DOCSLIB.ORG

On a -Analogue of Stancu Operators

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
On a -Analogue of Stancu Operators Cent. Eur. J. Math. • 8(1) • 2010 • 191-198 DOI: 10.2478/s11533-009-0057-9 Central European Journal of Mathematics On a q-analogue of Stancu operators Research Article Octavian Agratini1∗ 1 Faculty of Mathematics and Computer Science, Babeș-Bolyai University, Cluj-Napoca, Romania Received 8 June 2009; accepted 27 September 2009 Abstract: This paper is concerned with a generalization in q-Calculus of Stancu operators. Involving modulus of continuity and Lipschitz type maximal function, we give estimates for the rate of convergence. A probabilistic approach is presented and approximation properties are established. MSC: 41A36 Keywords: q-integers • q-Bernstein polynomials • Uniform convergence • Smoothness • Lipschitz-type maximal function © Versita Warsaw and Springer-Verlag Berlin Heidelberg. 1. Introduction Recently, G. Nowak [13] introduced a q-analogue of Stancu’s operators [19]. As special case, the Phillips q-polynomials [16] can be reobtained. The aim of this note is to present approximation properties of the mentioned class of operators. The work is motivated by the following. During the last decade, q-Calculus was intensively used for the construction of various generalizations of many classical approximation processes of positive type. The first researches have been achieved by A. Lupaș [11] and G.M. Phillips [16] who proposed q-variants of the original Bernstein operators. Gaining popularity, these new polynomials have been studied lately by a number of authors, see, e.g., [6, 8, 15, 21, 22]. Also, other classes of discrete operators have been extended in q-Calculus, for example: Meyer-König and Zeller operators [20], Bleimann, Butzer and Hahn operators [5], Szász-Mirakjan operators [4], Balázs-Szabados operators [7]. 2. Background and preliminary results At first we collect some facts regarding q-Calculus, see, e.g., [3, 9]. Let q > 0. For any n ∈ N0 = {0} ∪ N, the q-integer [n]q and the q-factorial [n]q! are respectively defined by n−1 n X j Y [n]q = q ; [n]q! = [j]q; n ∈ N; j=0 j=1 ∗ E-mail: [email protected] 191 On a q-analogue of Stancu operators h n i and [0]q = 0, [0]q! = 1. The q-binomial coefficients are denoted by and are defined by k q h n i [n] ! = q ; k = 0; 1; : : : ; n: k q [k]q![n − k]q! h n i n Clearly, for q = 1 one has [n]1 = n, [n]1! = n! and = , the ordinary binomial coefficients. k q k In the sequel we always will assume that q ∈ (0; 1). For f ∈ C([0; 1]), α ≥ 0 and each n ∈ N, in [13] have been defined the operators n q;α X q;α [k]q (Bn f)(x) = pn;k (x)f ; x ∈ [0; 1]; (1) [n]q k=0 where Qk−1 Qn−1−k s q;α h n i i=0 (x + α[i]q) s=0 (1 − q x + α[s]q) pn;k (x) = n−1 : k q Q i=0 (1 + α[i]q) From here on, an empty product is taken to be equal to 1. This class contains as special cases the following well-known q;0 q sequences of linear and positive operators. For α = 0, Bn ≡ Bn represents q-Bernstein operator introduced by Phillips 1;0 1;α hαi [16]. For q = 1 and α = 0, Bn ≡ Bn is the classical Bernstein polynomial. For q = 1, Bn ≡ Bn turns into Stancu operator [19] defined as follows n 1 X n k (Bhαif)(x) = x[k;−α](1 − x)[n−k;−α]f ; x ∈ [0; 1]: (2) n 1[n;−α] k n k=0 m−1 Y Here t[m;a] = (t − ja) represents the generalized factorial power with the step a, a ∈ R, m ∈ N. The following j=0 identities hold [13] q;α q;α (Bn e0)(x) = 1; (Bn e1)(x) = x; q;α 1 x(1 − x) (Bn e2)(x) = x(x + α) + ; x ∈ [0; 1]; (3) 1 + α [n]q where ej , j ∈ N, stands for the monomial of j-th degree and e0(x) = 1. It is worth to be mentioned that other two q-analogues of Stancu operators have been earlier introduced by A. Lupaș [12]. Remark 2.1. (i) On the basis of (1) one ascertains that the approximated function f is evaluated at non-equally spaced points satisfying the relations k [k]q h n i k kq = ∈ ; n ; 0 ≤ k ≤ n: [n]q k qn n nq q;α (ii) These operators interpolate the approximated function f at the endpoints of the interval, this means (Bn f)(τ) = f(τ), τ ∈ {0; 1}: q;α q;α (iii) The operators are non-expansive. Indeed, since Bn e0 = 1, we get kBn fk ≤ kfk, where k · k represents the norm of the uniform convergence of the space C([0; 1]). hαi q q;α (iv) As the original Bernstein and Stancu operators verify Bnf ≤ Bn f, in the same manner one has Bnf ≤ Bn f, for every α ≥ 0 and f. 192 O. Agratini q;α The operator Bn can be reintroduced by using probabilistic tools. Following [2, Section 5.2], let (Ω; F;P) be a probability space and Zq : N × [0; 1] → M2(Ω) a random scheme on [0; 1], where M2(Ω) stands for the space of all real square-integrable random variables on Ω. We consider [k]q q;α P Zq(n; x) = = pn;k (x); 0 ≤ k ≤ n; n ∈ N; x ∈ [0; 1]: [n]q As usual, the mathematical expectation and the variance of Zq(n; x) are denoted by E(Zq(n; x)) and V ar(Zq(n; x)), respectively. Setting PZq(n;x) the distribution of Zq(n; x) with respect to P, one has Z Z 1 q;α (Bn f)(x) = f ◦ Zq(n; x)dP = fdPZq(n;x) = E(f(Zq(n; x))); (4) Ω 0 x ∈ [0; 1], and, in harmony with (3), we obtain 1 + α[n]q E(Zq(n; x)) = x; V ar(Zq(n; x)) = x(1 − x): (5) (1 + α)[n]q Theorem 2.1 ([13]). qn;αn Let (qn)n≥1, (αn)n≥1 be real sequences such that 0 < qn < 1, αn ≥ 0, n ∈ N. Let Bn , n ∈ N, be defined as in (1). If lim qn = 1 and lim αn = 0, then for each f ∈ C([0; 1]) n n qn;αn lim(Bn f)(x) = f(x); uniformly in x ∈ [0; 1]: (6) n The proof of this result, see [13, Theorem 2.6], is based on Bohman-Korovkin criterion. In Section 4 we will present another proof based on the probabilistic look over the operators. A similar approach has been achieved by Il’inskii and Ostrovska for q-Bernstein operators [8, Lemma 1]. 3. Statement of the results q;α We explore the rate of convergence of Bn , 0 < q < 1, α > 0, n ∈ N, in terms of the modulus of continuity ω1(f; ·), where ω1(f; δ) = sup |f(x) − f(y)|, δ ≥ 0. x;y∈[0;1] |x−y|≤δ Theorem 3.1. q;α Let Bn , n ∈ N, be defined by (1). (i) If f ∈ C([0; 1]), then s ! q;α 3 1 + α[n]q |(Bn f)(x) − f(x)| ≤ ω1 f; ; x ∈ [0; 1]: (7) 2 (1 + α)[n]q (ii) If f is differentiable and f0 ∈ C([0; 1]), then s s ! q;α 3 1 + α[n]q 0 1 + α[n]q |(Bn f)(x) − f(x)| ≤ ω1 f ; ; x ∈ [0; 1]: (8) 4 (1 + α)[n]q (1 + α)[n]q For q = 1, the inequalities (7), (8) reduce to the known estimates of the order of approximation by Stancu operators, see [19, Th. 5.1, Th. 5.2]. Taking q = qn ∈ (0; 1) and αn ≥ 0 such that lim qn = 1, lim αn = 