Search

Research into vaccine hesitancy is a critical component of the public health enterprise, as rates of communicable diseases preventable by routine childhood immunization have been increasing in recent years. It is therefore important to estimate proportions of "never‐vaccinators" in various subgroups of the population in order to successfully target interventions to improve childhood vaccination rates. However, due to privacy issues, it may be difficult to obtain individual patient data (IPD) needed to perform the appropriate time‐to‐event analyses: state‐level immunization information services may only be willing to share aggregated data with researchers. We propose statistical methodology for the analysis of aggregated survival data that can accommodate a cured fraction based on a polynomial approximation of the mixture cure model log‐likelihood function relying only on summary statistics. We study the performance of the method through simulation studies and apply it to a real‐world data set from a study examining reminder/recall approaches to improve human papillomavirus (HPV) vaccination uptake. The proposed methods may be generalized for use when there is interest in fitting complex likelihood‐based models but IPD is unavailable due to data privacy or other concerns. [ABSTRACT FROM AUTHOR]

Series of the form $ \sum _{k=0}^{\infty }\frac {\prod _{i=1}^{p}\left (a_{i}\right) _{k}}{\prod _{j=1}^{q}\left (b_{j}\right) _{k}}w^{k}{}F_{k}(z) $ ∑ k = 0 ∞ ∏ i = 1 p (a i) k ∏ j = 1 q (b j) k w k F k (z) are considered, where $ F_{k}(z) $ F k (z) are special functions of hypergeometric type. Various types of series involving Bessel, Struve, incomplete gamma functions, and Laguerre, Hermite, Jacobi, Legendre, and Chebyshev polynomials are represented in terms of the Kampé de Fériet and generalized Horn hypergeometric functions of two variables. [ABSTRACT FROM AUTHOR]

This study investigates the vibrational behavior of three-phase polymer-glass fiber (FG)-carbon nanotubes (CNT) nanocomposites conical panels. Effects of the wavy shape of CNTs and the viscoelastic nature of the polymer matrix are considered. First and foremost, the equivalent material properties are estimated in the context of a three-step hierarchical approach. Then, the governing equations are attained by combining the first-order shear deformation theory (FSDT) and Hamilton's principle. Furthermore, the Ritz method is employed to extract the frequency responses with the help of Chebyshev polynomials for both simply supported and clamped ends panels. For the parametrical studies, the effect of different geometrical constraints, for instance, the effects of relaxation time and the effect of considering different waviness coefficients for the CNT, are reported. The findings of this paper reveal the significant effect of the viscoelastic properties of the polymer matrix on the estimation of the dimensionless frequency. For instance, under simply supported boundary conditions and a relaxation parameter of 60 s, the frequency of the structure reduces by approximately 23% after 100 s. [ABSTRACT FROM AUTHOR]

This paper dealt with the vibration characteristics of an auxetic rectangular plate under in-plane compression. Firstly, the equivalent bending stiffness matrix of a star-shaped auxetic plate was obtained using Castigliano's theorem and the homogenization technique. Then, employing the classical plate theory (CPT) in conjunction with the Rayleigh–Ritz method, the natural frequencies of auxetic plate were extracted. The chebyshev polynomial series has been selected to define the assumed displacement fields of the plate. Convergence study for the Rayleigh–Ritz method was conducted. The accuracy of the proposed mathematical model for the star-shaped auxetic plate was validated using the results from a finite element analysis (FEA). Effects of unit cell geometric parameters on the natural frequencies of the plate were examined. The auxeticity angle of star-shaped pattern had a significant effect on the natural frequencies. The present approach can be extended into other auxetic patterns, such as re-entrant bowtie auxetics. [ABSTRACT FROM AUTHOR]

Let G be an n-vertex graph and Q(G) be its signless Laplacian matrix. The Q-walk matrix of G, denoted by W Q (G) , is [ e , Q (G) e , … , Q n - 1 (G) e ] , where e is the all-one vector. Let G ∘ P m be the graph obtained from G and n copies of the path P m by identifying the i-th vertex of G with an endvertex of the i-th copy of P m for each i. We prove that, det W Q (G ∘ P m) = ± (det Q (G)) m - 1 (det W Q (G)) m holds for any m ≥ 2 . This gives a signless Laplacian counterpart of the following recently established identity (Wang et al. in Linear Multilinear Algebra 72:828–840, 2024. ): det W A (G ∘ P m) = ± (det A (G)) ⌊ m 2 ⌋ (det W A (G)) m , where A(G) is the adjacency matrix of G and W A (G) = [ e , A (G) e , … , A n - 1 (G) e ] . We also propose a conjecture to unify the above two equalities. [ABSTRACT FROM AUTHOR]

This paper is dedicated to the full discretization of linear wave equations, where the space discretization is carried out with a discontinuous Galerkin method on spatial meshes which are locally refined or have a large wave speed on only a small part of the mesh. Such small local structures lead to a strong Courant–Friedrichs–Lewy (CFL) condition in explicit time integration schemes causing a severe loss in efficiency. For these problems, various local time-stepping schemes have been proposed in the literature in the last years and have been shown to be very efficient. Here, we construct a quite general class of local time integration methods preserving a perturbed energy and containing local time-stepping and locally implicit methods as special cases. For these two variants we prove stability and optimal convergence rates in space and time. Numerical results confirm the stability behavior and show the proved convergence rates. [ABSTRACT FROM AUTHOR]

