sets based on a given shape. Secondly, we fit an implicit Chebyshev polynomial to the data set consisting of the obtained level sets together with all the boundary points on the original shape and call the obtained coefficients of the fitted implicit Chebyshev polynomial ICMs. Finally, we propose a new approach to derive geometric invariants based on ICMs. In addition, we also develop an algorithm for the determination of a suitable degree for implicit Chebyshev polynomials before representing a given shape. Experimental results show the ICMs are more efficient for representing complex shapes than CMs. [ABSTRACT FROM AUTHOR]

The Chebyshev problem on the square Π = {z = x + iy ∈ ℂ: max{∣x∣, ∣y∣} ≤ 1} of the complex plane ℂ is studied. Let p n ∈ P n be the set of algebraic polynomials of a given degree n with the unit leading coefficient. The problem is to find the smallest value τn(Π) of the uniform norm ∥pn∥C(π) of polynomials P n on the square Π and a polynomial with the smallest norm, which is called a Chebyshev polynomial (for the square). The Chebyshev constant τ (Q) = lim n → ∞ τ n (Q) n for the square is found. Thus, the logarithmic asymptotics of the least deviation τn(Π) with respect to the degree of a polynomial is found. The problem is solved exactly for polynomials of degrees from 1 to 7. The class of polynomials in the problem is restricted; more exactly, it is proved that, for n = 4m + s, 0 ≤ s ≤ 3, it is sufficient to solve the problem on the set of polynomials zsqm(z), q m ∈ P m . Effective two-sided estimates for the value of the least deviation τn (Π) with respect to n are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we develop Galerkin's residual-based numerical scheme for solving a system of Cauchy-type singular integral equations of index minus N using Chebyshev polynomials of the first and second kind, where N is the total number of Cauchy-type singular integral equations in the system. Without theoretical analysis, a numerical scheme is not justified. Therefore, first, we prove the well-posedness of the system of Cauchy-type singular integral equations with the help of the compactness of an operator. Further, we derive a theoretical error bound and the order of convergence. Also, we show that the resulting system of equations obtained by applying the algorithm is well-posed together with an explicit representation of the solution in matrix form. Finally, we give some illustrative examples to validate the theoretical error bounds numerically. • A residual based Galerkin's method using Chebyshev polynomials has been developed. • A system of Cauchy singular integral equations (SCSIE) has been numerically solved. • The well-posedness of the SCSIE has been shown using the operator theory. • A well-posedness of the system of the resulting algebraic equations has been shown. • The error bounds as well as the rate of convergence has been derived. • The conditions under which the exact solution of the SCSIE has been obtained. [ABSTRACT FROM AUTHOR]

As a measure of vertex importance according to the graph structure, PageRank has been widely applied in various fields. While many PageRank algorithms have been proposed in the past decades, few of them take into account whether the graph under investigation is directed or not. Thus, some important properties of undirected graph—symmetry on edges, for example—is ignored. In this paper, we propose a parallel PageRank algorithm specifically designed for undirected graphs that can fully leverage their symmetry. Formally, our algorithm extends the Chebyshev Polynomial approximation from the field of real function to the field of matrix function. Essentially, it reflects the symmetry on edges of undirected graph and the density of diagonalizable matrix. Theoretical analysis indicates that our algorithm has a higher convergence rate and requires less computation than the Power method, with a 50% higher convergence rate when the damping factor c = 0.85. Experiments on six datasets illustrate that our algorithm with 38 parallelism can be up to 43 times faster than the single-thread Power method. • We propose an approach of computing PageRank via Chebyshev Polynomial approximation. • Based on this approach, we propose a parallel PageRank algorithm and provide an accurate convergence rate. • We conduct numerical experiments to validate our theoretical findings, and the results align with our expectations. [ABSTRACT FROM AUTHOR]

