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• Vibration and damping analysis of ERF sandwich deep arch. • First-order shear deformable bidirectional FGM face sheets. • Chebyshev collocation method. • Electric field-dependent shear modulus. In this project, for the first time, the dynamic features of a deep sandwich arch with an electrorheological fluid (ERF) core are evaluated. The retentive face sheets of this structure are composed of novel bi-directionally functionally graded materials (FGMs) according to a dual-index power law. The curved beam's displacement field is predicted by applying the sandwich theory, which takes into account shear deformations and rotating inertias for all layers. When calculating the equivalent features of FGM face layers, the rule of mixtures is utilized as a homogenization technique. The ERF core's constitutive law is found to be in the pre-yield zone by considering the magnitudes of the electric field applied. Employing Hamilton's principle, the dynamic equations in terms of displacement components are derived. The discrete version of these equations is achieved by the use of the Chebyshev collocation tool. Parametric investigations are carried out after analyzing the efficacy and precision of the used approach and formulation via the presentation of verification cases. In the parametric section, the contribution of each geometric and material characteristic, as well as different boundary conditions, on the system frequencies and related mode shapes and loss factors, will be explored. [ABSTRACT FROM AUTHOR]

A mathematical model is developed to examine the effects of the Stefan blowing, second order velocity slip, thermal slip and microorganism species slip on nonlinear bioconvection boundary layer flow of a nanofluid over a horizontal plate embedded in a porous medium with the presence of passively controlled boundary condition. Scaling group transformations are used to find similarity equations of such nanobioconvection flows. The similarity equations are numerically solved with a Chebyshev collocation method. Validation of solutions is conducted with a Nakamura tri-diagonal finite difference algorithm. The effects of nanofluid characteristics and boundary properties such as the slips, Stefan blowing, Brownian motion and Grashof number on the dimensionless fluid velocity, temperature, nanoparticle volume fraction, motile microorganism, skin friction, the rate of heat transfer and the rate of motile microorganism transfer are investigated. This work is relevant to bio-inspired nanofluid-enhanced fuel cells and nano-materials fabrication processes. [ABSTRACT FROM AUTHOR]

In this paper, a new approach for solving space fractional order diffusion equations is proposed. The fractional derivative in this problem is in the Caputo sense. This approach is based on shifted Chebyshev polynomials of the fourth kind with the collocation method. The finite difference method is used to reduce the equations obtained by our approach for a system of algebraic equations that can be efficiently solved. Numerical results obtained with our approach are presented and compared with the results obtained by other numerical methods. The numerical results show the efficiency of the proposed approach. [ABSTRACT FROM AUTHOR]

The free vibration analysis of non-local orthotropic Kirchhoff plates has been investigated. Kirchhoff plates at the micro/nanoscale are modeled using Eringen's nonlocal elasticity theory, where the small scale effect is taken into consideration. The governing equations are derived using the nonlocal differential constitutive relations of Eringen. For this purpose, the resulted eigenvalue problem is solved numerically by applying the Chebyshev collocation method. The effects of the the Winkler modulus parameter, the shear modulus parameter, the aspect ratio, the taper, the nonlocal scale coefficient, and the boundary conditions on the natural frequencies have been studied. [ABSTRACT FROM AUTHOR]

Based on the Filon–Clenshaw–Curtis method for highly oscillatory integrals, and together with the Sommariva’s result (Sommariva, 2013) for Clenshaw–Curtis quadrature rule, we present a Chebyshev collocation method for a class of Fredholm integral equations with highly oscillatory kernels, whose unknown function is assumed to be less oscillatory than the kernel. In the proposed method, the Filon–Clenshaw–Curtis method is used to compute the involved oscillatory integrals, which makes the proposed method very precise. By solving only a small system of linear equations, we can obtain a very satisfactory numerical solution. The performance of the presented method is illustrated by several numerical examples. Compared with the method proposed by Li et al. (2012), this method enjoys a lower computational complexity. Furthermore, numerical examples show that the presented method has a competitive advantage on the accuracy compared with the method in Li et al. (2012). [ABSTRACT FROM AUTHOR]