0, we deduce lim[n]q = ∞ and relation (7) implies (6). n n n n 193 On a q-analogue of Stancu operators Remark 3.1. qn;αn p −1 Examining (7) one observes that the order of approximation of f by Bn is O( ([n]qn ) + αn depending on (qn)n − and (αn)n, two independent sequences. Since 0 < qn → 1 , there exist n0 ∈ N and a constant γ0 > 0 such that −1 n−1 −1 −1 ([n]qn ) ≤ (nqn ) < γ0n , for each n ≥ n0. To maximize 1/[n]qn it is necessary to have [n]qn ∼ n. This requirement is implied by the following: there exist n1 ∈ N and a constant γ1 > 0 such that qn > 1 − γ1/n for each n ≥ n1. Further −1 on, selecting√ (αn)n such that αn ∼ n , we deduce that, in any case, the order of approximation cannot be greater than O(1/ n), which represents exactly the order of approximation of f by the classical Bernstein operator Bn. q;α 1 q;α q;α m q;α q;α (m−1) q;α By definition, the m-th iterate of Bn is Bn := Bn , Bn := Bn ( Bn ), m = 2; 3;::: . Our next aim is to study m q;α the convergence of the iterates Bn as m tends to infinity. Theorem 3.2. q;α Let Bn , n ∈ N, be defined by (1). For any fixed n ∈ N, one has m q;α lim( Bn f)(x) = f(0) + (f(1) − f(0))x; f ∈ C([0; 1]); (9) m uniformly on [0; 1]. q We mention that for q-Bernstein polynomials Bn, n ∈ N, a detailed study of their iterates has been done by Ostrovska [14].
Load more

Recommended publications
  • IRJET-V3I8330.Pdf
    International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395 -0056 Volume: 03 Issue: 08 | Aug-2016 www.irjet.net p-ISSN: 2395-0072 A comprehensive study on different approximation methods of Fractional order system Asif Iqbal and Rakesh Roshon Shekh P.G Scholar, Dept. of Electrical Engineering, NITTTR, Kolkata, West Bengal, India P.G Scholar, Dept. of Electrical Engineering, NITTTR, Kolkata, West Bengal, India ---------------------------------------------------------------------***--------------------------------------------------------------------- Abstract - Many natural phenomena can be more infinite dimensional (i.e. infinite memory). So for analysis, accurately modeled using non-integer or fractional order controller design, signal processing etc. these infinite order calculus. As the fractional order differentiator is systems are approximated by finite order integer order mathematically equivalent to infinite dimensional filter, it’s system in a proper range of frequencies of practical interest proper integer order approximation is very much important. [1], [2]. In this paper four different approximation methods are given and these four approximation methods are applied on two different examples and the results are compared both in time 1.1 FRACTIONAL CALCULAS: INRODUCTION domain and in frequency domain. Continued Fraction Expansion method is applied on a fractional order plant model Fractional Calculus [3], is a generalization of integer order and the approximated model is converted to its equivalent calculus to non-integer order fundamental operator D t , delta domain model. Then step response of both delta domain l and continuous domain model is shown. where 1 and t is the lower and upper limit of integration and Key Words: Fractional Order Calculus, Delta operator the order of operation is .