This paper mainly studies numerical solution to the Volterra integral equation of the second kind. By using the roots of Chebyshev polynomial as collocation points, a new collocation method is proposed to solve the Volterra integral equation of the second kind. The proposed method firstly interpolates the product of the kernel function and the unknown solution at the roots of Chebyshev polynomial. Then, the Volterra integral equation is transformed into a system of linear algebra equations by properties of Chebyshev polynomials. Finally, the numerical solution of the Volterra integral equation is obtained by the Chebyshev polynomial interpolation. In addition, the error estimates of the proposed method are provided in a semi-posteriori sense; and numerical examples are given to show effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

The place of public key cryptography (PKC) in guaranteeing the security of wireless networks under human-centered IoT environments cannot be overemphasized. PKC uses the idea of paired keys that are mathematically dependent but independent in practice. In PKC, each communicating party needs the public key and the authorized digital certificate of the other party to achieve encryption and decryption. In this circumstance, a directory is required to store the public keys of the participating parties. However, the design of such a directory can be cost-prohibitive and time-consuming. Recently, identity-based encryption (IBE) schemes have been introduced to address the vast limitations of PKC schemes. In a typical IBE system, a third-party server can distribute the public credentials to all parties involved in the system. Thus, the private key can be harvested from the arbitrary public key. As a result, the sender could use the public key of the receiver to encrypt the message, and the receiver could use the extracted private key to decrypt the message. In order to improve systems security, new IBE schemes are solely desired. However, the complexity and cost of designing an entirely new IBE technique remain. In order to address this problem, this paper presents a provably secure IBE transformation model for PKC using conformable Chebyshev chaotic maps under the human-centered IoT environment. In particular, we offer a robust and secure IBE transformation model and provide extensive performance analysis and security proofs of the model. Finally, we demonstrate the superiority of the proposed IBE transformation model over the existing IBE schemes. Overall, results indicate that the proposed scheme posed excellent security capabilities compared to the preliminary IBE-based schemes.

The normal mode model is important in computational atmospheric acoustics. It is often used to compute the atmospheric acoustic field under a time-independent single-frequency sound source. Its solution consists of a set of discrete modes radiating into the upper atmosphere, usually related to the continuous spectrum. In this article, we present two spectral methods, the Chebyshev-Tau and Chebyshev-Collocation methods, to solve for the atmospheric acoustic normal modes, and corresponding programs are developed. The two spectral methods successfully transform the problem of searching for the modal wavenumbers in the complex plane into a simple dense matrix eigenvalue problem by projecting the governing equation onto a set of orthogonal bases, which can be easily solved through linear algebra methods. After the eigenvalues and eigenvectors are obtained, the horizontal wavenumbers and their corresponding modes can be obtained with simple processing. Numerical experiments were examined for both downwind and upwind conditions to verify the effectiveness of the methods. The running time data indicated that both spectral methods proposed in this article are faster than the Legendre-Galerkin spectral method proposed previously.

Abstract We investigate the existence of solutions for a sum-type fractional integro-differential problem via the Caputo differentiation. By using the shifted Legendre and Chebyshev polynomials, we provide a numerical method for finding solutions for the problem. In this way, we give some examples to illustrate our results.

Abstract We apply the Chebyshev polynomial-based differential quadrature method to the solution of a fractional-order Riccati differential equation. The fractional derivative is described in the Caputo sense. We derive and utilize explicit expressions of weighting coefficients for approximation of fractional derivatives to reduce a Riccati differential equation to a system of algebraic equations. We present numerical examples to verify the efficiency and accuracy of the proposed method. The results reveal that the method is accurate and easy to implement.

This report summarize and compares with each other various methods for computing the symmetrical Zolotarev Polynomial of the first kind and its spectrum. Suitable criteria are suggested for the comparison. The best numerical stability shows the method employing Chebyshev polynomial recurrence. In case of the polynomial spectrum computation the best method is the one using the difference backward recursion introduced by M. Vlcek. Both methods are able to generate the polynomial of high degree up to, at least, 2000, using 32-bit IEEE floating point arithmetics.