Electrocardiogram (ECG) signals generate massive volume of digital data, so they need to be suitably compressed for efficient transmission and storage. Polynomial approximations and polynomial interpolation have been used for ECG data compression where the data signal is described by polynomial coefficients only. Here, we propose approximation using hermite polynomial interpolation with chebyshev nodes for compressing ECG signals that consequently denoises them too. Recommended algorithm is applied on various ECG signals taken from MIT-BIH arrhythmia database without any additional noise as the signals are already contaminated with noise. Performance of the proposed algorithm is evaluated using various performance metrics and compared with some recent compression techniques. Experimental results prove that the proposed method efficiently compresses the ECG signals while preserving the minute details of important morphological features of ECG signal required for clinical diagnosis. [ABSTRACT FROM AUTHOR]

This paper presents a novel approach to microwave lowpass filter design that addresses several challenges in modern microwave systems, including size, performance, and cost. With that in mind, we propose using monic Chebyshev polynomials as the characteristic function for the filter to achieve a low passband insertion loss and maximum cutoff slope. The half-power cutoff frequency characterizes these filters, and the ripple factor is not considered a design parameter. To demonstrate the effectiveness of the monic Chebyshev approximation, a comparison is made with chained function (CF) filters designed for microwave applications. It has been shown that the monic Chebyshev filter outperforms CF filters in all applications, including microwaves. [ABSTRACT FROM AUTHOR]

This paper established a functional design of the spectral Tau method (STM) upon the first derivatives of Chebyshev polynomials (FDCHPs). A new linearization relation has been developed. Hence, this relation and another investigated one for integration have been used via the Tau method to execute the Tau integration analytically. Moreover, explicit algebraic systems for solving the Lane–Emden and Riccati equations and the contamination of a system of three artificial lakes are introduced. The convergence and error analyses are explored in depth. Some enlightening boundary value problems (BVPs) for real-life and physical applications, as singularly perturbed equations, are provided to guarantee that this approach is authentic, reliable, and appropriate. Accurate results are obtained using only a few number of retained nodes. The different outcomes, patterns, and better results showed that the FDCHPs differ from Chebyshev polynomials of the second kind. [ABSTRACT FROM AUTHOR]

One basic limitation of using the periodogram as a frequency estimator is that any of its significant peaks may result from a diffuse (or spread) frequency component rather than a pure one. Diffuse components are common in applications such as channel estimation, in which a given periodogram peak reveals the presence of a complex multipath distribution (unresolvable propagation paths or diffuse scattering, for example). We present a method to detect the presence of a diffuse component in a given peak based on analyzing the projection of the data vector onto the span of the signature's derivatives up to a given order. Fundamentally, a diffuse component is detected if the energy in the derivatives' subspace is too high at the peak's frequency, and its spread is estimated as the ratio between this last energy and the peak's energy. The method is based on exploiting the signature's Vandermonde structure through the properties of discrete Chebyshev polynomials. We also present an efficient numerical procedure for computing the data component in the derivatives' span based on barycentric interpolation. The paper contains a numerical assessment of the proposed estimator and detector. [ABSTRACT FROM AUTHOR]

The main goal of this study is to solve a class of linear and nonlinear Volterra integral equations (VIEs) of the third kind. The proposed technique estimates the solution of these equations using a technique based on MLS approximation and shifted Chebyshev polynomials. The approach is independent of the geometry of the domain and does not require any background meshes; therefore, it can be considered a meshless approach. The new method is effective and more flexible for many types of VIEs of the third kind, and its algorithm can be simply implemented on computers. The construction of a new technique for the proposed equations has been introduced. The convergence analysis is also studied. The convergence precision of the new approach is checked on classes of VIEs of the third kind, and the results validate the theoretical error estimates. Finally, numerical tests are provided to illustrate the accuracy and reliability of this approach. [ABSTRACT FROM AUTHOR]

Based on domain decomposition, a semi-analytical method (SAM) is applied to analyze the free vibration of double-curved shells of revolution with a general curvature radius made from graphene nanoplatelet (GPL)-reinforced porous composites. The mechanical properties of the GPL-reinforced composition are assessed with the Halpin–Tsai model. The double-curvature shell of revolution is broken down into segments along its axis in accordance with first-order shear deformation theory (FSDT) and the multi-segment partitioning technique, to derive the shell's functional energy. At the same time, interfacial potential is used to ensure the continuity of the contact surface between neighboring segments. By applying the least-squares weighted residual method (LWRM) and modified variational principle (MVP) to relax and achieve interface compatibility conditions, a theoretical framework for analyzing vibrations is developed. The displacements and rotations are described through Fourier series and Chebyshev polynomials, accordingly, converting a two-dimensional issue into a suite of decoupled one-dimensional problems. The obtained solutions are contrasted with those achieved using the finite element method (FEM) and other existing results, and the current formulation's validity and precision are confirmed. Example cases are presented to demonstrate the free vibration of GPL-reinforced porous composite double-curved paraboloidal, elliptical, and hyperbolical shells of revolution. The findings demonstrate that the natural frequency of the shell is related to pore coefficients, porosity, the mass fraction of the GPLs, and the distribution patterns of the GPLs. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce the hyperbolic generalized k-Horadam quaternions and octonions and investigate their algebraic properties. We present some properties and identities of these quaternions and octonions for generalized k-Horadam numbers. Moreover, we give some determinants related to the hyperbolic generalized k-Horadam quaternions and octonions. Finally, we evaluate its determinants through the Chebyshev polynomials of the second kind and give an illustrative example as well. [ABSTRACT FROM AUTHOR]