    [Show full text]
  • High Order Gradient, Curl and Divergence Conforming Spaces, with an Application to NURBS-Based Isogeometric Analysis
    High order gradient, curl and divergence conforming spaces, with an application to compatible NURBS-based IsoGeometric Analysis R.R. Hiemstraa, R.H.M. Huijsmansa, M.I.Gerritsmab aDepartment of Marine Technology, Mekelweg 2, 2628CD Delft bDepartment of Aerospace Technology, Kluyverweg 2, 2629HT Delft Abstract Conservation laws, in for example, electromagnetism, solid and fluid mechanics, allow an exact discrete representation in terms of line, surface and volume integrals. We develop high order interpolants, from any basis that is a partition of unity, that satisfy these integral relations exactly, at cell level. The resulting gradient, curl and divergence conforming spaces have the propertythat the conservationlaws become completely independent of the basis functions. This means that the conservation laws are exactly satisfied even on curved meshes. As an example, we develop high ordergradient, curl and divergence conforming spaces from NURBS - non uniform rational B-splines - and thereby generalize the compatible spaces of B-splines developed in [1]. We give several examples of 2D Stokes flow calculations which result, amongst others, in a point wise divergence free velocity field. Keywords: Compatible numerical methods, Mixed methods, NURBS, IsoGeometric Analyis Be careful of the naive view that a physical law is a mathematical relation between previously defined quantities. The situation is, rather, that a certain mathematical structure represents a given physical structure. Burke [2] 1. Introduction In deriving mathematical models for physical theories, we frequently start with analysis on finite dimensional geometric objects, like a control volume and its bounding surfaces. We assign global, ’measurable’, quantities to these different geometric objects and set up balance statements.
    [Show full text]
  • Numerical Integration and Differentiation Growth And
    Numerical Integration and Differentiation Growth and Development Ra¨ulSantaeul`alia-Llopis MOVE-UAB and Barcelona GSE Spring 2017 Ra¨ulSantaeul`alia-Llopis(MOVE,UAB,BGSE) GnD: Numerical Integration and Differentiation Spring 2017 1 / 27 1 Numerical Differentiation One-Sided and Two-Sided Differentiation Computational Issues on Very Small Numbers 2 Numerical Integration Newton-Cotes Methods Gaussian Quadrature Monte Carlo Integration Quasi-Monte Carlo Integration Ra¨ulSantaeul`alia-Llopis(MOVE,UAB,BGSE) GnD: Numerical Integration and Differentiation Spring 2017 2 / 27 Numerical Differentiation The definition of the derivative at x∗ is 0 f (x∗ + h) − f (x∗) f (x∗) = lim h!0 h Ra¨ulSantaeul`alia-Llopis(MOVE,UAB,BGSE) GnD: Numerical Integration and Differentiation Spring 2017 3 / 27 • Hence, a natural way to numerically obtain the derivative is to use: f (x + h) − f (x ) f 0(x ) ≈ ∗ ∗ (1) ∗ h with a small h. We call (1) the one-sided derivative. • Another way to numerically obtain the derivative is to use: f (x + h) − f (x − h) f 0(x ) ≈ ∗ ∗ (2) ∗ 2h with a small h. We call (2) the two-sided derivative. We can show that the two-sided numerical derivative has a smaller error than the one-sided numerical derivative. We can see this in 3 steps. Ra¨ulSantaeul`alia-Llopis(MOVE,UAB,BGSE) GnD: Numerical Integration and Differentiation Spring 2017 4 / 27 • Step 1, use a Taylor expansion of order 3 around x∗ to obtain 0 1 00 2 1 000 3 f (x) = f (x∗) + f (x∗)(x − x∗) + f (x∗)(x − x∗) + f (x∗)(x − x∗) + O3(x) (3) 2 6 • Step 2, evaluate the expansion (3) at x
    [Show full text]
  • Quadrature Formulas and Taylor Series of Secant and Tangent
    QUADRATURE FORMULAS AND TAYLOR SERIES OF SECANT AND TANGENT Yuri DIMITROV1, Radan MIRYANOV2, Venelin TODOROV3 1 University of Forestry, Sofia, Bulgaria [email protected] 2 University of Economics, Varna, Bulgaria [email protected] 3 Bulgarian Academy of Sciences, IICT, Department of Parallel Algorithms, Sofia, Bulgaria [email protected] Abstract. Second-order quadrature formulas and their fourth-order expansions are derived from the Taylor series of the secant and tangent functions. The errors of the approximations are compared to the error of the midpoint approximation. Third and fourth order quadrature formulas are constructed as linear combinations of the second- order approximations and the trapezoidal approximation. Key words: Quadrature formula, Fourier transform, generating function, Euler- Mclaurin formula. 1. Introduction Numerical quadrature is a term used for numerical integration in one dimension. Numerical integration in more than one dimension is usually called cubature. There is a broad class of computational algorithms for numerical integration. The Newton-Cotes quadrature formulas (1.1) are a group of formulas for numerical integration where the integrand is evaluated at equally spaced points. Let . The two most popular approximations for the definite integral are the trapezoidal approximation (1.2) and the midpoint approximation (1.3). The midpoint approximation and the trapezoidal approximation for the definite integral are Newton-Cotes quadrature formulas with accuracy . The trapezoidal and the midpoint approximations are constructed by dividing the interval to subintervals of length and interpolating the integrand function by Lagrange polynomials of degree zero and one. Another important Newton-Cotes quadrat -order accuracy and is obtained by interpolating the function by a second degree Lagrange polynomial.