In this present study, Chebyshev collocation method is utilized to solve buckling and vibration problems of functionally graded (FG) sandwich beams resting on two-parameter elastic foundation including Winkler and shear layer springs. The faces of FG sandwich beam are assumed to be made by functionally graded materials (FGMs) composing of ceramic and metal phases and the core of the beam is made from homogenous material. Timoshenko beam theory is employed to construct the governing equations of motion in order to cover the significant effects of shear deformation and rotary inertia. The beams with various boundary conditions are considered to find out their critical loadings and natural frequencies. An accuracy of the present solutions is confirmed by comparing with some available results in the literature. Moreover, many important parametric studies of layer and beam thickness ratios, material volume fraction index, spring constants, etc. are taken into investigation. According to numerical exercises, it is revealed that the spring constants of elastic foundation have significant impact on buckling and vibration results of such beams. [ABSTRACT FROM AUTHOR]

In this paper, we propose a numerical scheme to solve space fractional order diffusion equation. Our scheme uses shifted Chebyshev polynomials of the third kind. The fractional differential derivatives are expressed in terms of the Caputo sense. Moreover, Chebyshev collocation method together with the finite difference method are used to reduce these types of differential equations to a system of algebraic equations which can be solved numerically. Numerical approximations performed by the proposed method are presented and compared with the results obtained by other numerical methods. The results reveal that our method is a simple and effective numerical method. [ABSTRACT FROM AUTHOR]

• The instability of flow of an electrically conducting fluid in a channel filled with a saturated porous material is investigated. • The Brinkman-Darcy model has been used. • The effects of a uniform magnetic field and slip boundary conditions has been studied. • Two Chebyshev collocation techniques are used. We investigate the linear instability of a fully developed pressure-driven flow of an electrically conducting fluid in a porous channel using the Brinkman-Darcy model and the additional effects of a uniform magnetic field and slip boundary conditions. Two Chebyshev collocation techniques are used to solve the ordinary eigenvalue system governing the onset of convection. The critical Reynolds number, R e c , wavenumber, a c , and wave speed c c are found in terms of the porous parameter M , the dimensionless slip length, N 0 , and the Hartmann number H a. [ABSTRACT FROM AUTHOR]

Active reflection control system is a useful tool to improve the low-frequency performance of the sound absorption coating. However, the reflection reduction level of the previous system is unsatisfactory. It is of great value to evaluate the various factors that may affect the control performance, and to identify the main limitations, which in turn helps to develop an effective control system. For this purpose, the 2D (two-dimensional) time-domain simulation based on the Chebyshev collocation method is exploited to investigate the main influence factors in the control system. The simulation scheme is accurate enough to provide efficient and reliable solutions; in addition, it avoids the defects in experiments, where all the influence factors are mixed together so that they cannot be judged independently. Three factors are chosen to be analyzed one by one, and the design principles to improve the control performance are obtained: the accuracy of the time delay from the reflective surface to the virtual error sensor is the most important one, thus a FIR (Finite Impulse Response) filter should be introduced in the control algorithm, in order to predict an accurate reflected sound pressure; a compromise is required between the good performance and the compactness of the control system. [ABSTRACT FROM AUTHOR]

The purpose of this study is to apply the Chebyshev collocation method to linear second-order partial differential equations (PDEs) under the most general conditions. The method is given with a priori error estimate which is obtained by polynomial interpolation. The residual correction procedure is modified to the problem so that the absolute error may be estimated. Finally, the effectiveness of the method is illustrated in several numerical experiments such as Laplace and Poisson equations. Numerical results are overlapped with the theoretical results. [ABSTRACT FROM AUTHOR]

The free vibration analysis of non-local annular sector Mindlin plates has been investigated. Mindlin plates at the micro/nano-scale are modeled using Eringen׳s nonlocal elasticity theory, where the small scale effect is taken into consideration. The governing equations are derived using the nonlocal differential constitutive relations of Eringen. For this purpose, the resulted eigenvalue problem is solved numerically by applying the Chebyshev collocation method. The effects of the inner to outer radius ratio, the thickness to outer radius ratio, the nonlocal scale effect, and the boundary conditions on the natural frequencies have been studied. [ABSTRACT FROM AUTHOR]