In many situations, exact solutions to complex problems may be challenging or impossible to obtain, making approximation techniques necessary for making informed decisions. Since function implementation on FPGAs can be difficult and resource-consuming, therefore it would be a better idea to approximate them. In this study, the Chebyshev approximation technique is thoroughly investigated, and an FPGA implementation is proposed and analyzed. This implementation will be compared to other implementations of approximation techniques with parameters such as accuracy, speed, and design size taken into consideration. Applications of this FPGA design are also discussed and shown such as approximating the sigmoid function for machine learning in an efficient manner. This study proved the adequacy of the Chebyshev approximation and its accelerated FPGA implementation for various applications including machine learning, filter design, signal processing, and many other practical applications in engineering and science. [ABSTRACT FROM AUTHOR]

In this work, the tempered fractional derivative in the Caputo form is considered to define the tempered time fractional coupled Burger's equations. The Chebyshev polynomials, as a well-known family of polynomial basis functions, are used to establish a collocation scheme based on the Gauss points for this problem. To this end, the tempered fractional derivative matrix of these polynomials is obtained. By approximating the problem's solution via the Chebyshev polynomials (with some unknown coefficients) and employing the expressed tempered fractional derivative matrix, along with the collocation strategy based on the Gauss points, a nonlinear system of algebraic equations is obtained. By solving this system, the unknown coefficients, and subsequently the solution of the original tempered fractional problem are obtained. Some illustrative examples are considered to confirm the high accuracy of the designed procedure. [ABSTRACT FROM AUTHOR]

We strengthen the Weierstrass approximation theorem by proving that any real-valued continuous function on an interval I ⊂ R can be uniformly approximated by a real-valued polynomial whose only (possibly complex) critical points are contained in I . The proof uses a perturbed version of the Chebyshev polynomials and an application of the Brouwer fixed point theorem. [ABSTRACT FROM AUTHOR]

This paper studies wave localization in chains of finitely many resonators. There is an extensive theory predicting the existence of localized modes induced by defects in infinitely periodic systems. This work extends these principles to finite‐sized systems. We consider one‐dimensional, finite systems of subwavelength resonators arranged in dimers that have a geometric defect in the structure. This is a classical wave analog of the Su–Schrieffer–Heeger model. We prove the existence of a spectral gap for defectless finite dimer structures and find a direct relationship between eigenvalues being within the spectral gap and the localization of their associated eigenmode. Then, for sufficiently large‐size systems, we show the existence and uniqueness of an eigenvalue in the gap in the defect structure, proving the existence of a unique localized interface mode. To the best of our knowledge, our method, based on Chebyshev polynomials, is the first to characterize quantitatively the localized interface modes in systems of finitely many resonators. [ABSTRACT FROM AUTHOR]

In this paper, we focus on designing a well‐conditioned Galerkin spectral methods for solving a two‐sided fractional diffusion equations with drift in which the fractional operators are defined neither in Riemann–Liouville nor Caputo sense, and its physical meaning is clear. Based on the image spaces of Riemann–Liouville fractional integral operators on Lp([a, b]) space discussed in our previous work, after a step by step deduction, three kinds of Galerkin spectral formulations are proposed, the final obtained corresponding scheme of which shows to be well‐conditioned—the condition number of the stiff matrix can be reduced from O(N2α) to O(Nα), where N is the degree of the polynomials used in the approximation. Another point is that the obtained schemes can also be applied successfully to approximate fractional Laplacian with generalized homogeneous boundary conditions, whose fractional order α ∈ (0, 2), not only having to be limited to α ∈ (1, 2). Several numerical experiments demonstrate the effectiveness of the derived schemes. Besides, based on the numerical results, we can observe the behavior of mean first exit time, an interesting quantity that can provide us with a further understanding about the mechanism of abnormal diffusion. [ABSTRACT FROM AUTHOR]

In this paper we consider an orthonormal basis, generated by a tensor product of Fourier basis functions, half period cosine basis functions, and the Chebyshev basis functions. We deal with the approximation problem in high dimensions related to this basis and design a fast algorithm to multiply with the underlying matrix, consisting of rows of the non-equidistant Fourier matrix, the non-equidistant cosine matrix and the non-equidistant Chebyshev matrix, and its transposed. Using this, we derive the ANOVA (analysis of variance) decomposition for functions with partially periodic boundary conditions through using the Fourier basis in some dimensions and the half period cosine basis or the Chebyshev basis in others. We consider sensitivity analysis in this setting, in order to find an adapted basis for the underlying approximation problem. More precisely, we find the underlying index set of the multidimensional series expansion. Additionally, we test this ANOVA approximation with mixed basis at numerical experiments, and refer to the advantage of interpretable results. [ABSTRACT FROM AUTHOR]

With this paper we continue the investigations started in [6] and concerned with stability conditions for collocation-quadrature methods based on the zeros of classical Jacobi polynomials, not only Chebyshev polynomials. While in [6] we only proved the necessity of certain conditions, here we will show also their sufficiency in particular cases. [ABSTRACT FROM AUTHOR]

