Chebyshev series: Derivation and evaluation.

Previous article Next article On Double Chebyshev Series ApproximationN. K. BasuN. K. Basuhttps://doi.org/10.1137/0710045PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractMany of the standard practical techniques for Chebyshev polynomial approximation in one-variable problems have been extended to two-variable problems.[1] Ferdinand Freudenstein, Bi-variate, rectangular, optimum-interval interpolation, Math. Comp., 15 (1961), 288–291 MR0129532 0168.14903 CrossrefGoogle Scholar[2] C. W. Clenshaw, A note on the summation of Chebyshev series, Math. Tables Aids Comput., 9 (1955), 118–120 MR0071856 0065.05403 CrossrefGoogle Scholar[3] David Elliott, Error analysis of an algorithm for summing certain finite series, J. Austral. Math. Soc., 8 (1968), 213–221 MR0230452 0159.45104 CrossrefGoogle Scholar[4] David Elliott, The numerical solution of integral equations using Chebyshev polynomials, J. Austral. Math. Soc., 1 (1959/1960), 344–356 MR0128087 0096.10201 CrossrefGoogle Scholar[5] L. Fox and , E. T. Goodwin, The numerical solution of non-singular linear integral equations, Philos. Trans. Roy. Soc. London. Ser. A., 245 (1953), 501–534 MR0054355 0050.12902 CrossrefISIGoogle Scholar[6] Google Scholar[7] L. Fox and , I. B. Parker, Chebyshev polynomials in numerical analysis, Oxford University Press, London, 1968ix+205 MR0228149 0153.17502 Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Analytical and numerical investigation of Poiseuille flow through semi-elliptic annulusPhysics of Fluids, Vol. 34, No. 8 | 1 Aug 2022 Cross Ref Study of the free surface Green function at zero forward speed in finite water depth using Newman’s methodApplied Ocean Research, Vol. 120 | 1 Mar 2022 Cross Ref Comparison of existing methods for the calculation of the infinite water depth free-surface Green function for the wave–structure interaction problemApplied Ocean Research, Vol. 81 | 1 Dec 2018 Cross Ref Nondimensionalized Univariate Equation Characterizing Optimal State Feedback Control for Collision AvoidanceIEEE Transactions on Intelligent Transportation Systems, Vol. 19, No. 10 | 1 Oct 2018 Cross Ref The automatic solution of partial differential equations using a global spectral methodJournal of Computational Physics, Vol. 299 | 1 Oct 2015 Cross Ref Numerical proof of stability of roll waves in the small-amplitude limit for inclined thin film flowJournal of Differential Equations, Vol. 257, No. 8 | 1 Oct 2014 Cross Ref A new kind of double Chebyshev polynomial approximation on unbounded domainsBoundary Value Problems, Vol. 2013, No. 1 | 22 January 2013 Cross Ref On Approximate Solutions of Second-Order Linear Partial Differential EquationsApplied Mathematics, Vol. 03, No. 09 | 1 Jan 2012 Cross Ref A fitting formula for the falloff curves of unimolecular reactions, II: Tunneling effectsInternational Journal of Chemical Kinetics, Vol. 43, No. 1 | 19 October 2010 Cross Ref Direct numerical simulation of preferential particle concentration in decaying turbulence under the influence of magnetic fieldInternational Journal for Numerical Methods in Fluids, Vol. 419 | 1 Jan 2009 Cross Ref Large and small signal distortion analysis using modified Volterra seriesAnalog Integrated Circuits and Signal Processing, Vol. 54, No. 2 | 29 November 2007 Cross Ref Volterra Analysis Using Chebyshev Series2007 IEEE International Symposium on Circuits and Systems | 1 May 2007 Cross Ref Large and Small Signal Distortion Analysis Using Modified Volterra Series2006 NORCHIP | 1 Nov 2006 Cross Ref An efficient numerical approximation for the linear class of mixed integral equationsApplied Mathematics and Computation, Vol. 167, No. 2 | 1 Aug 2005 Cross Ref Chebyshev polynomial solutions of second-order linear partial differential equationsApplied Mathematics and Computation, Vol. 134, No. 1 | 1 Jan 2003 Cross Ref Chebyshev series solution of linear Fredholm integrodifferential equationsInternational Journal of Mathematical Education in Science and Technology, Vol. 29, No. 4 | 1 Jul 1998 Cross Ref Fast polynomial multiplication and convolutions related to the discrete cosine transformLinear Algebra and its Applications, Vol. 252, No. 1-3 | 1 Feb 1997 Cross Ref Chebyshev series solutions of Fredholm integral equationsInternational Journal of Mathematical Education in Science and Technology, Vol. 27, No. 5 | 9 July 2006 Cross Ref An algorithm for the evaluation of multidimensional (direct and inverse) discrete Chebyshev transformsJournal of Computational and Applied Mathematics, Vol. 8, No. 4 | 1 Dec 1982 Cross Ref Optimum acoustic design of free-running low speed propellersAircraft Systems and Technology Meeting | 16 August 1977 Cross Ref Clenshaw sums in several variablesConstructive Theory of Functions of Several Variables | 2 October 2006 Cross Ref Volume 10, Issue 3| 1973SIAM Journal on Numerical Analysis History Submitted:05 March 1971Published online:14 July 2006 InformationCopyright © 1973 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0710045Article page range:pp. 496-505ISSN (print):0036-1429ISSN (online):1095-7170Publisher:Society for Industrial and Applied Mathematics