Flexural vibration analysis of beams made of functionally graded materials (FGMs) with various boundary conditions is considered in this paper. Due to technical problems during FGM fabrication, porosities and micro-voids can be created inside FGM samples which may lead to the reduction in density and strength of materials. In this investigation, the FGM beams are assumed to have even and uneven distributions of porosities over the beam cross-section. The modified rule of mixture is used to approximate material properties of the FGM beams including the porosity volume fraction. In order to cover the effects of shear deformation, axial and rotary inertia, the Timoshenko beam theory is used to form the coupled equations of motion for describing dynamic behavior of the beams. To solve such a problem, Chebyshev collocation method is employed to find natural frequencies of the beams supported by different end conditions. Based on numerical results, it is revealed that FGM beams with even distribution of porosities have more significant impact on natural frequencies than FGM beams with uneven porosity distribution. [ABSTRACT FROM AUTHOR]

A Chebyshev collocation method is presented for the static analysis of the geometrically exact beams undergoing arbitrary large deflections using intrinsic formulation. In the intrinsic formulation of the governing equations of the spatial beams, neither displacement nor rotation variables appear. The proposed collocation technique is based on the Chebyshev points as the collocation points and the orthogonal Chebyshev polynomials as the trial functions. This method is successfully applied to the static solution of the strong form of the intrinsic governing equations of the nonlinear spatial beam and in order to show the applicability, accuracy and computational efficiency of the proposed method the numerical results are compared to the experimental, analytical and other numerical results for a number of test cases. [ABSTRACT FROM AUTHOR]

This paper investigates the dynamic response of Timoshenko beams made of functionally graded materials (FGMs). The beams are supported by various classical and non-classical boundary conditions. By using Timoshenko beam theory to establish the governing equations of motion for describing the vibration behavior of the beams, the significant effects of shear deformation and rotary inertia are taken into account. Different mathematical models that is the power law, exponential and Mori–Tanaka models are used to describe material composition across the beam thickness. To predict accurate vibration behavior of the beams, the Chebyshev collocation method (CCM) is applied to solve the vibration problem of such beams. According to the numerical results, it is revealed that the proposed modeling and analysis method can provide accurate frequency results of the beams as compared to some cases in the literature. New frequency results of the beams with different material compositions and boundary conditions are also presented for future comparison and development. [ABSTRACT FROM AUTHOR]

The purpose of this article is to examine the dynamic behavior of shear deformation beams subjected to high-speed thermal and mechanical loadings. The structure's theoretical equations are derived by employing the first-order shear deformation theory and nonlinear strain-displacement relationships. Loading is accomplished in two different time stages. Initially, the beam is subjected to stress and strain due to fast surface heating. We explore the circumstances that result in dynamic bending or buckling. After that, the bent beam is abruptly exposed to consistent pressure over time. For this reason, two-parameter mechanical loading is proposed. The first parameter indicates the final loading amount, while the second specifies the loading rate. As a consequence of this load, heat-induced displacements are decreased. Additionally, it is investigated whether mechanical loading results in snap-through in the structure based on the Budiansky criterion. The transient heat conduction equation is solved analytically, and the temperature profile is obtained over time. The nonlinear dynamic partial differential equations are then solved by the Chebyshev collocation, Newmark, and Newton-Raphson numerical methods. Moreover, the effect of a nonlinear viscoelastic foundation on the beam response is examined in this research. After demonstrating validation, some novel outcomes are provided to characterize the system's response in various situations. Dynamic response of a beam under two-step thermal and mechanical loading resting on the nonlinear elastic foundation. [Display omitted] • Nonlinear dynamic analysis of beams under two-step thermal and mechanical loads. • Utilizing a nonlinear viscoelastic foundation to control the beam response. • Investigation the occurrence of dynamic thermal buckling in beams under heat shock. • Dynamic snap-through buckling of beams subjected to two-parameter pressure loading using the Budiansky criterion. • Chebyshev method and Newmark and Newton-Raphson procedures to calculate the system response. [ABSTRACT FROM AUTHOR]