Expansion of measured dynamic response at a limited set of points is of great interest in the study of structural dynamic systems. Measurements at limited points on a structure can be expanded to unmeasured points by using System Equivalent Reduction Expansion Process (SEREP). However, a finite element model is required to obtain the mode shapes of the system. In this work, a non-model based expansion technique is proposed to expand the vibration characteristics at a sparse set of points to a much larger set of points without the use of a finite element model. Shape functions based on orthogonal polynomials can be used to decompose the deformation of vibrating structures into the summation of a set of shape functions with corresponding weighting coefficients. To obtain the vibration characteristics at an augmented set of points, measurements at a sparse set of points can be related to a set of shape functions which have the same mesh resolution as the sparse configuration. The vibration characteristics at an augmented set of points can then be reconstructed from the shape functions of the high resolution and scaled shape function weighting coefficients of the sparse configuration. Only a sparse set of measurements, geometry and coordinates of measured points are needed. In the work studied here, an analytical plate and a Base-Upright (BU) structure are used as samples to study the application of the proposed method. Both analysis and experiment are used to prove the concept. [ABSTRACT FROM AUTHOR]

The standard backpropagation algorithm has already proven its effectiveness in most of the potential problems, but the major limitation is entrapment of local minima and slow convergence rate. To address these issues, a modified backpropagation algorithm has been proposed by adding a third term called inertia, the physical component used to accelerate the network towards the convergence without getting stuck into local minima. The Chebyshev polynomial form is a convenient method for expanding a function in a linear independent term. Inertia has been expanded using Chebyshev polynomial which is used as a third term in weight updation. The performance of the proposed algorithm outperforms the standard backpropagation algorithm (SBP) and the backpropagation algorithm with momentum (SBPM). The proposed algorithm was tested with the standard benchmark problems such as XOR problem, parity checking problem and dataset from UCI machine learning repository such as iris flower classification, wheat classification, breast cancer detection and wine classification. Experimental results show that the addition of the third parameter called inertia in the backpropagation algorithm gave better performance and faster convergence rate compared to the SBP and SBPM. [ABSTRACT FROM AUTHOR]

The union of near-field communication (NFC) and mobile devices has led to significant changes in payment systems over recent years. Currently, NFC-based services are the leading form of mobile payment method. In particular, many companies that use electronic payment services are adopting NFC systems to replace credit cards. Additionally, the safety of communication has been enhanced by using standard techniques to activate NFC services. The properties of mobile NFC payments provide a business model for the Internet of Things (IoT) environment. However, electronic payment methods based on NFC are still vulnerable to various security threats. One example is the case of credit card data hacking under the KS X 6928 standard. In particular, the security level of the NFC payment method in passive mode is limited by the storage, power consumption, and computational capacity of the low-cost tags. Recently, chaotic encryption based on Chebyshev polynomials has been used to address certain security issues. Our proposed scheme is based on the Chebyshev chaotic map, unlike traditional encryption protocols that apply complex cryptography algorithms. Considering the tag limitations, the hash, XOR, and bitwise operations in the proposed scheme provide high-level security for payment environments. We propose a security-enhanced transaction scheme based on Chebyshev polynomials for mobile payment services in an IoT device environment considering the signature record-type definition and KS X 6928 standard. [ABSTRACT FROM AUTHOR]

In this paper, a Chebyshev polynomial-based functional link artificial neural network (CFLANN) technique for Wyner–Ziv (WZ) frame estimation in a distributed video coding framework is proposed. The estimated WZ frame at the decoder is also referred to as the side information (SI). The proposed scheme (CFLANN-SI) works in two phases, namely, training and testing. The network is trained offline, and to achieve better generalization, the training (input, target) patterns are created across several video sequences constituting varied motion behavior. It estimates the SI frame using adjacent key frames as inputs. The training convergence characteristics of CFLANN-SI is observed to be faster with reduced mean square error as compared to a multi-layer perceptron-based prediction scheme. It is also observed that once the model is trained, it is capable of estimating SI for rest of the incoming WZ frames of the video sequences as well as for the video sequences which are not considered during the learning phase. The proposed scheme is evaluated with respect to different parameters, namely, rate-distortion, peak-signal-to-noise-ratio, the number of parity requests made per estimated frame, decoding time requirement and so on. Comparative analysis shows that the present CFLANN-SI technique generates better SI in resemblance to the competent schemes, in terms of the subjective quality improvement as well as the objective quality gains. Further, to substantiate that the present scheme provides a significant improvement over that of the benchmark techniques, a statistical analysis tool is used with a significance level of 5%. [ABSTRACT FROM AUTHOR]

