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In this paper, Chebyshev collocation method is applied to fractional Riccati differential equation (FRDE) using the shifted Chebyshev polynomials of the third kind. Approximate analytical solution of FRDE is considered as Chebyshev series expansion. The fractional derivative is described in the Caputo sense. Using properties of Chebyshev polynomials FRDE with initial condition is reduced to a nonlinear system of algebraic equations which solved by the Newton iteration method. The accuracy and efficiency of the proposed method is illustrated by numerical examples.

The topic of numerical methods for solving fractional differential equations (FDEs) with Riemann-Liouville (RL) derivatives has not received extensive attention compared to the ones for solving FDEs with Caputo derivatives. There is, also, not a sophisticated method to approximate fractional-order derivatives of a function in the sense of RL. In this paper, a new representation of FDEs with fractional-order initial conditions is given, which can be solved in a proposed spectral collocation framework. For this purpose, a new operational matrix of left-sided RL fractional differentiation is constructed to approximate the left-sided RL derivative operator at Chebyshev-Gauss-Lobatto points. In numerical examples, the advantages of using the proposed operational matrix in calculating fractional derivatives of a function or solving FDEs with RL derivatives are illustrated.

Based on the idea of collocation methods, a spline-based integration method for the stability prediction of systems governed by time-periodic delay differential equations (DDEs) is presented. A B-spline curve is adopted to approximate the solution of the DDEs using the direct integration technique. The stability of the dynamic system is then predicted using the Floquet theory based on the established state transition matrix. The proposed method can apply to time-periodic DDEs where the delay and the period are incommensurate, and the proposed method is extended to study the stability of milling processes as well. For stability analysis of variable pitch cutters, the proposed method use one or multiple B-spline curves to approximate the solution of the DDEs according to the relationship between the cutter angles and the radial immersion, which can describe exactly the free vibration of the cutter. Compared with the numerical integration method and the Chebyshev collocation method, the simulation results demonstrate that the proposed technique is more efficient and accurate for stability prediction at low radial immersion cases in milling processes with variable pitch cutters.

Temporal instability of liquid-metal flow in a square duct is investigated using a two-dimensional Chebyshev collocation method. In this study, the flow is subjected to a transverse magneti...

The stability and free vibration analyses of single and double composite Timoshenko beams have been investigated. The closed-section beams are subjected to constant axially compressive or tensile forces. The double beams are assumed to be connected by a layer of elastic translational and rotational springs. The coupled governing partial differential equations of motion are discretized, and the resulted eigenvalue problem is solved numerically by applying the Chebyshev spectral collocation method. The effects of the elastic layer parameters, the axial forces, the slenderness ratio, the bending–torsional coupling, and the boundary conditions on the critical buckling loads, mode shapes, and natural transverse frequencies have been studied. A parametric study was performed, and the obtained results revealed different features, which hopefully can be useful for single- and double-beam-like engineering structures.

Bu çalışma üç ana bölüm şeklinde düzenlenmiştir. Birinci bölümde Burgers denklemi, denklemin analitik ve sayısal çözümleri üzerine literatür bilgileri verilmiştir. ikinci bölümde denklemin analitik çözümleri sunulmuş, son bölümde ise Burgers denklemi için Chebyshev polinomlarına dayalı bir sıralama yöntemi sunulmuş ve bu yöntemin uygulanabilirliğini ve verimliliğini göstermek için çeşitli örnekler ele alınmıştır. This study is organized as three main chapters. In the first chapter, literatures on the Burgers equation and their analytical and numerical solutions are given. In the second chapter, the analytical solutions of the Burgers equation are given and in the last chapter, a collocation method based on the Chebyshev polynomials is presented for the Burgers equations and some examples are discussed to demonstrate the accuracy, applicability and the efficiency of this method. 96