In this article, an implementation of an efficient numerical method for solving the linear fractional Klein-Gordon equation (LFKGE) is introduced. The fractional derivative is described in the Caputo sense. The method is based upon a combination between the properties of the Chebyshev approximations and finite difference method (FDM). The proposed method reduces LFKGE to a system of ODEs, which is solved using FDM. Special attention is given to study the convergence analysis and deduce an error upper bound of the proposed method. Numerical example is given to show the validity and the accuracy of the proposed algorithm. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The stability of fluid flow in a horizontal layer of Brinkman porous medium with fluid viscosity different from effective viscosity is investigated. A modified Orr-Sommerfeld equation is derived and solved numerically using the Chebyshev collocation method. The critical Reynolds number Re c , the critical wave number α c and the critical wave speed c c are computed for various values of porous parameter and ratio of viscosities. Based on these parameters, the stability characteristics of the system are discussed in detail. Streamlines are presented for selected values of parameters at their critical state. [ABSTRACT FROM AUTHOR]

The main purpose of this work is to provide a numerical approach for the delay partial differential equations based on a spectral collocation approach. In this research, a rigorous error analysis for the proposed method is provided. The effectiveness of this approach is illustrated by numerical experiments on two delay partial differential equations. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

SUMMARY This paper addresses the accuracy of numerical methods to compute the transient energy growth of plane Poiseuille flow. We show that using the Chebyshev collocation method to discretize the linearized governing equations in the wall-normal direction can introduce numerical problems when computing the energy evolution of the flow. We demonstrate that spurious eigenmodes of the discretized linear operator and numerical integration errors are the possible sources of the numerical problems, and we also show that spurious eigenvalues with negative real parts of large magnitude can affect the calculation of energy growth. These difficulties can be avoided by using a spectral Galerkin method where the basis functions satisfy the boundary conditions. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The purpose of this study is to present a new collocation method for the solution of second-order, linear partial differential equations (PDEs) under the most general conditions. The method has improved from Chebyshev matrix method, which has been given for solving of ordinary differential, integral and integro-differential equations. The method is based on the approximation by the truncated bivariate Chebyshev series. PDEs and conditions are transformed into the matrix equations, which corresponds to a system of linear algebraic equations with the unknown Chebyshev coefficients, via Chebyshev collocation points. Combining these matrix equations and then solving the system yields the Chebyshev coefficients of the solution function. Finally, the effectiveness of the method is illustrated in several numerical experiments and error analysis is performed. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with some theoretical and numerical results for Volterra Integral Algebraic Equations (IAEs) of index-1 with weakly singular kernels. This type of equations typically has solutions whose derivatives are unbounded at the left endpoint of the interval of integration. For overcoming this non-smooth behavior of solutions, using the appropriate coordinate transformation the primary system is changed into a new IAEs which its solutions have better regularity. An effective numerical method based on the Chebyshev collocation scheme is designed and its convergence analysis is provided. Our numerical experiments show that the theoretical results are in good accordance with actual convergence rates obtained by the given algorithm. [Copyright &y& Elsevier]

Abstract: We present a simple technique for avoiding physically spurious eigenmodes that often occur in the solution of hydrodynamic stability problems by the Chebyshev collocation method. The method is demonstrated on the solution of the Orr–Sommerfeld equation for plane Poiseuille flow. Following the standard approach, the original fourth-order differential equation is factorised into two second-order equations using a vorticity-type auxiliary variable with unknown boundary values which are then eliminated by a capacitance matrix approach. However the elimination is constrained by the conservation of the structure of matrix eigenvalue problem, it can be done in two basically different ways. A straightforward application of the method results in a couple of physically spurious eigenvalues which are either huge or close to zero depending on the way the vorticity boundary conditions are eliminated. The zero eigenvalues can be shifted to any prescribed value and thus removed by a slight modification of the second approach. [Copyright &y& Elsevier]