    [Show full text]
  • Interpolation Polynomials Which Give Best Order Of
    transactions of the american mathematical society Volume 200, 1974 INTERPOLATIONPOLYNOMIALS WHICH GIVE BEST ORDER OF APPROXIMATIONAMONG CONTINUOUSLY DIFFERENTIABLE FUNCTIONSOF ARBITRARYFIXED ORDER ON [-1, +1] by A. K. VARMA Dedicated to Professor P. Turan ABSTRACT. The object of this paper is to show that there exists a poly- nomial Pn(x) of degree < 2n — 1 which interpolates a given function exactly at the zeros of nth Tchebycheff polynomial and for which \\f —Pn\\ < Ckwk(iln> f) where wfc(l/n, /) is the modulus of continuity of / of fcth order. 1. Introduction. The classical theorems of D. Jackson extend the Weierstrass approximation theorem by giving quantitative information on the degree of ap- proximation to a continuous function in terms of its smoothness. Specifically, Jackson proved Theorem 1. Let f(x) be continuous for \x\ < 1 and have modulus of continuity w(t). Then there exists a polynomial P(x) of degree n at most suchthat \f(x)-P(x)\ <Aw(l/n) for \x\ < 1, where A is a positive numer- ical constant. In 1951 S. B. Steckin [3] made an important generalization of the Jackson theorem. Theorem 2 (S. B. Steckin). Let k be a positive integer; then there exists a positive constant Ck such that for every fE C[-\, +1] we can find an algebraic polynomial Pn(x) of degree n so that (1.1) Hf-Pn\\<Ckwk(l/n,f), where wk(l/n, f) is the modulus of continuity of f(x) of kth order. During personal conversation in Poznan, S. B. Steckin raised the following Presented to the Society, April 20, 1973; received by the editors March 23, 1973 and, in revised form, December 7, 1973.
    [Show full text]
  • "FOSLS on Curved Boundaries"
    Approximated Flux Boundary Conditions for Raviart-Thomas Finite Elements on Domains with Curved Boundaries and Applications to First-Order System Least Squares von der Fakultät für Mathematik und Physik der Gottfried Wilhelm Leibniz Universität Hannover zur Erlangung des akademischen Grades Doktor der Naturwissenschaften Dr. rer. nat. genehmigte Dissertation von Dipl.-Math. Fleurianne Bertrand geboren am 06.06.1987 in Caen (Frankreich) 2014 2 Referent: Prof. Dr. Gerhard Starke Universität Duisburg-Essen Fakultät für Mathematik Thea-Leymann-Straße 9 45127 Essen Korreferent: Prof. Dr. James Adler Department of Mathematics Tufts University Bromfield-Pearson Building 503 Boston Avenue Medford, MA 02155 Korreferent: Prof. Dr. Joachim Escher Institut für Angewandte Mathematik Gottfried Wilhelm Leibniz Universität Hannover Welfengarten 1 30167 Hannover Tag der Promotion: 15.07.2014 Acknowledgement: This work was supported by the German Research Foundation (DFG) under grant STA402/10-1. 3 Abstract Optimal order convergence of a first-order system least squares method using lowest-order Raviart-Thomas elements combined with linear conforming elements is shown for domains with curved boundaries. Parametric Raviart-Thomas elements are presented in order to retain the optimal order of convergence in the higher-order case in combination with the isoparametric scalar elements. In particular, an estimate for the normal flux of the Raviart-Thomas elements on interpolated boundaries is derived in both cases. This is illustrated numerically for the Pois- son problem on the unit disk. As an application of the analysis derived for the Poisson problem, the effect of interpolated interface condition for a stationary two-phase flow problem is then studied. Keywords: Raviart-Thomas, Curved Boundaries, First-Order System Least Squares 4 Zusammenfassung Für Gebiete mit gekrümmten Rändern wird die optimale Konvergenzordnung einer Least Squares finite Elemente Methode für Systeme erster Ordnung mit Raviart-Thomas Elementen niedrig- ster Ordnung und linearen konformen Elementen gezeigt.