Bu tez çalışmasında en önemli evolüsyon denklemlerinden biri olan adveksiyon difüzyon denklemlerinin temsil ettiği model problemlerin çözümleri için nümerik algoritmalar geliştirildi. Adveksiyon difüzyon denklemleri akışkanlar dinamiği, kütle transferi, enerji transferi ve sürekli olasılıksal süreçlerin fiziksel olgularını modeller. Kompakt sonlu fark yöntemi ve Chebyshev spektral kolekasyon yöntemi, lineer ve lineer olmayan adveksiyon difüzyon denklemlerinin uzaysal türevlerinin ayrıklaştırılması için kullanıldı.Uzaysal değişkenlerin başarılı bir şekilde ayrıştırılmasından sonra, denklemlerin tam ayrıklaştırılmış çözümlerini elde edebilmek için, dördüncü mertebe açık Runge–Kutta metodu ve ikinci mertebe kapalı Crank– Nicolson metodu zaman integrasyonu için kullanıldı. Önerilen yöntemlerin etkinliğinin analiz edilebilmesinde, çeşitli adveksiyon difüzyon denklemleri için üretilen nümerik çözümler analitik çözümlerle ve literatür ile karşılaştırıldı. Ortaya konan yöntemlerin model problemleri çözmekte etkili sonuçlar verdiği gösterildi. Model problemlerin fiziksel davranışlarının sunulabilmesi için kayda değer örnekler ele alındı. Adveksiyonun baskın olduğu durumlarda dahi önerilen yöntemlerin yüksek doğrulukta ve osilasyon içermeyen çözümler üretebildiği görüldü. In this study we construct numerical algorithms to solve model problems describedby one of the most important evolution equations, advection–diffusion equation (ADE). The ADE models a variety of physical phenomena in fluid dynamics,mass transfer, energy transfer and even in continuous stochastic processes. Compact finite difference scheme and Chebyshev spectral collocation method are presented for the spatial discretization of the ADE by considering both linear and nonlinear processes. After successful discretization in space, to solve the ADE, we choose appropriate time integration techniques both including explicit Runge–Kutta method of order four and implicit Crank–Nicolson method. To analyze the efficiency of the proposed methods, numerical results of the present schemes for various ADE problems have been computed and compared with exact solutions and with available literature. The proposed approaches have been shown to be capable of solving the model equations effectively. Challenging examples have been taken to illustrate the physical behaviour of the model in detail. The computed results have been seen to be highly accurate and be oscillation free even if we consider advection dominated cases. 89

The onset of convection in the Rayleigh-Bénard problem for a monatomic rarefied gas at small Knudsen number has been investigated. Compressibility-induced density variations have been considered without imposing any restriction on the magnitude of temperature difference. A linear temporal stability analysis has been conducted for a compressible slip-flow model considering a Maxwellian gas and the dispersion relation is calculated using a Chebyshev collocation method. A neutral stability curve obtained in the Froude-Knudsen number plane marks transition to convection from a pure conduction state. The critical wave number observed for the onset of convection is in good agreement with the existing literature. A comparison of two molecular interaction models: hard-sphere and Maxwellian gas, a more realistic model, for predicting the boundaries of the convection domain has been presented here which is expected to be useful for future studies on related topics using more realistic gas models.

In this paper, we use chebyshev polynomial basis functions to solve the Fredholm and Volterra integro-differential equations. We directly calculate integrals and other terms are calculated by approximating the functions with the Chebyshev polynomials. This method affects the computational aspect of the numerical calculations. Application of the method on some examples show its accuracy and efficiency.

An annular sector model for the telephone cord buckles of elastic thin films on rigid substrates is established, in which the von Karman plate equations in polar coordinates are used for the elastic thin film and a discrete version of the Griffith criterion is applied to determine the shape and scale of the parameters. A numerical algorithm combining the Newmark-β scheme and the Chebyshev collocation method is designed to numerically solve the problem in a quasi-dynamic process. Numerical results are presented to show that the numerical method works well and the model agrees well with physical observations, especially successfully simulated for the first time the telephone cord buckles with two humps along the ridge of each section of a buckle.