It has long been recognized that Fibonacci-type recurrence relations can be used to define a set of versatile polynomials $$\{ p_{n} (z)\}$$ that have Fibonacci numbers and Chebyshev polynomials as special cases. We show that a tridiagonal matrix, which can be factored into the product $$AB$$ of two special matrices $$A$$ and $$B$$ , is associated with these polynomials. We apply tools that have been developed to study the supersymmetry of Hamiltonians that have a tridiagonal matrix representation in a basis to derive a set of partner polynomials $$\{ p_{n}^{( + )} (z)\}$$ associated with the matrix product $$BA$$ . We find that special cases of these polynomials share similar properties with the Fibonacci numbers and Chebyshev polynomials. As a result, we find two new sum rules that involve the Fibonacci numbers and their product with Chebyshev polynomials.

An operational matrix for the integration of Chebyshev vector whose elements are Chebyshev polynomials is first derived in this paper. Using the operational matrix of integration, a time‐invariant linear system can be represented by a set of simultaneous linear algebraic equations with the approximation of time functions by Chebyshev series. Analysis and parameter identification of linear system can thus be made by solving the algebraic equations.

In this paper, a numerical approach is proposed for solving one dimensional nonlinear time-space-fractional reaction-advection-diffusion equation with Dirichlet boundary conditions. The fractional derivatives are described in the conformable sense. The numerical scheme is based on shifted Chebyshev polynomials of the fourth kind. The unknown function is written as Cheby-shev series with m terms. The nonlinear space fractional reaction-advection-diffusion equation is reduced to a system of nonlinear ordinary differential equations by using the properties of Chebyshev polynomials and conformable fractional calculus.The finite difference method is applied to solve this system. Finally, numerical example is presented to confirm the reliability and effectiveness of the proposed approach.

We denote by and the gamma and the incomplete gamma functions, respectively. In this paper we prove some monotonicity results for the gamma function and extend, to , a lower bound established by Elbert and Laforgia (2000) for the function , with , only for .

در این تحقیق به منظور تعیین بهترین منحنی تولید شیر در یک دوره شیردهی از 260709 رکورد روز آزمون تولید شیر متعلق به 30903 رأس گاو هلشتاین شکم اول استان خراسان استفاده شد. اطلاعات مورد استفاده مربوط به گاو‌های 216 گله بود که طی سالهای 1367 الی 1385 زایش داشتند. توابع مورد استفاده شامل تابع گامای ناقص (وود)، تابع ویلمینک، تابع علی‌شفر و تابع لژاندر (با درجه برازش پنجم) بود. معیارهای مورد نظر برای مقایسه توابع شامل مجموع مربعات خطا، میانگین مجموع مربعات خطا، ضریب تعیین، ضریب تعیین تصحیح شده ، ضریب همبستگی و فاکتور تورم واریانس بود. براساس تمام معیارها به استثناء فاکتور تورم واریانس تابع علی- شفر و براساس فاکتور تورم واریانس تابع لژاندر به نحو مطلوب‌تری با داده‌های تولید شیر مطابقت داشتند.

