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Studies in Computational Intelligence 624 George A. Anastassiou Ioannis K. Argyros Intelligent Numerical Methods: Applications to Fractional Calculus Studies in Computational Intelligence Volume 624 Series editor Janusz Kacprzyk, Polish Academy of Sciences, Warsaw, Poland e-mail: [email protected] About this Series The series “Studies in Computational Intelligence” (SCI) publishes new develop- mentsandadvancesinthevariousareasofcomputationalintelligence—quicklyand with a high quality. The intent is to cover the theory, applications, and design methods of computational intelligence, as embedded in the fields of engineering, computer science, physics and life sciences, as well as the methodologies behind them. The series contains monographs, lecture notes and edited volumes in computational intelligence spanning the areas of neural networks, connectionist systems, genetic algorithms, evolutionary computation, artificial intelligence, cellular automata, self-organizing systems, soft computing, fuzzy systems, and hybrid intelligent systems. Of particular value to both the contributors and the readership are the short publication timeframe and the worldwide distribution, which enable both wide and rapid dissemination of research output. More information about this series at http://www.springer.com/series/7092 George A. Anastassiou Ioannis K. Argyros (cid:129) Intelligent Numerical Methods: Applications to Fractional Calculus 123 George A.Anastassiou Ioannis K.Argyros Department ofMathematical Sciences Department ofMathematical Sciences TheUniversity of Memphis Cameron University Memphis,TN Lawton, OK USA USA ISSN 1860-949X ISSN 1860-9503 (electronic) Studies in Computational Intelligence ISBN978-3-319-26720-3 ISBN978-3-319-26721-0 (eBook) DOI 10.1007/978-3-319-26721-0 LibraryofCongressControlNumber:2015955857 SpringerChamHeidelbergNewYorkDordrechtLondon ©SpringerInternationalPublishingSwitzerland2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart 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 orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. 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 authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor foranyerrorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper SpringerInternationalPublishingAGSwitzerlandispartofSpringerScience+BusinessMedia (www.springer.com) This monograph is dedicated to: Angela, Christopher, Gus, Michael, and Peggy Preface It is a well-known fact that there exist functions that have nowhere first order derivative, but possess continuous Riemann-Liouville and Caputo fractional derivatives of all orders less than one, e.g., the famous Weierstrass function, see Chap. 1, [9, 18], p. 50. ThisstrikingphenomenonmotivatedtheauthorstostudyNewton-likeandother similar numerical methods, which involve fractional derivatives and fractional integral operators, for the first time studied in the literature. All for the purpose to solve numerically equations whose associated functions can be also non- differentiable in the ordinary sense. That is among others extending the classical Newton method theory which requires usual differentiability offunction. In this monograph we present the complete recent work of the past three years of the authors on Numerical Analysis and Fractional Calculus. It is the natural outgrowthoftheirrelatedpublications.Chaptersareself-containedandcanberead independently and several advanced courses can be taught out of this book. An extensivelistofreferencesisgivenperchapter.ThetopicscoveredarefromAtoZ of this research area, all studied for the first time by the authors. The list of presented topics of our related studies follows. Newton-like methods on generalized Banach spaces and applications in fractional calculus. Semilocal convergence of Newton-like methods under general conditions with applications in fractional calculus. Ontheconvergenceofiterativemethodswithapplicationsingeneralizedfractional calculus. A fixed point technique for some iterative algorithm with applications to generalized right fractional calculus. Approximating fixed points with applications in k-fractional calculus. Generalized g-fractional calculus and iterative methods. A unified convergence analysis for a certain family of iterative algorithms with applications to fractional calculus. vii viii Preface A convergence analysis for extended iterative algorithms with applications to fractional and vector calculus. A convergence analysis for a certain family of extended iterative methods with applications to modified fractional calculus. A convergence analysis for secant-like methods with applications to modified fractional calculus. Semilocal convergence of secant-type methods with applications to modified g- fractional calculus. On the convergence of secant-like algorithms with applications to generalized fractional calculus. Generalized g-fractional calculus of Canavati-type and secant-like methods. Aconvergenceanalysisforsomeiterativealgorithmswithapplicationstofractional calculus. Convergence for iterative methods on Banach spaces of a convergence structure with applications to fractional calculus. LocalconvergenceanalysisofinexactGauss–Newtonmethodforsingularsystems of equations under majorant and center-majorant condition. The asymptotic mesh independence principle of Newton’s method under weaker conditions. Ball convergence of a sixth