The Experts below are selected from a list of 360 Experts worldwide ranked by ideXlab platform
R M Hafez - One of the best experts on this subject based on the ideXlab platform.
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Chebyshev collocation treatment of volterra fredholm integral equation with error analysis
Arabian Journal of Mathematics, 2020Co-Authors: Y H Youssri, R M HafezAbstract:This work reports a collocation algorithm for the numerical solution of a Volterra–Fredholm integral equation (V-FIE), using shifted Chebyshev collocation (SCC) method. Some properties of the shifted Chebyshev polynomials are presented. These properties together with the shifted Gauss–Chebyshev nodes were then used to reduce the Volterra–Fredholm integral equation to the solution of a matrix equation. Nextly, the error analysis of the proposed method is presented. We compared the results of this algorithm with others and showed the accuracy and potential applicability of the given method.
Hakan Ali Cirpan - One of the best experts on this subject based on the ideXlab platform.
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a set of new Chebyshev kernel functions for support vector machine pattern classification
Pattern Recognition, 2011Co-Authors: Sedat Ozer, Chi H Chen, Hakan Ali CirpanAbstract:In this study, we introduce a set of new kernel functions derived from the generalized Chebyshev polynomials. The proposed generalized Chebyshev polynomials allow us to derive different kernel functions. By using these polynomial functions, we generalize recently introduced Chebyshev kernel function for vector inputs and, as a result, we obtain a robust set of kernel functions for Support Vector Machine (SVM) classification. Thus in this study, besides clarifying how to apply the Chebyshev kernel functions on vector inputs, we also increase the generalization capability of the previously proposed Chebyshev kernels and show how to derive new kernel functions by using the generalized Chebyshev polynomials. The proposed set of kernel functions provides competitive performance when compared to all other common kernel functions on average for the simulation datasets. The results indicate that they can be used as a good alternative to other common kernel functions for SVM classification in order to obtain better accuracy. Moreover, test results show that the generalized Chebyshev kernel approaches to the minimum support vector number for classification in general.
John P. Boyd - One of the best experts on this subject based on the ideXlab platform.
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numerical experiments on the accuracy of the Chebyshev frobenius companion matrix method for finding the zeros of a truncated series of Chebyshev polynomials
Journal of Computational and Applied Mathematics, 2007Co-Authors: John P. Boyd, Daniel H GallyAbstract:For a function f(x) that is smooth on the interval [email protected]?[a,b] but otherwise arbitrary, the real-valued roots on the interval can always be found by the following two-part procedure. First, expand f(x) as a Chebyshev polynomial series on the interval and truncate for sufficiently large N. Second, find the zeros of the truncated Chebyshev series. The roots of an arbitrary polynomial of degree N, when written in the form of a truncated Chebyshev series, are the eigenvalues of an NxN matrix whose elements are simple, explicit functions of the coefficients of the Chebyshev series. This matrix is a generalization of the Frobenius companion matrix. We show by experimenting with random polynomials, Wilkinson's notoriously ill-conditioned polynomial, and polynomials with high-order roots that the Chebyshev companion matrix method is remarkably accurate for finding zeros on the target interval, yielding roots close to full machine precision. We also show that it is easy and cheap to apply Newton's iteration directly to the Chebyshev series so as to refine the roots to full machine precision, using the companion matrix eigenvalues as the starting point. Lastly, we derive a couple of theorems. The first shows that simple roots are stable under small perturbations of magnitude @e to a Chebyshev coefficient: the shift in the root x"* is bounded by @e/df/dx(x"*)+O(@e^2) for sufficiently small @e. Second, we show that polynomials with definite parity (only even or only odd powers of x) can be solved by a companion matrix whose size is one less than the number of nonzero coefficients, a vast cost-saving.