The problem of solving the equivalent resistance between two points for resistor networks has important significance in physics. This paper mainly changes and rewrites the formula for calculating the resistance between two points of an unconventional m × n cylindrical resistor network with a zero resistor axis and any two left and right boundaries. To enhance the efficiency of calculating the equivalent resistance between two points, Chebyshev polynomials and hyperbolic cosine functions are employed to represent the new formula. And in the inference process, the famous discrete cosine transform of the third kind (DCT-III) is used to process the matrix. We give the equivalent resistance formula for several special cases, and display them by a three-dimensional graph. Subsequently, the calculation efficiency of the original formula and the rewritten formula are compared. At the end of the paper, a heuristic algorithm suitable for robot path planning on cylindrical environment is proposed. [ABSTRACT FROM AUTHOR]

The main idea of this work is to present a numerical method based on Vieta‐Fibonacci polynomials (VFPs) for finding approximate solutions of fractal‐fractional (FF) pantograph differential equations and a system of differential equations. Although the presented scheme can be applied to any fractional integral, we focus on the Caputo, Atangana‐Baleanu, and Caputo‐Fabrizio integrals with due to their privileges. To carry out the method, first, we introduce FF integral operators in the Caputo, Atangana‐Baleanu, and Caputo‐Fabrizio senses. Then, by applying the Vieta‐Fibonacci polynomials and their FF integral operators together with the collocation method, the problem becomes reduced to a system of algebraic equations that can be solved by Mathematical software. In the presented scheme, acceptable approximate solutions are achieved by employing only a few number of the basic functions. Moreover, the error analysis of the presented method is investigated. Finally, the accuracy of the presented method is examined through the numerical examples. The proposed scheme is implemented for some famous systems of FF differential equations, such as memristor, which is a fundamental circuit element so called universal charge‐controlled mem‐element, convective fluid motion in rotating cavity, and Lorenz chaotic system. [ABSTRACT FROM AUTHOR]

This paper explores geometric circulant matrices whose entries are Chebyshev polynomials of the first or second kind. Motivated by our previous research on r − circulant matrices and Chebyshev polynomials, we focus on calculating the Frobenius norm and deriving estimates for the spectral norm bounds of these matrices. Our analysis reveals that this approach yields notably improved results compared to previous methods. To validate the practical significance of our research, we apply it to existing studies on geometric circulant matrices involving the generalized k − Horadam numbers. The obtained results confirm the effectiveness and utility of our proposed approach. [ABSTRACT FROM AUTHOR]

Due to the similar characteristics of chaotic systems and cryptography, public key encryption algorithms based on chaotic systems are worth in-depth research and have high value for the future. Chebyshev polynomials have good properties and are often used in the design of public key algorithms. This paper improves the Bose Multi-Chaotic Public Key Cryptographic Algorithm (BMPKC) by applying Chebyshev polynomials. The proposed algorithm (CMPKC- k i ) introduces the selective coefficient k i based on the properties of Chebyshev polynomials, allowing the special functions that need to be negotiated in the original system to be freely and randomly chosen as Chebyshev polynomials, and can also be expanded to m levels. The improved cryptographic algorithm also utilizes chaotic hash functions and logistic mapping to generate pseudo-random sequences and overcomes shortcomings of the Bose algorithm by iteratively iterating the selected Chebyshev polynomials based on the number of 0s or 1s in the pseudo-random sequence, thus providing better security. Analysis and software testing results indicate that this algorithm has strong robustness against brute force attacks, achieving a higher attack time for breaking the private key compared to the CEPKC, BMPKC, and CMPKC algorithms. Compared to the CMPKC algorithm, our proposal algorithm achieves better performance in the encryption and decryption phases. Furthermore, we combine this Multi-Chaotic System Key Exchange Protocol with the Advanced Encryption Standard (AES) algorithm, while providing a demonstration, offering more possibilities for practical applications of this system. [ABSTRACT FROM AUTHOR]

In this study, we designed multi-section UWB Wilkinson power dividers for both equal and unequal power divider. The key innovation is successfully matching these power dividers using the Chebyshev method. The proposed method achieves very low insertion losses and high transmission coefficients across the ports. For the three-section equal power divider, Chebyshev polynomials were used to model the reflection coefficient, and the characteristic impedance for each section was calculated symmetrically. The design achieved a fractional bandwidth of 106% for a 20 dB return loss, with an insertion loss of 0.3 dB. For unequal multi-section dividers, power division ratios of 1.5:1 and 2:1 were chosen. Theoretical microwave calculations were used to determine the input and output transmission line impedances. Chebyshev approximation was then applied to design the three-section unequal power dividers. The return losses for these unequal dividers were measured at 17 dB and 15 dB, with fractional bandwidths of 120% and 126%. Both circuits had insertion losses of 0.25 dB. The consistent results from analytical calculations, simulations, and measurements confirm the effectiveness of the Chebyshev method for both equal and unequal power division. [ABSTRACT FROM AUTHOR]

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]