Throughout applied mathematics, one encounters three closely analogous canonical settings associated with the names of Fourier, Laurent, and Chebyshev. In fact, if we impose certain symmetries in the Fourier and Laurent cases, the analogies become equivalences. The Chebyshev setting is the one of central interest in this book, concerning a variable x and a function f defined on [−1, 1]:

We construct asymptotic expansions for the normalised incomplete gamma function Q ( a , z ) = Γ ( a , z ) / Γ ( a ) Q(a,z)=\Gamma (a,z)/\Gamma (a) that are valid in the transition regions, including the case z ≈ a z\approx a , and have simple polynomial coefficients. For Bessel functions, these types of expansions are well known, but for the normalised incomplete gamma function they were missing from the literature. A detailed historical overview is included. We also derive an asymptotic expansion for the corresponding inverse problem, which has importance in probability theory and mathematical statistics. The coefficients in this expansion are again simple polynomials, and therefore its implementation is straightforward. As a byproduct, we give the first complete asymptotic expansion as a → − ∞ a\to -\infty of the unique negative zero of the regularised incomplete gamma function γ ∗ ( a , x ) \gamma ^*(a,x) .

Considering the equations for some functions involving the first or the second derivatives of the biconfluent Heun function, we construct two expansions of the solutions of the biconfluent Heun equation in terms of incomplete Beta functions. The first series applies single Beta functions as expansion functions, while the second one involves a combination of two Beta functions. The coefficients of expansions obey four- and five-term recurrence relations, respectively. It is shown that the proposed technique is potent to produce series solutions in terms of other special functions. Two examples of such expansions in terms of the incomplete Gamma functions are presented.

Four topics from calculus will be examined in this chapter: truncated series, formal power series, approximation theory, and limits From the examples it will be clear that various truncated series expansions such as Taylor series, Laurent series, Puisseux series, and Chebyshev series are available in Maple. Pade and Chebyshev-Pade are part of Maple’s numerical approximation package, which is called numapprox. When appropriate, series expansions are used in computing limits [72].

In this paper, we present numerical technique for solving the Riccati equation by using operational matrix method with Chebyshev polynomials. The method consists of expanding the required approximate solution as truncated Chebyshev series. Using operational matrix method, we reduce the problem to a set of algebraic equations. Some numerical examples are given to demonstrate the validity and applicability of the method. The method is easy to implement and produces very accurate results.

The object of this study was to test the incomplete gamma function previously developed as a descriptor for grapite disintegration. Samples of disintegrating granite were taken from several sites at three locations in a hot semiarid environment in south‐central Africa (southern Rhodesia). These samples were from surface rock, profile corestones, and disintegrating rock at the base of the soil profile. Particlesize distributions were obtained using the standard sieving and pipette method. For materials > 10 mm, Martin's diameter was measured directly. The incomplete gamma function was fitted to these data, and also to U.S. and French data from the literature, for granite in an apparently similar state of disintegration. The values for the degree‐of‐disintegration parameter were fairly uniform. The mode of disintegration appeared the same regardless of site. The two‐parameter incomplete gamma distribution was found to be a reasonably good descriptor of the particle‐size distribution of disintegrating granite. The mean size appeared to be a sensitive and useful single‐valued descriptor for these distributions.

A new analytical formula for the Arrhenius integral has been proposed by using incomplete gamma functions, which is simple, accurate and reliable. The proposed formula has compared with several published the Arrhenius integral approaches, and is in agreement with the other approaches. Compared with the other approximate formulas proposed in the literature, our proposed method gives more accurate values in the precision of the activation energy as a function of x and temperature.

This paper reviews some properties of the gamma function, particularly the incomplete gamma function and its complement, as a function of the Laplace variable s. The utility of these functions in the solution of initialization problems in fractional-order system theory is demonstrated.

Chapter 4 - Special functions