In this paper, an efficient numerical method is introduced for solving the fractional (Caputo sense) Fisher equation. We use the spectral collocation method which is based upon Chebyshev approximations. The properties of Chebyshev polynomials of the third kind are used to reduce the proposed problem to a system of ODEs, which is solved by the finite difference method (FDM). Some theorems about the convergence analysis are stated and proved. A numerical simulation is given and the results are compared with the exact solution.

We study the electrically conducting fluid stability of magnetohydrodynamic flow between parallel plates by Chebyshev collocation method by applied transverse magnetic field. Temporal growth is obtained by the governing equations. The results show that the dominating factor is the change in shape of the undisturbed velocity profile caused by the magnetic field, which depends only on the Hartmann number. The stability equations is solved by QZ-algorithm to find the eigenvalue problem. The numerical calculation show that Magnetic field with particular magnitude destabilizes Couette flow while other magnitude stabilize the flow. It is also analyzed that Rec decreases rapidly to the minimum value for Hartmann number Ha greater than 3.887 and increases steadily from Hartmann number Ha (>5.559). It is observed that critical Reynolds number Rec is larger as Hartmann number decreases from Ha=3.887. The two critical Reynolds numbers in the elliptic curves are found for different values of Hartmann number.

Two-dimensional boundary characteristic orthonormal polynomials are used in Rayleigh-Ritz method to study the title problem. In general, it is found that this method gives better results than the other traditional method such as boundary integral equation methods, Spline methods, Chebyshev collocation method, Frobenius method etc. The thickness is taken to be linearly varying in two orthogonal directions. Comparisons in particular cases have been made with the existing results in the literature. Convergence of frequencies of at least up to five significant figures is obtained. Results showing the variation in frequencies with taper parameters and aspect ratios are presented in tabular form. Mode shapes are shown using three-dimensional graphs of plates in displaced configurations.

In this paper, we discuss the multi-interval Chebyshev (MIC) method for analog and radio-frequency (RF) circuit simulations. The method has been applied in a wide variety of circuit analyses including periodic steady-state, time-varying small signal, and cyclostationary noise analyses for both driven and autonomous circuits. In contrast to traditional analog/RF circuit simulations using either low-order time-domain integration methods or harmonic balance methods, we show that the MIC method can efficiently achieve high accuracy on strongly nonlinear circuits possessing waveforms with rapid transitions. In addition, it has the same flexibility in simulating frequency-dependent components as the harmonic balance methods but can use similar preconditioning strategies and matrix-implicit Krylov-subspace solvers as time-domain techniques, leading to good convergence on nonlinear problems.

An efficient technique for solving parabolic partial integrodifferential equation is presented. This technique is based on Chebyshev polynomials and finite difference method.A priorierror estimate for the proposed technique is deduced. Some examples are presented to illustrate the validity and efficiency of the presented method.

Asymptotic method was adopted to obtain a receptivity model for a pipe Poiseuille flow under periodical pressure, the wall of the pipe with a bump. Bi-orthogonal eigen-function systems and Chebyshev collocation method were used to resolve the problem. Various spatial modes and the receptivity coefficients were obtained. The results show that different modes dominate the flow in different stages, which is comparable with the phenomena observed in experiments.

OZTURK, Yalcin/0000-0002-4142-5633 WOS: 000366539300035 Computer viruses are serious problems for individual and corporate computer systems, and thus many studies have investigated how to avoid their deleterious effects by creating anti-virus programs that act as vaccines in personal computers or strategic network nodes. An alternative approach to combat the propagation of viruses is establishing preventive policies based on the overall operation of a system, which can be modeled with population models similar to those used in epidemiological studies. In this study, we propose an approximate solution to computer viruses, which is a modified version of the Susceptible-Infected-Removed model. We use the shifted Chebyshev collocation method to solve the computer virus dynamical model. This method transforms the system of nonlinear differential equations and the given conditions into a matrix equation, which corresponds to a system of nonlinear algebraic equations. We also present some numerical results. (C) 2015 Elsevier Inc. All rights reserved. Scientific Research Project of Mugla Sitki Kocman UniversityMugla Sitki Kocman University [12/93] The authors would like to thank the anonymous referee for their time, effort, and extensive comments on the revised version of the manuscript. This research was supported by the Scientific Research Project of Mugla Sitki Kocman University under No. 12/93.

