Applied Mathematics and Approximation Theory-George A. Anastassiou, Oktay DumanSpringer Internationa

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Unformatted text preview: Advances in Intelligent Systems and Computing 441 George A. Anastassiou Oktay Duman Editors Intelligent Mathematics II: Applied Mathematics and Approximation Theory Advances in Intelligent Systems and Computing Volume 441 Series editor Janusz Kacprzyk, Polish Academy of Sciences, Warsaw, Poland e-mail: [email protected] About this Series The series “Advances in Intelligent Systems and Computing” contains publications on theory, applications, and design methods of Intelligent Systems and Intelligent Computing. Virtually all disciplines such as engineering, natural sciences, computer and information science, ICT, economics, business, e-commerce, environment, healthcare, life science are covered. The list of topics spans all the areas of modern intelligent systems and computing. The publications within “Advances in Intelligent Systems and Computing” are primarily textbooks and proceedings of important conferences, symposia and congresses. They cover significant recent developments in the field, both of a foundational and applicable character. An important characteristic feature of the series is the short publication time and world-wide distribution. This permits a rapid and broad dissemination of research results. Advisory Board Chairman Nikhil R. Pal, Indian Statistical Institute, Kolkata, India e-mail: [email protected] Members Rafael Bello, Universidad Central “Marta Abreu” de Las Villas, Santa Clara, Cuba e-mail: [email protected] Emilio S. Corchado, University of Salamanca, Salamanca, Spain e-mail: [email protected] Hani Hagras, University of Essex, Colchester, UK e-mail: [email protected] László T. Kóczy, Széchenyi István University, Győr, Hungary e-mail: [email protected] Vladik Kreinovich, University of Texas at El Paso, El Paso, USA e-mail: [email protected] Chin-Teng Lin, National Chiao Tung University, Hsinchu, Taiwan e-mail: [email protected] Jie Lu, University of Technology, Sydney, Australia e-mail: [email protected] Patricia Melin, Tijuana Institute of Technology, Tijuana, Mexico e-mail: [email protected] Nadia Nedjah, State University of Rio de Janeiro, Rio de Janeiro, Brazil e-mail: [email protected] Ngoc Thanh Nguyen, Wroclaw University of Technology, Wroclaw, Poland e-mail: [email protected] Jun Wang, The Chinese University of Hong Kong, Shatin, Hong Kong e-mail: [email protected] More information about this series at George A. Anastassiou Oktay Duman • Editors Intelligent Mathematics II: Applied Mathematics and Approximation Theory 123 Editors George A. Anastassiou Department of Mathematical Sciences University of Memphis Memphis, TN USA Oktay Duman Department of Mathematics TOBB University of Economics and Technology Ankara Turkey ISSN 2194-5357 ISSN 2194-5365 (electronic) Advances in Intelligent Systems and Computing ISBN 978-3-319-30320-8 ISBN 978-3-319-30322-2 (eBook) DOI 10.1007/978-3-319-30322-2 Library of Congress Control Number: 2016932751 © Springer International Publishing Switzerland 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland Dedicated to World Peace! AMAT 2015 Conference, TOBB University of Economics and Technology, Ankara, Turkey, May 28–31, 2015 George A. Anastassiou and Oktay Duman Ankara, Turkey, May 29, 2015 Preface Applied Mathematics and Approximation Theory (AMAT) is an international conference that brings together researchers from all areas of Applied Mathematics and Approximation Theory held every 3 or 4 years. This special volume is devoted to the proceedings having to do more with applied mathematics contents