This paper introduces an efficient algorithm for finding the dominant generalized eigenvectors of a pair of symmetric matrices. Combining tools from approximation theory and convex optimization, we develop a simple scalable algorithm with strong theoretical performance guarantees. More precisely, the algorithm retains the simplicity of the well-known power method but enjoys the asymptotic iteration complexity of the powerful Lanczos method. Unlike these classic techniques, our algorithm is designed to decompose the overall problem into a series of subproblems that only need to be solved approximately. The combination of good initializations, fast iterative solvers, and appropriate error control in solving the subproblems lead to a linear running time in the input sizes compared to the superlinear time for the traditional methods. The improved running time immediately offers acceleration for several applications. As an example, we demonstrate how the proposed algorithm can be used to accelerate canonical correlation analysis, which is a fundamental statistical tool for learning of a low-dimensional representation of high-dimensional objects. Numerical experiments on real-world datasets confirm that our approach yields significant improvements over the current state of the art. [ABSTRACT FROM AUTHOR]

In this paper, we develop a new method to produce explicit formulas for the number τ (n) of spanning trees in the undirected circulant graphs C n (s 1 , s 2 , ... , s k) and C 2 n (s 1 , s 2 , ... , s k , n). Also, we prove that in both cases the number of spanning trees can be represented in the form τ (n) = p n a (n) 2 , where a (n) is an integer sequence and p is a prescribed natural number depending on the parity of n. Finally, we find an asymptotic formula for τ (n) through the Mahler measure of the associated Laurent polynomial L (z) = 2 k − ∑ j = 1 k (z s j + z − s j ). [ABSTRACT FROM AUTHOR]

In recent years, there has been an incredible enthusiasm towards fractional order partial differential equations because of their incessant presence alongside different fields. Fractional derivatives offer an in-depth and precise analysis of the models of the systems. Particularly, fractional order telegraph equations (FOTE) have been taken into consideration and solved by plenty of researchers, using different techniques. In this paper, we present a novel approach and technique to solve fractional telegraph equation by fusion of cubic radial basis function and Chebyshev polynomials with the aid of Kronecker product. The numerical examples have been considered to verify the accuracy and also to demonstrate the performance of the new approach. [ABSTRACT FROM AUTHOR]

Cryptography is the most peculiar way to secure data and most of the encryption algorithms are mainly used for textual data and not suitable for transmission data such as images. It is seen that the generation of secure key in Image cryptography has been a challenging task in the way of providing secured key generation for the transmitted data. In order to aid secured key generation in this context, an optimized secret key generation based on Chebyshev polynomial with Adaptive Firefly (FF) optimization technique is proposed. The optimized key is utilized with process of shuffling, diffusion, and swapping to get a better encrypted image. At the receiver end, reverse process is applied with optimized key to retrieve the original input image. The efficiency of our proposed method is assessed by the exhaustive experimental study. The results show that the proposed methodology provided correlation coefficient of 0.21, Number of Pixels Change Rate (NPCR) of 0.996, Unified Average Changing Intensity (UACI) of 0.3346 and Information Entropy of 7.995 as compared with the existing methods. [ABSTRACT FROM AUTHOR]