The Chebyshev-collocation method is briefly presented in a manner favorable for solving ODEs in geophysics depending on geocentric distance and singular at the Earth center. The main advantage of this method is that it is not needed to treat the ODEs particularly at the Earth center as is traditionally done when using the Runge-Kutta method. Comparisons of spheroidal free oscillation periods and load Love numbers with earlier works show very close agreement.

航空宇宙技術研究所 16-18 Jun. 1999 東京 日本, National Aerospace Laboratory 16-18 Jun. 1999 Tokyo Japan, チャンネル流の層流-乱流遷移を並列計算機上で直接数値シミュレーションを行った。シミュレーションは、基本流と様々なレイノルズ数に対応する異なるエネルギーレベルの小さな撹乱の和で与えられる初期流れで開始する。超臨界レイノルズ数の場合、以下の2つの機構により遷移が起こる。ひとつは初期撹乱が非常に小さい場合、流れの中で成長するトルミーン・シュリヒティング波であり、もうひとつは初期撹乱が大きい場合、流れの方向に軸を持つ縦渦の過渡的増幅である。一方、亜臨界レイノルズ数では、初期撹乱が非常に大きい場合にのみ過渡的増幅により遷移が起こる。過渡的増幅型遷移を引き起こす初期撹乱のエネルギーレベルは、レイノルズ数の-7/2乗に比例することが見出された。, Laminar-turbulent transition of channel flow is directly simulated on a parallel computer. The simulations are started with initial flows given as the basic flow plus small disturbances with different energy levels for various Reynolds numbers. In the case of supercritical Reynolds numbers, the transition is triggered by two mechanisms: one is Tollmien-Schlichting waves growing in the flow when initial disturbances are very small, and the other is transient growth of stream-wise vortices when initial disturbances are rather large. On the other hand, for subcritical Reynolds numbers, transition is triggered by the transient growth only when initial disturbances are considerably large. It is found that the energy levels of initial disturbances, which can trigger the transient-growth type transition, are proportional to -7/2 power of the Reynolds number., 資料番号: AA0001961054, レポート番号: NAL SP-44

A new preconditioning method is investigated to reduce the roundoff error in computing derivatives using Chebyshev collocation methods(CCM). Using this preconditioning causes ratio of roundoff error of preconditioning method and CCM becomes small when N gets large. Also for accuracy enhancement of differentiation we use a domain decomposition approach. Error analysis shows that for this domain decomposition method error reduces proportional to the length of subintervals. Numerical results show that using domain decomposition and preconditioning simultaneously, gives super accurate approximate values for first derivative of the function and good approximate values for moderately high derivatives.

In this study, a matrix method called the Chebyshev collocation method is presented for numerically solving the linear integro-differential equations by a truncated Chebyshev series. Using the Chebyshev collocation points, this method transforms the integro-differential equation to a matrix equation which corresponds to a system of linear algebraic equations with unknown Chebyshev coefficients. Therefore this allows us to make use of the computer. Also the method can be used for differential and integral equations. To illustrate the method, it is applied to certain linear differential, integral, and integro-differential equations, and the results are compared.

In this paper we in part recreate simulations in the 8:1 differentially heated cavity that were carried out by Xin and Le Quere [1]. Their method uses second order time stepping scheme and Chebyshev collocation method for spatial dimensions. Our results with the same method differ for about 10−4 from theirs. We further performed simulations using a method based on trigonometric functions for spatial dimensions. Results using the latter method differ for about few percent compared to results in [1].