presented in the 3rd AMAT Conference held at TOBB Economics and Technology University, during May 28–31, 2015 in Ankara, Turkey. We are particularly indebted to the Organizing Committee and the Scientific Committee for their great efforts. We also appreciate the plenary speakers: George A. Anastassiou (University of Memphis, USA), Martin Bohner (Missouri University of Science and Technology, USA), Alexander Goncharov (Bilkent University, Turkey), Varga Kalantarov (Koç University, Turkey), Gitta Kutyniok (Technische Universität Berlin, Germany), Choonkil Park (Hanyang University, South Korea), Mircea Sofonea (University of Perpignan, France), Tamaz Vashakmadze (Tbilisi State University, Georgia). We would like also to thank the anonymous reviewers who helped us select the best articles for inclusion in this proceedings volume, and also the authors for their valuable contributions. Finally, we are grateful to “TOBB University of Economics and Technology” for hosting this conference and providing all of its facilities, and also to “Central Bank of Turkey” for financial support. November 2015 George A. Anastassiou Oktay Duman vii Contents Bivariate Left Fractional Polynomial Monotone Approximation . . . . . . George A. Anastassiou 1 Bivariate Right Fractional Pseudo-Polynomial Monotone Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . George A. Anastassiou 15 Nonlinear Approximation: q-Bernstein Operators of Max-Product Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Oktay Duman 33 A Two Dimensional Inverse Scattering Problem for Shape and Conductive Function for a Dielectic Cylinder. . . . . . . . . . . . . . . . . Ahmet Altundag 57 Spinning Particle in Interaction with a Time Dependent Magnetic Field: A Path Integral Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hilal Benkhelil and Mekki Aouachria 73 New Complexity Analysis of the Path Following Method for Linear Complementarity Problem . . . . . . . . . . . . . . . . . . . . . . . . . El Amir Djeffal, Lakhdar Djeffal and Farouk Benoumelaz 87 Branch and Bound Method to Resolve the Non-convex Quadratic Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 R. Benacer and Boutheina Gasmi Rogue Wave Solutions for the Myrzakulov-I Equation . . . . . . . . . . . . . 119 Gulgassyl Nugmanova Fuzzy Bilevel Programming with Credibility Measure. . . . . . . . . . . . . . 127 Hande Günay Akdemir A New Approach of a Possibility Function Based Neural Network . . . . 139 George A. Anastassiou and Iuliana F. Iatan ix x Contents Elementary Matrix Decomposition Algorithm for Symmetric Extension of Laurent Polynomial Matrices and Its Application in Construction of Symmetric M-Band Filter Banks . . . . . . . . . . . . . . . 151 Jianzhong Wang Solution of Equation for Ruin Probability of Company for Some Risk Model by Monte Carlo Methods . . . . . . . . . . . . . . . . . . . . . . . . . 169 Kanat Shakenov Determinant Reprentation of Dardoux Transformation for the (2+1)-Dimensional Schrödinger-Maxwell-Bloch Equation . . . . . . 183 K.R. Yesmahanova, G.N. Shaikhova, G.T. Bekova and Zh.R. Myrzakulova Numerical Solution of Nonlinear Klein-Gordon Equation Using Polynomial Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 Jalil Rashidinia and Mahmood Jokar A New Approach in Determining Lot Size in Supply Chain Using Game Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 Maryam Esmaeili Voronovskaya Type Asymptotic Expansions for Multivariate Generalized Discrete Singular Operators . . . . . . . . . . . . . . . . . . . . . . . 233 George A. Anastassiou and Merve Kester Variational Analysis of a Quasistatic Contact Problem . . . . . . . . . . . . . 