Wrinkling is a typical failure mode of sandwich structures with thin skins, where the bending deformation of face skins occurs simultaneously in conjunction with transverse stretching deformation of core layer. To analyze the three-dimensional (3D) deformation of wrinkling, a model is required to possess a capability describing completely different deformations at each ply. Therefore, by utilizing the Chebyshev polynomial at each layer of sandwich panels, this paper focuses on proposing a higher-order model to individually illustrate deformations of the skins and the core. By means of the proposed model, a refined triangular element has been constructed. By analyzing the stability of a sandwich plate without stiffeners, the finite-element formulation has been verified by comparing it to the quasi-3D elasticity solutions. In addition, it is anticipated that wrinkling failure can be effectively restrained by designing the stiffener layout. To this end, the influence of stiffener layouts on wrinkling behaviors of sandwich plates has been investigated in detail, and it is found that the capability of resisting the wrinkling deformation can be obviously improved by designing reasonable stiffener layouts. Moreover, the translation between buckling and wrinkling behaviors of sandwich panels with different stiffeners has been also explored, which can help to better understand wrinkling deformation mechanism. In summary, the proposed model can be used to design a reasonable arrangement of stiffeners to resist the wrinkling failure of sandwich panels. [ABSTRACT FROM AUTHOR]

We consider periodic Jacobi operators and obtain upper and lower estimates on the sizes of the spectral bands. Our proofs are based on estimates on the logarithmic capacities and connections between the Chebyshev polynomials and logarithmic capacity of compact subsets of the real line. [ABSTRACT FROM AUTHOR]

In this paper, a novel inline network realization of non-resonating node (NRN) and dangling resonator (R) pair direct synthesis approach is presented. Arbitrary prescribed transmission zeros at real frequencies are realized through independently controlling the dangling resonators of the NRN-R pairs with impedance inverters between adjacent nodes. The bandpass element values for fourth order acoustic wave filter with a center frequency at 98 MHz are obtained through execution and mapping of the synthesis results based on the Generalized Chebyshev polynomials. The layout of prototype is presented. Finally, the prototype is constructed and measured using network analyzer to validate the proposed concept in realizing the BAW filter using Butterworth Lowpass Van Dyke (BVD) model. The filter has an insertion loss of 3.45 dB, a return loss of 8.5 dB, and two transmission zeros (TZs). In terms of implementation, this synthesis technique allows flexibility to synthesize and realize a conventional filter with transmission zeros as an inline network without cross couplings. This offers advantages in terms of size and cost reduction in filter production due to the inline resonator arrangement and reduced sensitivity in the design and tuning process. [ABSTRACT FROM AUTHOR]

This research work aims to theoretically examine the influence of various factors on the three-dimensional nanofluid flow. The study includes parameters such as temperature ratio coefficient, Prandtl numbers, Schmidt, Soret, Dufour, Biot, expansion ratio coefficient, Power index, and nanoparticle volume fraction parameter, as well as the effect of non-linear thermal radiation and magnetic parameter on the behavior of the nanofluid. These characteristics significantly impact the flow of the three-dimensional boundary layer in the presence of an expansion plate. To facilitate the investigation, we have selected nanofluids that contain water-based copper and aluminum oxide for this study. We have developed a model of a system of partial differential equations (SYS-PDEs) with non-linear terms. Based on selected similarity equations, the SYS-PDEs with non-linear terms has been transformed into a system of ordinary differential equations (SYS-ODEs) whose terms are non-linear. To approximate and solve the obtained SYS-ODEs, we utilized a modified spectral Chebyshev polynomials' first derivative pseudo-Galerkin spectral method. Additionally, we conducted an error analysis discussion to ensure the credibility of our results. We presented our analysis in graphical form and provided comments on each figure along with the effects of the various parameters studied. Consequently, we concluded that the power low is an essential factor affecting the flow's behavior, such as the nanofluid's velocity, temperature, and concentration. [ABSTRACT FROM AUTHOR]

We present a novel numerical approach for solving nonlinear constrained optimal control problems (NCOCPs). Instead of directly solving the NCOCPs, we start by linearizing the constraints and dynamic system, which results in a sequence of sub-problems. For each sub-problem, we use finite number of Chebyshev polynomials to estimate the control and state vectors. To eliminate the errors at non-collocation points caused by conventional collocation methods, we additionally estimate the coefficient functions involved in the linear constraints and dynamic system by Chebyshev polynomials. By leveraging the characteristics of Chebyshev polynomials, the approximate sub-problem is changed into an equivalent nonlinear optimization problem with linear equality constraints. Consequently, any feasible point of the approximate sub-problem will satisfy the constraints and dynamic system throughout the entire time scale. To validate the efficacy of the new method, we solve three examples and assess the accuracy of the method through the computation of its approximation error. Numerical results obtained show that our approach achieves lower approximation error when compared to the Chebyshev pseudo-spectral method. The proposed method is particularly suitable for scenarios that require high-precision approximation, such as aerospace and precision instrument production. • A sequence of sub-problems is constructed by linearizing the dynamic system and constraints. • Each sub-problem can be transformed into an equivalent nonlinear optimization problem with linear equality constraints through Chebyshev polynomials. • Any feasible solution of the approximate sub-problem will satisfy its dynamic system and constraints over the entire time horizon. • Global convergence of the proposed method is established. [ABSTRACT FROM AUTHOR]

Fourth-order fractional Sturm-Liouville problems are studied in this work. The numerical simulation uses the pseudospectral method, utilizing Chebyshev cardinal polynomials. The presented algorithm is implemented after converting the desired equation into an associated integral equation and gives us a linear system of algebraic equations. Then, we can find the eigenvalues by calculating the roots of the corresponding characteristic polynomial. What is most striking is that the proposed scheme accurately solves this type of equation. Numerical experiments confirm this claim. [ABSTRACT FROM AUTHOR]