Convection in an infinite horizontal slot subject to spatially distributed heating has been investigated for a wide range of Prandtl numbers. The primary flow response consists of convection in the form of rolls. When heating wave number alpha is sufficiently large the convection is found to be limited to a layer adjacent to lower wall and a uniform conductive layer emerges at upper section of the slot. Conditions leading to the emergence of secondary convection have been identified using linear stability of the above primary convection. The secondary convection gives rise to longitudinal, or transverse, or oblique rolls; or to oscillatory mode at onset. Three mechanisms of instability have been identified. For small and moderate alpha parametric resonance leads to the pattern of instability that is locked-in with the pattern of heating. The second mechanism is associated with the formation of patterns of vertical temperature gradients and patterns of primary convection currents, operates approximately in the same range of alpha. The third mechanism operates for large alpha where instability is driven by uniform mean vertical temperature gradient created by primary convection and fluid response becoming similar to that found in the case of uniformly heated wall. A rapid stabilization of the oblique rolls is observed when alpha is reduced sufficiently, with the oscillatory mode taking the dominant role. As alpha becomes very small, secondary motions concentrate around the hot spots. When an external flow is introduced into the slot, the heating assists in reduction of the overall drag.

In order to compare numerical and analytical results for the free vibration analysis of Kirchhoff plates with both partially and completely damaged boundaries, the Chebyshev collocation and perturbation methods are utilized in this paper, where the damaged boundaries are represented by distributed translational and torsional springs. In the Chebyshev collocation method, the convergence studies are performed to determine the sufficient number of the grid points used. In the perturbation method, the small perturbation parameter is defined in terms of the damage parameter of the plate, and a sequence of recurrent linear boundary value problems is obtained which is further solved by the separation of variables technique. The results of the two methods are in good agreement for small values of the damage parameter as well as with the results in the literature for the undamaged case. The case of mixed damaged boundary conditions is also treated by the Chebyshev collocation method.

In this study, matrix representation of the Chebyshev collocation method for partial differential equation has been represented and applied to solve magnetohydrodynamic (MHD) flow equations in a rectangular duct in the presence of transverse external oblique magnetic field. Numerical solution of velocity and induced magnetic field is obtained for steady-state, fully developed, incompressible flow for a conducting fluid inside the duct. The Chebyshev collocation method is used with a reasonable number of collocations points, which gives accurate numerical solutions of the MHD flow problem. The results for velocity and induced magnetic field are visualized in terms of graphics for values of Hartmann number H≤1000. Copyright © 2010 John Wiley & Sons, Ltd.

By the use of the Chebyshev series, a direct computational method for solving the higher order nonlinear differential equations has been developed in this paper. This method transforms the nonlinear differential equation into the matrix equation, which corresponds to a system of nonlinear algebraic equations with unknown Chebyshev coefficients, via Chebyshev collocation points. The solution of this system yields the Chebyshev coefficients of the solution function. An algorithm for this nonlinear system is also proposed in this paper. The method is valid for both initial-value and boundary-value problems. Several examples are presented to illustrate the accuracy and effectiveness of the method. © 2010 Elsevier Inc. All rights reserved.

In order to compare numerical and analytical results for the free vibration analysis of Kirchoff plates with both mixed and damaged boundaries, the Chebyshev collocation and perturbation methods are utilized in this paper, where the damaged boundaries are represented by distributed translational and torsional springs. In the Chebyshev collocation method, the convergence studies are performed to determine the sufficient number of the grid points used. In the analytical method, the small perturbation parameter is defined in terms of the damage parameter of the plate, and a sequence of recurrent linear boundary value problems is obtained which is further solved by the separation of variables technique. The results of the two methods are in good agreement for small values of the damage parameter as well as with the results in the literature for the undamaged case. This study can lead to an efficient technique for structural health monitoring (SHM).Copyright © 2010 by ASME