In this research, a modified rational interpolation method for the numerical solution of initial value problem is presented. The proposed method is obtained by fitting the classical rational interpolation formula in Chebyshev polynomials leading to a new stability function and new scheme. Three numerical test problems are presented in other to test the efficiency of the proposed method. The numerical result for each test problem is compared with the exact solution. The approximate solutions are show competitiveness with the exact solutions of the ODEs throughout the solution interval. [ABSTRACT FROM AUTHOR]

In this work, we obtain an upper bound estimate for the second Hankel determinant of a subclass 𝒩 σ μ (λ , t) of analytic bi-univalent function class σ which is associated with Chebyshev polynomials in the open unit disk. [ABSTRACT FROM AUTHOR]

Attitude computation is of vital importance for a variety of applications. Based on the functional iteration of the Rodrigues vector integration equation, the RodFIter method can be advantageously applied to analytically reconstruct the attitude from discrete gyroscope measurements over the time interval of interest. It is promising to produce ultra-accurate attitude reconstruction. However, the RodFIter method imposes a high computational load and does not lend itself to an onboard implementation. In this paper, a fast approach to significantly reduce RodFIter's computation complexity is presented while maintaining almost the same accuracy of attitude reconstruction. It reformulates the Rodrigues vector iterative integration in terms of the Chebyshev polynomial iteration. Due to the excellent property of Chebyshev polynomials, the fast RodFIter is achieved by means of appropriate truncation of Chebyshev polynomials, with provably guaranteed convergence. Moreover, simulation results validate the speed and accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

Dynamic analysis of soft-landing is very important for the design of lunar lander. At present, the determined landing attitude and speed are considered while not considering the uncertainty of these parameters in the analysis of soft-landing dynamics. Based on Chebyshev interval analysis method, an analysis process of landing dynamic interval based on nonlinear finite-element model is proposed for the dynamic characteristics of landing process. The upper and lower bounds of dynamic response are calculated using Chebyshev method and compared with the simulated results of Monte Carlo method. Comparative result shows that the analyzed results of Chebyshev interval analysis method can fully cover those of Monte Carlo method, and the dynamic interval is not enlarged. The influence of truncation order on the analytic error of dynamic interval was analyzed. The analyzed result shows that the truncation order has little influence on analysis error. Chebyshev method has the advantage of high accuracy and efficiency. [ABSTRACT FROM AUTHOR]

In this paper, a Galerkin-type approximation using induced rational functions of Chebyshev polynomials is proposed and analyzed in order to determine the solution of the heat equation over a whole R. We have shown by numerical tests that these new rational functions are very well adapted to approximations of PDEs in unbounded domain. [ABSTRACT FROM AUTHOR]

Highlights • Study of structure-borne sound transmission of cascaded plate–cavity system. • Application of Chebyshev-Ritz method. • Comparable numerical accuracy as compared to finite element method. • Numerical examples are simulated for different configurations. • Better prediction of sound transmission via tuning of cascaded plate–cavity system. Abstract A Chebyshev-Ritz based analytical model is proposed to investigate I-junction within the structural–acoustic model of a cascaded rectangular plate–cavity system. By considering of the structural interconnection force and the moment at edges and structural-acoustic interaction on the interface, the structural and acoustic systems are coupled. Two-dimensional and three-dimensional Chebyshev Polynomial series are used to present the unknown panel displacements and the sound pressure field variable inside the cavities, respectively. The effectiveness and correctness of the proposed model on an I-junction in a typical marine offshore platform are verified with those calculated from Finite Element Analysis. The influence of boundary conditions, structural coupling, plate properties, and size of the source-to-receiver cavities on the offshore platform on structure-borne sound transmission are analyzed and addressed. Numerical examples are simulated for several different configurations. It is shown that the boundary conditions, structural coupling manner, plate properties, and the volume ratio of the source-to-receiver cavity will change the structure-borne sound transmission characteristics of the cascaded rectangular plate–cavity system. With the proposed approach, a better prediction can be obtained for structure-borne sound transmission via proper tuning of the cascaded rectangular plate–cavity system on the offshore platform. [ABSTRACT FROM AUTHOR]