This work aims to provide a new Galerkin algorithm for solving the fractional RayleighStokes equation (FRSE). We select the basis functions for the Galerkin technique to be appropriate orthogonal combinations of the second kind of Chebyshev polynomials (CPs). By implementing the Galerkin approach, the FRSE, with its governing conditions, is converted into a matrix system whose entries can be obtained explicitly. This system can be obtained by expressing the derivatives of the basis functions in terms of the second-kind CPs and after computing some definite integrals based on some properties of CPs of the second kind. A thorough investigation is carried out for the convergence analysis. We demonstrate that the approach is applicable and accurate by providing some numerical examples. [ABSTRACT FROM AUTHOR]

Nonlinear optimal control problems governed by variable-order fractional integro-differential equations constitute an important subgroup of optimal control problems. This group of problems is often difficult or impossible to solve analytically because of the variable-order fractional derivatives and fractional integrals. In this article, we utilized the expansion of Lagrange polynomials in terms of Chebyshev polynomials and the power series of Chebyshev polynomials to find an approximate solution with high accuracy. Subsequently, by employing collocation points, the problem was transformed into a nonlinear programming problem. In addition, variable-order fractional derivatives in the Caputo sense were represented by a new operational matrix, and an operational matrix represented fractional integrals. As a result, the mentioned integro-differential optimal control problem becomes a nonlinear programming problem that can be easily solved with the repetitive optimization method. In the end, the proposed method is illustrated by numerical examples that demonstrate its efficiency and accuracy. [ABSTRACT FROM AUTHOR]

Fractional differential operators are inherently non-local, so global methods, such as spectral methods, are well suited for handling these non-local operators. Long-time integration of differential models such as chaotic dynamical systems poses specific challenges and considerations that make multi-domain numerical methods advantageous when dealing with such problems. This study proposes a novel multi-domain pseudospectral method based on the first kind of Chebyshev polynomials and the Gauss–Lobatto quadrature for fractional initial value problems.The proposed technique involves partitioning the problem's domain into non-overlapping sub-domains, calculating the fractional differential operator in each sub-domain as the sum of the 'local' and 'memory' parts and deriving the corresponding differentiation matrices to develop the numerical schemes. The linear stability analysis indicates that the numerical scheme is absolutely stable for certain values of arbitrary non-integer order and conditionally stable for others. Numerical examples, ranging from single linear equations to systems of non-linear equations, demonstrate that the multi-domain approach is more appropriate, efficient and accurate than the single-domain scheme, particularly for problems with long-term dynamics. [ABSTRACT FROM AUTHOR]

The present study introduced modifications to the standard Adomian decomposition method (ADM) by combining the Taylor series with orthogonal polynomials, such as Legendre polynomials and the first and second kinds of Chebyshev polynomials. These improvements can be applied to solve fractional differential equations with initial-value problems in the Caputo sense. The approaches are based on the use of orthogonal polynomials, which are essential components in approximation theories. The study carefully analyzed their respective absolute error differences, highlighting the computational benefits of the proposed modifications, which offer improved accuracy and require fewer computational steps. The effectiveness and accuracy of the approach were validated through numerical examples, confirming its efficiency and reliability.

In this work, the tempered fractional derivative in the Caputo form is considered to define the tempered time fractional coupled Burger’s equations. The Chebyshev polynomials, as a well-known family of polynomial basis functions, are used to establish a collocation scheme based on the Gauss points for this problem. To this end, the tempered fractional derivative matrix of these polynomials is obtained. By approximating the problem’s solution via the Chebyshev polynomials (with some unknown coefficients) and employing the expressed tempered fractional derivative matrix, along with the collocation strategy based on the Gauss points, a nonlinear system of algebraic equations is obtained. By solving this system, the unknown coefficients, and subsequently the solution of the original tempered fractional problem are obtained. Some illustrative examples are considered to confirm the high accuracy of the designed procedure.

In this work, we focus on a class of Ψ− fractional integro-differential equations (Ψ-FIDEs) involving Ψ-Caputo derivative. The objective of this paper is to derive the numerical solution of Ψ-FIDEs in the truncated Bell series. Firstly, Ψ-FIDEs by using the definition of Ψ− Caputo derivative is converted into a singular integral equation. Then, a computational procedure based on the Bell polynomials, Gauss-Legendre quadrature rule, and collocation method is developed to effectively solve the singular integral equation. The convergence of the approximation obtained in the presented strategy is investigated. Finally, the effectiveness and superiority of our method are revealed by numerical samples. The results of the suggested approach are compared with the results obtained by extended Chebyshev cardinal wavelets method (EChCWM). [ABSTRACT FROM AUTHOR]

In this study, we explore the effectiveness of elliptic partial differential equations (PDEs) in two and three dimensional space based on Helmholtz equation for simulations of acoustic sound in a very complex environment of propagation. For this purpose, we use an advance and robust numerical technique by utilizing the properties of shifted Chebyshev spectral collocation method. This technique is an extension of the traditional Chebyshev polynomials, incorporating a shift in their argument to enhance flexibility across a wider domain, while retaining an extraordinary numerical characteristic such as orthogonality and spectral convergence making them exceptionally effective in finding the approximate solutions. The exponential order of convergence of the proposed approach is shown both through theoretical and numerical approaches. We provide a number of numerical experiments to verify the theoretical results. The spectral convergence has been substantially enhanced by these numerical examples. The exponential order is further validated by numerical error behaviour in both L2 and L∞ norms. [ABSTRACT FROM AUTHOR]