    [Show full text]
  • Order of Approximation of Functions of Two Variables by New Type Gamma Operators1 Aydın Izgi˙
    General Mathematics Vol. 17, No. 1 (2009), 23–32 Order of approximation of functions of two variables by new type gamma operators1 Aydın Izgi˙ Abstract The theorems on weighetd approximation and order of approxi- mation of continuous functions of two variables by new type Gamma operators on all positive square region are established. 2000 Mathematics Subject Classification: 41A25,41A36 Key words and phrases: Order of approximation,weighted approximation,new type Gamma operators. 1 Introduction Lupa¸sand M¨uller[11] introduced the sequence of linear positive ope- rators {Gn},Gn : C(0, ∞) → C(0, ∞) defined by ∞ n G (f; x)= g (x, u)f( )du n n u Z0 xn+1 G is called Gamma operator, where g (x, u)= e−xuun,x> 0. n n n! 1Received 15 February, 2008 Accepted for publication (in revised form) 21 March, 2008 23 24 Aydın Izgi˙ Mazhar [12] used same gn(x, u) of Gamma operator and introduced the following sequence of linear positive operators: ∞ ∞ Fn(f; x) = gn(x, u)du gn−1(u, t)f(t)dt Z0 Z0 ∞ (2n)!xn+1 tn−1 = f(t)dt, n > 1,x> 0 n!(n − 1)! (x + t)2n+1 Z0 for any f which the last integral is convergent. Now we will modify the ope- rators Fn(f; x) as the following operators An(f; x) (see [9]) which confirm 2 2 An(t ; x) = x . Many linear operators Ln (f; x) confirm, Ln (1; x) = 1, 2 2 Ln (t; x)= x but don’t confirm Ln (t ; x)= x (see [2],[10]).
    [Show full text]
  • Order of Approximation by Linear Combinations of Positive Linear Operators
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOLKhAL Ok APPKOXIMATION THEORY 45. 375-382 ( 19x5 1 Order of Approximation by Linear Combinations of Positive Linear Operators B. WOOD Order of uniform approximation is studled for linear combmatlons due to May and Rathore of Baskakov-type operators and recent methods of Pethe. The order of approximation is estimated in terms of a higher-order modulus of continuity of the function being approximated. ( lY8C4cakmK Pm,. ln1. 1. INTRODUCTION Let c[O, CC) denote the set of functions that are continuous and boun- ded on the nonnegative axis. For ,f’E C[O, x8) we consider two classes of positive linear operators. DEFINITION 1.1. Let (d,,),,. N, d,,: [0, h] + R (h > 0) be a sequence of functions having the following properties: (i) q5,,is infinitely differentiable on [O,h]; (ii) 4,,(O)= 1; (iii) q5,,is completely monotone on [0, h], i.e., (~ 1 )“@],“‘(.Y)30 for .YE [0, h] and k E No; (iv) there exists an integer C’ such that (6)y(.Y)=n$q ,“(.Y) for .YE [0, h], k EN, n E N, and n > max(c, 0). For ,f’~ c[O, x ). x E [0, h], and n E N, define 376 B. WOOD The positive operators (1.1) specialize well-known methods of Baskakov [ 1] and Schurer 191. Recently Lehnhoff [S] has studied uniform approximation properties of ( 1.1 ). DEFINITION 1.2. Let O( .t) = C;-=,, ui J’. /y/ < Y, with (I,, = 1.