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a test based on conversion to the bernstein polynomial basis for an interval to be free of zeros applicable to polynomials in Chebyshev form and to transcendental functions approximated by Chebyshev series
Applied Mathematics and Computation, 2007Co-Authors: John P. BoydAbstract:Abstract Polynomials of high degree are much less vulnerable to roundoff error when expressed as truncated Chebyshev series rather than the usual power series form. Recent articles have developed subdivision methods in which all real roots on the canonical Chebyshev interval, x ∈ [−1, 1], are found by subdividing the interval and finding the roots of separate Chebyshev series of moderate degree on each subdomain. This strategy can be applied either to polynomials or to transcendental functions if the latter are analytic on the search interval and thus have rapidly convergent Chebyshev polynomial approximations. The last step is to compute the eigenvalues of the Chebyshev–Frobenius companion matrix for each local polynomial. Here, we propose a simple strategy for flagging some subdomains as “zero-free” so that eigensolving can be omitted. The test requires conversion of the polynomial from Chebyshev form to a Bernstein polynomial basis. The interval is zero-free if all coefficients in the Bernstein basis are of the same sign. We give the conversion matrices for various small and moderate N and quote Rababah’s formulas for general N . We show how to exploit parity so as to halve the cost of the conversion for large N to about N 2 floating point operations. We show that the conversion matrices have condition numbers that are approximately (5/8)2 N .
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Computing the zeros, maxima and inflection points of Chebyshev, Legendre and Fourier series: solving transcendental equations by spectral interpolation and polynomial rootfinding
Journal of Engineering Mathematics, 2006Co-Authors: John P. BoydAbstract:Recently, both companion-matrix methods and subdivision algorithms have been developed for finding the zeros of a truncated spectral series. Since the Chebyshev or Legendre coefficients of derivatives of a function f ( x ) can be computed by trivial recurrences from those of the function itself, it follows that finding the maxima, minima and inflection points of a truncated Chebyshev or Fourier series f _ N ( x ) is also a problem of finding the zeros of a polynomial when written in truncated Chebyshev series form, or computing the roots of a trigonometric polynomial. Widely scattered results are reviewed and a few previously unpublished ideas sprinkled in. There are now robust zerofinders for all species of spectral series. A transcendental function f ( x ) can be approximated arbitrarily well on a real interval by a truncated Chebyshev series f _ N ( x ) of sufficiently high degree N . It follows that through Chebyshev interpolation and Chebyshev rootfinders, it is now possible to easily find all the real roots on an interval for any smooth transcendental function.
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computing real roots of a polynomial in Chebyshev series form through subdivision with linear testing and cubic solves
Applied Mathematics and Computation, 2006Co-Authors: John P. BoydAbstract:Abstract An arbitrary polynomial of degree N , f N ( x ), can always be represented as a truncated Chebyshev polynomial series (“Chebyshev form”). This representation is much better conditioned than the usual “power form” of a polynomial. We describe a new method for finding the real roots of f N ( x ) in Chebyshev form. The canonical interval, x ∈ [−1, 1], is subdivided into N s subintervals. Each is tested for zeros using the error bound for linear interpolation. On “zero-possible” intervals, f N is approximated by a cubic polynomial, whose roots are then found by the usual sixteenth century formulas.
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pseudospectral methods on a semi infinite interval with application to the hydrogen atom a comparison of the mapped fourier sine method with laguerre series and rational Chebyshev expansions
Journal of Computational Physics, 2003Co-Authors: John P. Boyd, Chitra Rangan, P H BucksbaumAbstract:The Fourier-sine-with-mapping pseudospectral algorithm of Fattal et al. [Phys. Rev. E 53 (1996) 1217] has been applied in several quantum physics problems. Here, we compare it with pseudospectral methods using Laguerre functions and rational Chebyshev functions. We show that Laguerre and Chebyshev expansions are better suited for solving problems in the interval r ∈ [0, ∞] (for example, the Coulomb-Schrodinger equation), than the Fourier-sinemapping scheme. All three methods give similar accuracy for the hydrogen atom when the scaling parameter L is optimum, but the Laguerre and Chebyshev methods are less sensitive to variations in L. We introduce a new variant of rational Chebyshev functions which has a more uniform spacing of grid points for large r, and gives somewhat better results than the rational Chebyshev functions of Boyd [J. Comp. Phys. 70 (1987) 63].