order iterative method with one parameter for solving equations under weak conditions. Improved semilocal convergence of Broyden’s method with regularly continuous divided differences. Left general fractional monotone approximation theory. Right general fractional monotone approximation. Univariate left general high order fractional monotone approximation. Univariate right general high order fractional monotone approximation theory. Advanced fractional Taylor’s formulae. Generalized Canavati type fractional Taylor’s formulae. The last two topics were developed to be used in several chapters of this monograph. The book’s results are expected to find applications in many areas of applied mathematics, stochastics, computer science, and engineering. As such this monograph is suitable for researchers, graduate students, and seminars in the above subjects, also to be in all science and engineering libraries. The preparation of the book took place during 2014–2015 in Memphis, Tennessee and Lawton, Oklahoma, USA. WewouldliketothankProf.AlinaLupasofUniversityofOradea,Romania,for checking and reading the manuscript. October 2015 George A. Anastassiou Ioannis K. Argyros Contents 1 Newton-Like Methods on Generalized Banach Spaces and Fractional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Generalized Banach Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Semilocal Convergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Special Cases and Applications . . . . . . . . . . . . . . . . . . . . . . . 8 1.5 Applications to Fractional Calculus. . . . . . . . . . . . . . . . . . . . . 14 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2 Semilocal Convegence of Newton-Like Methods and Fractional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2 Generalized Banach Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.3 Semilocal Convergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.4 Special Cases and Applications . . . . . . . . . . . . . . . . . . . . . . . 30 2.5 Applications to Fractional Calculus. . . . . . . . . . . . . . . . . . . . . 30 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3 Convergence of Iterative Methods and Generalized Fractional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.2 Generalized Banach Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.3 Semilocal Convergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.4 Special Cases and Applications . . . . . . . . . . . . . . . . . . . . . . . 46 3.5 Applications to Generalized Fractional Calculus. . . . . . . . . . . . 47 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4 Fixed Point Techniques and Generalized Right Fractional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.2 Generalized Banach Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.3 Semilocal Convergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ix x Contents 4.4 Special Cases and Applications . . . . . . . . . . . . . . . . . . . . . . . 64 4.5 Applications to Generalized Right Fractional Calculus . . . . . . . 65 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 5 Approximating Fixed Points and k-Fractional Calculus . . . . . . . . . 75 5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.2 Generalized Banach Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.3 Semilocal Convergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.4 Special Cases and Applications . . . . . . . . . . . . . . . . . . . . . . . 82 5.5 Applications to k-Fractional Calculus . . . . . . . . . . . . . . . . . . . 83 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6 Iterative Methods and Generalized g-Fractional Calculus. . . . . . . . 95 6.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6.2 Generalized Banach Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . 96 6.3 Applications to g-Fractional Calculus . . . . . . . . . . . . . . . . . . . 98 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 7 Unified Convergence Analysis for Iterative Algorithms and Fractional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 7.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 7.2 Convergence Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 7.3 Applications to Fractional Calculus. . . . . . . . . . . . . . . . . . . . . 114 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 8 Convergence Analysis for Extended Iterative Algorithms and Fractional and Vector Calculus . . . . . . . . . . . . . . . . . . . . . . . 127 8.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 8.2 Convergence Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 8.3 Applications to Fractional and Vector Calculus . . . . . . . . . . . . 134 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 9 Convergence Analysis for Extended Iterative Algorithms and Fractional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 9.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 9.2 Convergence Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 9.3 Applications to Fractional Calculus. . . . . . . . . . . . . . . . . . . . . 156 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 10 Secant-Like Methods and Fractional Calculus. . . . . . . . . . . . . . . . 163 10.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 10.2 Convergence Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 10.3 Applications to Right Fractional Calculus . . . . . . . . . . . . . . . . 169 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174