    [Show full text]
  • A Method for the Numerical Integration of Differ­ Ential Equations of Second Order Without Explicit First Derivatives 1
    Journal of Research of the Notional Burea u of Sta ndards Vol. 54, No. 3, March 1955 Research Paper 2572 A Method for the Numerical Integration of Differ­ ential Equations of Second Order Without Explicit First Derivatives 1 Rene de Vogelaere 2 This is a fourth-order step-by-step method based on difference formulas. The ease of a single equat ion is discLlssed in detail. The use of this method for au tomatic compute r is considered. 1. Introduction This is a fourth-order step-by-step method based on difference formulas. The s tart of t he computation is easy. Only the case of a sin gle differential equation (i2y dx2 j(x,y) (1) will be co nsidered. The ge neralization to a set of differential equations i,k= 1,2, . ,n is immed iate. Application to electronic computing is discussed. 2. General Formulas We first note some general formulas that may be deduced, for instance, from t he expres­ sions (1.32) , (l.38), (1.39) , and (4.15) of L . Collatz [1] :3 i XiIj(X )dX2= (2) j(X)- (2) j (0)- (l) j(O)x = h2 ~P 2' P(U) /1 Pj_p (2) fJ ~ O = h2 ~ (- l)PP2; p( - u)/1 Pj o (3) fJ ~ O where x= uh, (4) and .x (") j = ja (n-I!.fdx (O) j = j , n = 1,2. Finally, (I) 1 21 41 (1)12- ) 0-- h(2:j1 +} 1. /1;0-90/1;1 -1 + .. .). (5) I Work perfo rmed in part when the author was a guest worker at the J\ational Bureau of Standards, ).'ovember 1953, and in part at the Uni· versity of Louvain , Belgium, 194b, toward the author's D issertation.
    [Show full text]
  • Chebfun Guide 1St Edition for Chebfun Version 5 ! ! ! ! ! ! ! ! ! ! ! ! ! ! Edited By: Tobin A
    ! ! ! Chebfun Guide 1st Edition For Chebfun version 5 ! ! ! ! ! ! ! ! ! ! ! ! ! ! Edited by: Tobin A. Driscoll, Nicholas Hale, and Lloyd N. Trefethen ! ! ! Copyright 2014 by Tobin A. Driscoll, Nicholas Hale, and Lloyd N. Trefethen. All rights reserved. !For information, write to [email protected].! ! ! MATLAB is a registered trademark of The MathWorks, Inc. For MATLAB product information, please write to [email protected]. ! ! ! ! ! ! ! ! Dedicated to the Chebfun developers of the past, present, and future. ! Table of Contents! ! !Preface! !Part I: Functions of one variable! 1. Getting started with Chebfun Lloyd N. Trefethen! 2. Integration and differentiation Lloyd N. Trefethen! 3. Rootfinding and minima and maxima Lloyd N. Trefethen! 4. Chebfun and approximation theory Lloyd N. Trefethen! 5. Complex Chebfuns Lloyd N. Trefethen! 6. Quasimatrices and least squares Lloyd N. Trefethen! 7. Linear differential operators and equations Tobin A. Driscoll! 8. Chebfun preferences Lloyd N. Trefethen! 9. Infinite intervals, infinite function values, and singularities Lloyd N. Trefethen! 10. Nonlinear ODEs and chebgui Lloyd N. Trefethen Part II: Functions of two variables (Chebfun2) ! 11. Chebfun2: Getting started Alex Townsend! 12. Integration and differentiation Alex Townsend! 13. Rootfinding and optimisation Alex Townsend! 14. Vector calculus Alex Townsend! 15. 2D surfaces in 3D space Alex Townsend ! ! Preface! ! This guide is an introduction to the use of Chebfun, an open source software package that aims to provide “numerical computing with functions.” Chebfun extends MATLAB’s inherent facilities with vectors and matrices to functions and operators. For those already familiar with MATLAB, much of what Chebfun does will hopefully feel natural and intuitive. Conversely, for those new to !MATLAB, much of what you learn about Chebfun can be applied within native MATLAB too.