245 Mircea Sofonea Efficient Lower Bounds for Packing Problems in Heterogeneous Bins with Conflicts Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 Mohamed Maiza, Mohammed Said Radjef and Lakhdar Sais Global Existence, Uniqueness and Asymptotic Behavior for a Nonlinear Parabolic System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 Naima Aïssa and H. Tsamda Mathematical Analysis of a Continuous Crystallization Process . . . . . . . 283 Amira Rachah and Dominikus Noll Wave Velocity Estimation in Heterogeneous Media. . . . . . . . . . . . . . . . 303 Sharefa Asiri and Taous-Meriem Laleg-Kirati Asymptotic Rate for Weak Convergence of the Distribution of Renewal-Reward Process with a Generalized Reflecting Barrier . . . . 313 Tahir Khaniyev, Başak Gever and Zulfiye Hanalioglu Contents xi Bias Study of the Naive Estimator in a Longitudinal Linear Mixed-Effects Model with Measurement Error and Misclassification in Covariates. . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 Jia Li, Ernest Dankwa and Taraneh Abarin Transformations of Data in Deterministic Modelling of Biological Networks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 Melih Ağraz and Vilda Purutçuoğlu Tracking the Interface of the Diffusion-Absorption Equation: Theoretical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 Waleed S. Khedr Tracking the Interface of the Diffusion-Absorption Equation: Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 Waleed S. Khedr Analyzing Multivariate Cross-Sectional Poisson Count Using a Quasi-Likelihood Approach: The Case of Trivariate Poisson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 Naushad Mamode Khan, Yuvraj Sunecher and Vandna Jowaheer Generalized Caputo Type Fractional Inequalities . . . . . . . . . . . . . . . . . 423 George A. Anastassiou Basic Iterated Fractional Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . 455 George A. Anastassiou Contributors Taraneh Abarin Department of Mathematics and Statistics, Memorial University, St John’s, Canada Melih Ağraz Middle East Technical University, Ankara, Turkey Naima Aïssa USTHB, Laboratoire AMNEDP, Algiers, Algeria Hande Günay Akdemir Department of Mathematics, Giresun University, Giresun, Turkey Ahmet Altundag Istanbul Sabahattin Zaim University, Istanbul, Turkey George A. Anastassiou Department of Mathematical Sciences, University of Memphis, Memphis, TN, USA Mekki Aouachria University of Hadj lakhdar, Batna, Algeria Sharefa Asiri Computer, Electrical and Mathematical Sciences and Engineering Division, King Abdullah University of Science and Technology (KAUST), Thuwal, Kingdom of Saudi Arabia G.T. Bekova L.N. Gumilyov Eurasian National University, Astana, Kazakhstan R. Benacer University of Hadj Lakhder Batna, Batna, Algeria Hilal Benkhelil University of Hadj lakhdar, Batna, Algeria Farouk Benoumelaz Department of Mathematics, University of Hadj Lakhdar, Batna, Algeria Ernest Dankwa Department of Mathematics and Statistics, Memorial University, St John’s, Canada El Amir Djeffal Department of Mathematics, University of Hadj Lakhdar, Batna, Algeria xiii xiv Contributors Lakhdar Djeffal Department of Mathematics, University of Hadj Lakhdar, Batna, Algeria Oktay Duman Department of Mathematics, TOBB Economics and Technology University, Ankara, Turkey Maryam Esmaeili Alzahra University, Tehran, Iran Boutheina Gasmi University of Hadj Lakhder Batna, Batna, Algeria Başak Gever TOBB University of Economics and Technology, Ankara, Turkey Zulfiye Hanalioglu Karabuk University, Karabuk, Turkey Iuliana F. Iatan Department of Mathematics and Computer Science, Technical University of Civil Engineering, Bucharest, Romania Mahmood Jokar School of Mathematics, Iran University of Science and Technology, Tehran, Iran Vandna