We develop a linearly scaling variant of the force coupling method [K. Yeo and M. R. Maxey, J. Fluid Mech. 649, 205–231 (2010)] for computing hydrodynamic interactions among particles confined to a doubly periodic geometry with either a single bottom wall or two walls (slit channel) in the aperiodic direction. Our spectrally accurate Stokes solver uses the fast Fourier transform in the periodic xy plane and Chebyshev polynomials in the aperiodic z direction normal to the wall(s). We decompose the problem into two problems. The first is a doubly periodic subproblem in the presence of particles (source terms) with free-space boundary conditions in the z direction, which we solve by borrowing ideas from a recent method for rapid evaluation of electrostatic interactions in doubly periodic geometries [Maxian et al., J. Chem. Phys. 154, 204107 (2021)]. The second is a correction subproblem to impose the boundary conditions on the wall(s). Instead of the traditional Gaussian kernel, we use the exponential of a semicircle kernel to model the source terms (body force) due to the presence of particles and provide optimum values for the kernel parameters that ensure a given hydrodynamic radius with at least two digits of accuracy and rotational and translational invariance. The computation time of our solver, which is implemented in graphical processing units, scales linearly with the number of particles, and allows computations with about a million particles in less than a second for a sedimented layer of colloidal microrollers. We find that in a slit channel, a driven dense suspension of microrollers maintains the same two-layer structure as above a single wall, but moves at a substantially lower collective speed due to increased confinement. [ABSTRACT FROM AUTHOR]

Fractional differential operators are inherently non-local, so global methods, such as spectral methods, are well suited for handling these non-local operators. Long-time integration of differential models such as chaotic dynamical systems poses specific challenges and considerations that make multi-domain numerical methods advantageous when dealing with such problems. This study proposes a novel multi-domain pseudospectral method based on the first kind of Chebyshev polynomials and the Gauss–Lobatto quadrature for fractional initial value problems.The proposed technique involves partitioning the problem’s domain into non-overlapping sub-domains, calculating the fractional differential operator in each sub-domain as the sum of the ‘local’ and ‘memory’ parts and deriving the corresponding differentiation matrices to develop the numerical schemes. The linear stability analysis indicates that the numerical scheme is absolutely stable for certain values of arbitrary non-integer order and conditionally stable for others. Numerical examples, ranging from single linear equations to systems of non-linear equations, demonstrate that the multi-domain approach is more appropriate, efficient and accurate than the single-domain scheme, particularly for problems with long-term dynamics.

This work aims to provide a new Galerkin algorithm for solving the fractional Rayleigh-Stokes equation (FRSE). We select the basis functions for the Galerkin technique to be appropriate orthogonal combinations of the second kind of Chebyshev polynomials (CPs). By implementing the Galerkin approach, the FRSE, with its governing conditions, is converted into a matrix system whose entries can be obtained explicitly. This system can be obtained by expressing the derivatives of the basis functions in terms of the second-kind CPs and after computing some definite integrals based on some properties of CPs of the second kind. A thorough investigation is carried out for the convergence analysis. We demonstrate that the approach is applicable and accurate by providing some numerical examples.

Doppler through-wall radar faces two challenges when locating targets concealed behind walls: (1) precisely determining the instantaneous frequency of the target within the frequency aliasing region and (2) reducing the impact of the wall on positioning by determining accurate wall parameters. To address these issues, this paper introduces a target localization algorithm that combines the Hough transform and support vector regression-BP neural network. First, a multiview fusion model framework is proposed for through-wall target detection, which enables the auxiliary estimation of wall parameter information by acquiring target positions from different perspectives. Second, a high-precision extraction and estimation algorithm for the instantaneous frequency curve of the target is proposed by combining the differential evolutionary algorithm and Chebyshev interpolation polynomials. Finally, a target motion trajectory compensation algorithm based on the Back Propagation (BP) neural network is proposed using the estimated wall parameter information, which suppresses the distorting effect of obstacles on target localization results and achieves the accurate localization of the target behind a wall. Experimental results indicate that compared with the conventional short-time Fourier method, the developed algorithm can accurately extract target instantaneous frequency curves within the time-frequency aliasing region. Moreover, it successfully reduces the impact caused by walls, facilitating the precise localization of multiple targets behind walls, and the overall localization accuracy is improved ～85%.

For one-dimensional inhomogeneous (with respect to the spatial variable) linear parabolic equations, a combined approach is used, dividing the original problem into two subproblems. The first of them is an inhomogeneous one-dimensional Poisson problem with Dirichlet-Robin boundary conditions, the search for a solution of which is based on the Chebyshev collocation method. The method was developed based on previously published algorithms for solving ordinary differential equations, in which the solution is sought in the form of an expansion in Chebyshev polynomials of the 1st kind on Gauss-Lobatto grids, which allows the use of discrete orthogonality of polynomials. This approach turns out to be very economical and stable compared to traditional methods, which often lead to the solution of poorly defined systems of linear algebraic equations. In the described approach, the successful use of integration matrices allows complete elimination of the need to deal with ill-conditioned matrices. The second, homogeneous problem of thermal conductivity is solved by the method of separation of variables. In this case, finding the expansion coefficients of the desired solution in the complete set of solutions to the corresponding Sturm-Liouville problem is reduced to calculating integrals of known functions. A simple technique for constructing Chebyshev interpolants of integrands allows to calculate the integrals by summing interpolation coefficients.