    [Show full text]
  • Approximating Piecewise-Smooth Functions
    Approximating Piecewise-Smooth Functions Yaron Lipman David Levin Tel-Aviv University Abstract We consider the possibility of using locally supported quasi-interpolation operators for the approximation of univariate non-smooth functions. In such a case one usually expects the rate of approximation to be lower than that of smooth functions. It is shown in this paper that prior knowledge of the type of ’singularity’ of the function can be used to regain the full approximation power of the quasi-interpolation method. The singularity types may include jumps in the derivatives at unknown locations, or even singularities of the form (x − s)a , with unknown s and a. The new approx- imation strategy includes singularity detection and high-order evaluation of the singularity parameters, such as the above s and a. Using the ac- quired singularity structure, a correction of the primary quasi-interpolation approximation is computed, yielding the final high-order approximation. The procedure is local, and the method is also applicable to a non-uniform data-point distribution. The paper includes some examples illustrating the high performance of the suggested method, supported by an analysis prov- ing the approximation rates in some of the interesting cases. 1 Introduction High-quality approximations of piecewise-smooth functions from a discrete set of function values is a challenging problem with many applications in fields such as numerical solutions of PDEs, image analysis and geometric model- ing. A prominent approach to the problem is the so-called essentially non- oscillatory (ENO) and subcell resolution (SR) schemes introduced by Harten [9]. The ENO scheme constructs a piecewise-polynomial interpolant on a uni- form grid which, loosely speaking, uses the smoothest consecutive data points 1 2 in the vicinity of each data cell.
    [Show full text]
  • A New Second Order Approximation for Fractional Derivatives with Applications
    SQU Journal for Science, 2018, 23(1), 43-55 DOI: http://dx.doi.org/10.24200/squjs.vol23iss1pp43-55 Sultan Qaboos University A New Second Order Approximation for Fractional Derivatives with Applications Haniffa M. Nasir* and Kamel Nafa Department of Mathematics and Statistics, College of Science, Sultan Qaboos University, P.O. Box 36, Al-Khod 123, Sultanate of Oman. *Email: [email protected] ABSTRACT: We propose a generalized theory to construct higher order Grünwald type approximations for fractional derivatives. We use this generalization to simplify the proofs of orders for existing approximation forms for the fractional derivative. We also construct a set of higher order Grünwald type approximations for fractional derivatives in terms of a general real sequence and its generating function. From this, a second order approximation with shift is shown to be useful in approximating steady state problems and time dependent fractional diffusion problems. Stability and convergence for a Crank-Nicolson type scheme for this second order approximation are analyzed and are supported by numerical results. Keywords: Grünwald approximation, Generating function, Fractional diffusion equation, Steady state fractional equation, Crank-Nicolson scheme, Stability and convergence. تقريب جديد من الدرجة الثانية للمشتقات الكسرية مع تطبيقاتها حنيفة محمد ناصر* و كمال نافع الملخص: نقترح في هذا البحث تعميما لنظرية بناء درجة عالية من نوع جرنوالد التقريبية للمشتقات الكسرية. نستخدم هذا التعميم لتبسيط براهين الدرجات المتعلقة بأنواع من التقريبات المعروفة للمشتقات الكسرية. كما نقوم ببناء مجموعة بدرجة عالية نوع جرنوالد التقريبية للمشتقات الكسرية باستخدام متتالية حقيقية عامة والدالة المولدة لها. وبهذا تم برهان أن التقريب من الدرجة الثانية مع اﻻنسحاب مفيد في تقريب مسائل الحالة الثابتة ومسائل اﻻنتشار الكسري المرتبطة بالزمن.
    [Show full text]