Ali H. Bhrawy - One of the best experts on this subject based on the ideXlab platform.
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a new formula for fractional integrals of Chebyshev polynomials application for solving multi term fractional differential equations
Applied Mathematical Modelling, 2013Co-Authors: Ali H. Bhrawy, M M Tharwat, Ahmet YildirimAbstract:Abstract A new explicit formula for the integrals of shifted Chebyshev polynomials of any degree for any fractional-order in terms of shifted Chebyshev polynomials themselves is derived. A fast and accurate algorithm is developed for the solution of linear multi-order fractional differential equations (FDEs) by considering their integrated forms. The shifted Chebyshev spectral tau (SCT) method based on the integrals of shifted Chebyshev polynomials is applied to construct the numerical solution for such problems. The method is then tested on examples. It is shown that the SCT yields better results.
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the operational matrix of fractional integration for shifted Chebyshev polynomials
Applied Mathematics Letters, 2013Co-Authors: Ali H. Bhrawy, Abdulaziz AlofiAbstract:Abstract A new shifted Chebyshev operational matrix (SCOM) of fractional integration of arbitrary order is introduced and applied together with spectral tau method for solving linear fractional differential equations (FDEs). The fractional integration is described in the Riemann–Liouville sense. The numerical approach is based on the shifted Chebyshev tau method. The main characteristic behind the approach using this technique is that only a small number of shifted Chebyshev polynomials is needed to obtain a satisfactory result. Illustrative examples reveal that the present method is very effective and convenient for linear multi-term FDEs.
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efficient Chebyshev spectral methods for solving multi term fractional orders differential equations
Applied Mathematical Modelling, 2011Co-Authors: E H Doha, Ali H. Bhrawy, S S EzzeldienAbstract:Abstract In this paper, we state and prove a new formula expressing explicitly the derivatives of shifted Chebyshev polynomials of any degree and for any fractional-order in terms of shifted Chebyshev polynomials themselves. We develop also a direct solution technique for solving the linear multi-order fractional differential equations (FDEs) with constant coefficients using a spectral tau method. The spatial approximation with its fractional-order derivatives (described in the Caputo sense) are based on shifted Chebyshev polynomials T L , n ( x ) with x ∈ (0, L ), L > 0 and n is the polynomial degree. We presented a shifted Chebyshev collocation method with shifted Chebyshev–Gauss points used as collocation nodes for solving nonlinear multi-order fractional initial value problems. Several numerical examples are considered aiming to demonstrate the validity and applicability of the proposed techniques and to compare with the existing results.
Sedat Ozer - One of the best experts on this subject based on the ideXlab platform.
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a set of new Chebyshev kernel functions for support vector machine pattern classification
Pattern Recognition, 2011Co-Authors: Sedat Ozer, Chi H Chen, Hakan Ali CirpanAbstract:In this study, we introduce a set of new kernel functions derived from the generalized Chebyshev polynomials. The proposed generalized Chebyshev polynomials allow us to derive different kernel functions. By using these polynomial functions, we generalize recently introduced Chebyshev kernel function for vector inputs and, as a result, we obtain a robust set of kernel functions for Support Vector Machine (SVM) classification. Thus in this study, besides clarifying how to apply the Chebyshev kernel functions on vector inputs, we also increase the generalization capability of the previously proposed Chebyshev kernels and show how to derive new kernel functions by using the generalized Chebyshev polynomials. The proposed set of kernel functions provides competitive performance when compared to all other common kernel functions on average for the simulation datasets. The results indicate that they can be used as a good alternative to other common kernel functions for SVM classification in order to obtain better accuracy. Moreover, test results show that the generalized Chebyshev kernel approaches to the minimum support vector number for classification in general.