This study utilizes a spectral tau method to acquire an accurate numerical solution of the time-fractional diffusion equation. The central point of this approach is to use double basis functions in terms of certain Chebyshev polynomials, namely Chebyshev polynomials of the seventh-kind and their shifted ones. Some new formulas concerned with these polynomials are derived in this study. A rigorous error analysis of the proposed double expansion further corroborates our research. This analysis is based on establishing some inequalities regarding the selected basis functions. Several numerical examples validate the precision and effectiveness of the suggested method. [ABSTRACT FROM AUTHOR]

This study presents a meshless modeling approach to design variable-stiffness laminates considering manufacturing constraints. The governing equations are derived using lamination parameters and first-order shear deformation theory. The solution approach uses Chebyshev polynomials and Galerkin's method to obtain the discretized equations of motion. The developed framework was used to maximize the fundamental frequency of composite plates. The variable-stiffness designs provided up to 28.4% higher frequencies compared to optimum constant-stiffness laminates, although the actual level of improvement depends on the number of layers. Finally, manufacturable fiber paths were obtained considering the allowed fiber curvature, which can also reduce the frequency values. [ABSTRACT FROM AUTHOR]

In this article, a nonlinear geometrically exact (GeX) hybrid-mixed four-node solid-shell element under non-conservative loading using the sampling surfaces (SaS) method is developed. The term "geometrically exact" means that the parametrization of the middle surface is known and, therefore, the coefficients of the first and second fundamental forms and the Christoffel symbols are taken exactly at element nodes. The SaS formulation is based on the choice of N SaS parallel to the middle surface and located at Chebyshev polynomial nodes in order to introduce the displacements of these surfaces as fundamental shell unknowns. Such a choice of unknowns with the use of Lagrange polynomials of degree N–1 in the through-thickness distributions of displacements, strains, and stresses leads to an efficient higher-order shell formulation. To develop the hybrid-mixed four-node solid-shell element, which is free of shear and membrane locking, the Hu–Washizu variational principle is utilized. The proposed GeX hybrid-mixed solid-shell element subjected to follower loads shows superior performance in the case of coarse meshes and allows the use of only one load step in most of the considered numerical examples. It is established that the difference between the second Piola-Kirchhoff and Cauchy stresses in some non-conservative problems for isotropic and composite shells undergoing finite rotations can be significant. [ABSTRACT FROM AUTHOR]

For T n (x) = cos n arccos x , x ∈ [ - 1 , 1 ] , the n-th Chebyshev polynomial of the first kind, we study the quantity τ n , k : = | T n (k) (ω n , k) | T n (k) (1) , 1 ≤ k ≤ n - 2 , where T n (k) is the k-th derivative of T n and ω n , k is the largest zero of T n (k + 1) . Since the absolute values of the local extrema of T n (k) increase monotonically towards the end-points of [ - 1 , 1 ] , the value τ n , k shows how small is the largest critical value of T n (k) relative to its global maximum T n (k) (1) . This is a continuation of our (joint with Alexei Shadrin) paper "On the largest critical value of T n (k) ", SIAM J. Math. Anal.50(3), 2018, 2389–2408, where upper bounds and asymptotic formuae for τ n , k have been obtained on the basis of the Schaeffer–Duffin pointwise upper bound for polynomials with absolute value not exceeding 1 in [ - 1 , 1 ] . We exploit a 1996 result of Knut Petras about the weights of the Gaussian quadrature formulae associated with the ultraspherical weight function w λ (x) = (1 - x 2) λ - 1 / 2 to find an explicit (modulo ω n , k ) formula for τ n , k 2 . This enables us to prove a lower bound and to refine the previously obtained upper bounds for τ n , k . The explicit formula admits also a new derivation of the asymptotic formula approximating τ n , k for n → ∞ . The new approach is simpler, without using deep results about the ordinates of the Bessel function, and allows to better analyze the sharpness of the estimates. [ABSTRACT FROM AUTHOR]

In this paper, we address the problem of the divergent sums of general arithmetic functions over the set of primes. In classical analytic number theory, the sum of the logarithm of the prime numbers plays a crucial role. We consider the sums of powers of the logarithm of primes and its connection with Riemann's zeta function (z.f.). This connection is achieved through the second Chebyshev function of order n, which can be estimated by exploiting the symmetry properties of Riemann's zeta function. Finally, a heuristic approach to evaluating more general sums is also given. [ABSTRACT FROM AUTHOR]

In this paper, a nonlinear mechanical system of ordinary differential equations (ODEs) with multi-point boundary conditions is considered by a novel type of reproducing kernel Hilbert space method (RKHSM). To begin, we define the unknown variables in terms of the reproducing kernel function. The roots of the Shifted Chebyshev polynomials (SCPs) are then utilized to collocate the resulting system. Finally, Newton's iterative method is employed to find the unknown expansion coefficients. The solutions of this system of equations, which arise from the flow of an electrically conducting nanofluid over an impermeable stretching cylinder, are numerically analyzed, and convergence analysis is discussed to demonstrate the reliability of the presented method (PM). Tables and figures are provided to further discuss the solutions and assess the effectiveness of the method in comparison to other techniques in the literature. [ABSTRACT FROM AUTHOR]