The objective of this paper is the development of a new numerical technique for the free vibration analysis of isotropic rectangular and annular Mindlin plates with damaged boundaries. For this purpose, the Chebyshev collocation method is applied to obtain the natural frequencies of Mindlin plates with damaged clamped boundary conditions, where the governing equations and boundary conditions are discretized by the presented method and put into matrix vector form. The damaged boundaries are represented by distributed translational and torsional springs. In the present study the boundary conditions are coupled with the governing equation to obtain the eigenvalue problem. Convergence studies are carried out to determine the sufficient number of grid points used. First, the results obtained for the undamaged plates are verified with previous results in the literature. Subsequently, the results obtained for the damaged Mindlin plate indicate the behavior of the natural vibration frequencies with respect to the severity of the damaged boundary. This analysis can lead to an efficient technique for structural health monitoring of structures in which joint or boundary damage plays a significant role in the dynamic characteristics. The results obtained from the Chebychev collocation solutions are seen to be in excellent agreement with those presented in the literature. [ABSTRACT FROM PUBLISHER]

Spectral methods seek the solution to a differential equation in terms of series of known smooth function. The Chebyshev series possesses the exponential-convergence property regardless of the imposed boundary condition, and therefore is suited for the regional modeling. We propose a new domain-decomposed Chebyshev collocation method which facilitates an efficient parallel implementation. The boundary conditions for the individual sub-domains are exchanged through one grid interval overlapping. This approach is validated using the one dimensional advection equation and the inviscid Burgers' equation. We further tested the vortex formation and propagation problems using two-dimensional nonlinear shallow water equations. The domain decomposition approach in general gave more accurate solutions compared to that of the single domain calculation. Moreover, our approach retains the exponential error convergence and conservation of mass and the quadratic quantities such as kinetic energy and enstrophy. The efficiency of our method is greater than one and increases with the number of processors, with the optimal speed up of 29 and efficiency 3.7 in 8 processors. Efficiency greater than one was obtained due to the reduction the degrees of freedom in each sub-domain that reduces the spectral operational count and also due to a larger time step allowed in the sub-domain method. The communication overhead begins to dominate when the number of processors further increases, but the method still results in an efficiency of 0.9 in 16 processors. As a result, the parallel domain-decomposition Chebyshev method may serve as an efficient alternative for atmospheric and oceanic modeling. [ABSTRACT FROM AUTHOR]

Abstract: This paper proposes a new numerical technique for the free vibration analysis of non-rotating and rotating Timoshenko beams with damaged boundaries. For this purpose, the Chebyshev collocation method is applied to obtain the natural frequencies and mode shapes. The damaged boundaries of the Timoshenko beam are represented by distributed translational and torsional springs. In the present study the boundary conditions are coupled with the governing equation to formulate the eigenvalue problem. [Copyright &y& Elsevier]

The Chebyshev collocation method in [21] to solve stiff initial-value problems is generalized by using arbitrary degrees of interpolation polynomials and arbitrary collocation points. The convergence of this generalized Chebyshev collocation method is shown to be independent of the chosen collocation points. It is observed how the stability region does depend on collocation points. In particular, A-stability is shown by taking the mid points of nodes as collocation points. [ABSTRACT FROM AUTHOR]

Abstract: The morphology of telephone cord buckles of elastic thin films can be used to evaluate the initial residual stress and interface toughness of the film–substrate system. The maximum out-of-plane displacement , the wavelength and amplitude of the wave buckles can be measured in physical experiments. Through , , and , the buckle morphology is obtained by an annular sector model established using the von Karman plate equations in polar coordinates for the elastic thin film. The mode-mix fracture criterion is applied to determine the shape and scale parameters. A numerical algorithm combining the Newmark- scheme and the Chebyshev collocation method is adopted to numerically solve the problem in a quasi-dynamic process. Numerical experiments show that the numerical results agree well with physical experiments. [Copyright &y& Elsevier]

In this paper, we present error corrected Euler methods for solving stiff initial value problems, which not only avoid unnecessary iteration process that may be required in most implicit methods but also have such a good stability as all implicit methods possess. The proposed methods use a Chebyshev collocation technique as well as an asymptotical linear ordinary differential equation of first-order derived from the difference between the exact solution and the Euler's polygon. These methods with or without the Jacobian are analyzed in terms of convergence and stability, in particular, it is proved that the proposed methods have a convergence order up to 4 regardless of the usage of the Jacobian. Numerical tests are given to support the theoretical analysis as evidences. [ABSTRACT FROM AUTHOR]