Jowaheer University of Mauritius, Moka, Mauritius Merve Kester Department of Mathematical Sciences, University of Memphis, Memphis, TN, USA Naushad Mamode Khan University of Mauritius, Moka, Mauritius Tahir Khaniyev TOBB University of Economics and Technology, Ankara, Turkey; Institute of Control Systems, Azerbaijan National Academy of Sciences, Baku, Azerbaijan Waleed S. Khedr Erasmus-Mundus MEDASTAR Project, Universidad de Oviedo, Oviedo, Asturias, Spain Taous-Meriem Laleg-Kirati Computer, Electrical and Mathematical Sciences and Engineering Division, King Abdullah University of Science and Technology (KAUST), Thuwal, Kingdom of Saudi Arabia Jia Li Department of Mathematics and Statistics, Memorial University, St John’s, Canada Mohamed Maiza Laboratoire de Mathématiques Appliquées-Ecole Militaire Polytechnique, Algiers, Algeria Zh.R. Myrzakulova L.N. Gumilyov Eurasian National University, Astana, Kazakhstan Dominikus Noll Institut de Mathématiques de Toulouse, Université Paul Sabatier, Toulouse, France Gulgassyl Nugmanova L.N. Gumilyov Eurasian National University, Astana, Kazakhstan Vilda Purutçuoğlu Middle East Technical University, Ankara, Turkey Contributors xv Amira Rachah Institut de Mathématiques de Toulouse, Université Paul Sabatier, Toulouse, France Mohammed Said Radjef Laboratoire de Modélisation et Optimisation des Systémes, Université de Béjaia, Béjaia, Algeria Jalil Rashidinia School of Mathematics, Iran University of Science and Technology, Tehran, Iran Lakhdar Sais Centre de Recherche en Informatique de Lens CNRS-UMR8188, Université d’Artois, Lens, France G.N. Shaikhova L.N. Kazakhstan Gumilyov Eurasian National University, Astana, Kanat Shakenov Al-Farabi Kazakh National University, Almaty, Kazakhstan Mircea Sofonea University of Perpignan Via Domitia, Perpignan, France Yuvraj Sunecher University of Technology, Port Louis, Mauritius H. Tsamda USTHB, Laboratoire AMNEDP, Algiers, Algeria Jianzhong Wang Department of Mathematics and Statistics, Sam Houston State University, Huntsville, TX, USA K.R. Yesmahanova L.N. Gumilyov Eurasian National University, Astana, Kazakhstan Bivariate Left Fractional Polynomial Monotone Approximation George A. Anastassiou   Abstract Let f ∈ C r, p [0, 1]2 , r, p ∈ N , and let L ∗ be a linear left fractional mixed partial differential operator such that L ∗ ( f ) ≥ 0, for all (x, y) in a critical there exists region of [0, 1]2 that depends on L ∗ . Then   a sequence of two-dimensional polynomials Q m 1 ,m 2 (x, y) with L ∗ Q m 1 ,m 2 (x, y) ≥ 0 there, where m 1 , m 2 ∈ N such that m 1 > r , m 2 > p, so that f is approximated left fractionally simultaneously and uniformly by Q m 1 ,m 2 on [0, 1]2 . This restricted left fractional approximation is accomplished quantitatively by the use of a suitable integer partial derivatives two-dimensional first modulus of continuity. 1 Introduction The topic of monotone approximation started in [5] has become a major trend in approximation theory. A typical problem in this subject is: given a positive integer k, approximate a given function whose kth derivative is ≥0 by polynomials having this property. In [2] the authors replaced the kth derivative with a linear differential operator of order k. We mention this motivating result. Theorem 1 Let h, k, p be integers, 0 ≤ h ≤ k ≤ p and let f be  a real function, f ( p) continuous in [−1, 1] with modulus of continuity ω f ( p) , x there. Let a j (x), j = h, h + 1, ..., k be real functions, defined and bounded on [−1, 1] and assume ah (x) is either ≥ some number α > 0 or ≤ some number β < 0 throughout [−1, 1]. Consider the operator  j  k  d a j (x) L= d xj j=h G.A. Anastassiou (B) Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA e-mail: [email protected] © Springer International Publishing Switzerland 2016 G.A. Anastassiou and O. Duman (eds.), Intelligent Mathematics II: Applied Mathematics and Approximation Theory, Advances in Intelligent Systems and Computing 441, DOI 10.1007/978-3-319-30322-2_1 1 2 G.A. Anastassiou and suppose, throughout [−1, 1], L ( f ) ≥ 0. (1) Then, for every integer n ≥ 1, there is a real polynomial Q n (x) of degree ≤ n such that L (Q n ) ≥ 0 throughout [−1, 1]  and max | f (x) − Q n (x)| ≤ Cn k− p −1≤x≤1 ω f ( p)  1 , , n where C is independent of n or f . We need   Definition 2 (see Stancu [6]) Let f ∈ C [0, 1]2 , [0, 1]2 = [0, 1] × [0, 1], where (x1 , y1 ), (x2 , y2 ) ∈ [0, 1]2 and δ1 , δ2 ≥ 0. The first modulus of continuity of f is defined as follows: ω1 ( f, δ1 , δ2 ) = sup |x1 −x2 |≤δ1 |y1 −y2 |≤δ2 | f (x1 , y1 ) − f (x2 , y2 )| . Definition 3 Let f be a real-valued function defined on [0, 1]2 and let m, n be two positive integers. Let Bm,n be the Bernstein (polynomial) operator of order (m, n) given by (2) Bm,n ( f ; x, y)       n m  i j m n , · = f · · x i · (1 − x)m−i · y j · (1 − y)n− j . i j m n i=0 j=0 For integers r, s ≥ 0, we denote by f (r,s) the differential operator of order (r, s), given by ∂ r +s f (x, y) . f (r,s) (x, y) = ∂ x r ∂ ys We use Theorem 4 (Badea and Badea [3]). It holds that (k,l) (k,l)  − Bm,n f f ∞   1 1 ≤ t (k, l) · ω1 f (k,l) ; √ ,√ m−k n −l k (k − 1) l (l − 1) · f (k,l) ∞ , + max , m n (3) Bivariate Left Fractional Polynomial Monotone Approximation 3 where m > k ≥ 0, n > l ≥ 0 are integers, f is a real-valued function on [0, 1]2 such that f (k,l) is continuous, and t is a positive real-valued function on Z+ = {0, 1, 2, ...}. Here ·∞ is the supremum norm on [0, 1]2 .   Denote C r, p [0, 1]2 := { f : [0, 1]2 → R; f (k,l) is continuous for 0 ≤ k ≤ r , 0 ≤ l ≤ p}. In [1] the author proved the following main motivational result. Theorem 5 Let h 1 ,h 2 , v1 , v2 , r, p be integers, 0 ≤ h 1 ≤ v1 ≤ r , 0 ≤ h 2 ≤ v2 ≤ p and let f ∈ C r, p [0, 1]2 . Let αi, j (x, y), i = h 1 , h 1 + 1, ..., v1 ; j = h 2 , h 2 + 1, ..., v2 be real-valued functions, defined and bounded in [0, 1]2 and assume αh 1 h 2 is either ≥ α > 0 or ≤ β < 0 throughout [0, 1]2 . Consider the operator v1  v2  L= αi j (x, y) i=h 1 j=h 2 ∂ i+ j ∂xi ∂y j (4) and suppose that throughout [0, 1]2 , L ( f ) ≥ 0. Then for integers m, n with  m > r , n > p, there exists a polynomial Q m,n (x, y) of degree (m, n) such that L Q m,n (x, y) ≥ 0 throughout [0, 1]2 and (k,l) f − Q (k,l) m,n ∞ ≤ Pm,n (L , f ) k,l + Mm,n ( f), (h 1 − k)! (h 2 − l)! (5) all (0, 0) ≤ (k, l) ≤ (h 1 , h 2 ). Furthermore we get (k,l) f − Q (k,l) m,n ∞ k,l ≤ Mm,n ( f), (6) for all (h 1 + 1, h 2 + 1) ≤ (k, l) ≤ (r, p). Also (6) is true whenever 0 ≤ k ≤ h 1 , h 2 + 1 ≤ l ≤ p or h 1 + 1 ≤ k ≤ r , 0 ≤ l ≤ h 2 . Here  k,l Mm,n ≡ (k,l) 1 1 ;√ ,√ ( f ) ≡ t (k, l) · ω1 f m−k n −l k (k − 1) l (l − 1) , · f (k,l) ∞ + max m n k,l Mm,n and Pm,n ≡ Pm,n (L , f ) ≡ v1  v2  i=h 1 j=h 2 i, j li j · Mm,n ,  (7) (8) 4 G.A. Anastassiou where t is a positive real-valued function on Z2+ and li j ≡ sup (x,y)∈[0,1]2 −1 α h 1 h 2 (x, y) · αi j (x, y) < ∞. (9) In this article we extend Theorem 5 to the fractional level. Indeed here L is replaced by L ∗ , a linear left Caputo fractional mixed partial differential operator. Now the monotonicity property is only true on a critical region of [0, 1]2 that depends on L ∗ parameters. Simultaneous fractional convergence remains true on all of [0, 1]2 . We need   Definition 6 Let α1 , α2 > 0; α = (α1 , α2 ), f ∈ C [0, 1]2 and let x = (x1 , x2 ), t = (t1 , t2 ) ∈ [0, 1]2 . We define the left mixed Rie...
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