In this article, a new method is presented for the solution of high-order linear partial differential equations (PDEs) with variable coefficients under the most general conditions. The method is based on the approximation by the truncated double Chebyshev series. PDE 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. Some numerical results are included to demonstrate the validity and applicability of the method. (C) 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 25:610-621, 2009

In this article, a new method is presented for the solution of high-order linear partial differential equations (PDEs) with variable coefficients under the most general conditions. The method is based on the approximation by the truncated double Chebyshev series. PDE 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. Some numerical results are included to demonstrate the validity and applicability of the method. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009

The main purpose of this article is to present an approximation method for higher order linear Fredholm-Volterra integro-differential equations (FVIDE) in the most general form under the mixed conditions in terms of Chebyshev polynomials. This method transforms FVIDE and the conditions into the matrix equations which correspond to a system of linear algebraic equations with unknown Chebyshev coefficients. Finally, some examples are presented to illustrate the method and the results are discussed. (c) 2006 Elsevier Inc. All rights reserved. C1 Pamukkale Univ, Fac Sci, Dept Math, Denizli, Turkey.

oz Bu çalışma, uygulamalı matematik ve özellikle mühendislikte sıkça kullanılan nümerik çözüm tekniklerinden Runge-Kutta ve Spektral Chebyshev yöntemlerinin irdelenmesi ve bu güçlü tekniklerin modern akışkanlar tarihinin temellerinde olan düzgün bir tabaka üzerinde oluşan akışa uygulanması ile ilgili olup dört bölümden oluşmaktadır. Birinci bölüm, nümerik çözüm metodlarına olan gereksinim, başlangıç-değer prob lemlerine yaklaşık çözüm metodların uygulanabilme koşulları ve en basit nümerik metot olan Euler yöntemi tanıtılarak, hata tahmini ve kararlılığı ile hidrodinamik kararlılık teorisini içermektedir. Ancak bu çalışmamızda incelediğimiz problemin nümerik çözüm koşullarını sağladığı bilinmektedir. ikinci ve üçüncü bölümde sırasıyla, kullanacağımız nümerik çözüm yöntemlerinden Runge-Kutta ve Chebyshev sıralama yöntemleri tanıtıldı. Dördüncü ve son bölümde, akışkanlar mekaniğinin temel bir problemi olan düzgün yüzeylerdeki akışın matematiksel modellemesi olarak bilinen Blasius denklemi tanıtılarak, daha önceki bölümlerde bahsedilen nümerik metotlarla çözümü elde edildi. Nümerik çözüm için gerekli olan algoritmalar F77 ve F90 bilimsel program lama dili üzerine kuruldu. Son olarakta hidrodinamik kararlılık teorisi ışığında bu denklemin kararlılığı için Orr-Sommerfeld denklemi elde edilerek nümerik çözümü yapıldı ve Blasius denkleminin kararlılığı ile ilgili grafikler verildi. 1& ABSTRACT This study deals with the examination of Runge- Kutta and Spectral Cheby- shev Collocation processes of frequenly used numerical methods in applied math ematics and especially in engineering and application of these methods to the flow over a flat plate which is a basic problem in the modern fluid dynamics. The first chapter includes the neccessity and applicability condition to initial value problem of numerical solution methods, introduced Euler method which is the simplest numerical solution technic and Euler method's error estimation and stability with hyrodynamic stability theory. But it was known that the problem which we are interested in this study satisfies the condition of the numerical solution beforehand. In the second and third chapter, Runge-Kutta and Chebyshev Collocation nu merical solution methods which we will use are introduced respectively. In the fourth chapter, a basic problem of the fluid dynamics which is known as Blasius equation and is mathematical formulization of flow over the fiat plate is introduced and obtained solutions with advertised methods. All the algorithms in this thesis were constructed on the scientific programming languages F77 and F90. Finally, considering the hydrodynamic stability theory we got the Orr-Sommerfeld equation to solve by numerical methods for stability of Blasius equation and we gave related graphics. u 48