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Fractional Calculus and Fractional Processes with Applications to Financial Economics

Fractional Calculus and Fractional Processes with Applications to Financial Economics
  • Author : Hasan Fallahgoul,Sergio Focardi,Frank Fabozzi
  • Publisher :Unknown
  • Release Date :2016-10-06
  • Total pages :118
  • ISBN : 9780128042847
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Summary : Fractional Calculus and Fractional Processes with Applications to Financial Economics presents the theory and application of fractional calculus and fractional processes to financial data. Fractional calculus dates back to 1695 when Gottfried Wilhelm Leibniz first suggested the possibility of fractional derivatives. Research on fractional calculus started in full earnest in the second half of the twentieth century. The fractional paradigm applies not only to calculus, but also to stochastic processes, used in many applications in financial economics such as modelling volatility, interest rates, and modelling high-frequency data. The key features of fractional processes that make them interesting are long-range memory, path-dependence, non-Markovian properties, self-similarity, fractal paths, and anomalous diffusion behaviour. In this book, the authors discuss how fractional calculus and fractional processes are used in financial modelling and finance economic theory. It provides a practical guide that can be useful for students, researchers, and quantitative asset and risk managers interested in applying fractional calculus and fractional processes to asset pricing, financial time-series analysis, stochastic volatility modelling, and portfolio optimization. Provides the necessary background for the book's content as applied to financial economics Analyzes the application of fractional calculus and fractional processes from deterministic and stochastic perspectives

Mathematical Economics

Mathematical Economics
  • Author : Vasily E. Tarasov
  • Publisher :Unknown
  • Release Date :2020-06-03
  • Total pages :278
  • ISBN : 9783039361182
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Summary : This book is devoted to the application of fractional calculus in economics to describe processes with memory and non-locality. Fractional calculus is a branch of mathematics that studies the properties of differential and integral operators that are characterized by real or complex orders. Fractional calculus methods are powerful tools for describing the processes and systems with memory and nonlocality. Recently, fractional integro-differential equations have been used to describe a wide class of economical processes with power law memory and spatial nonlocality. Generalizations of basic economic concepts and notions the economic processes with memory were proposed. New mathematical models with continuous time are proposed to describe economic dynamics with long memory. This book is a collection of articles reflecting the latest mathematical and conceptual developments in mathematical economics with memory and non-locality based on applications of fractional calculus.

Fractional-order Modeling of Nuclear Reactor: From Subdiffusive Neutron Transport to Control-oriented Models

Fractional-order Modeling of Nuclear Reactor: From Subdiffusive Neutron Transport to Control-oriented Models
  • Author : Vishwesh Vyawahare,Paluri S. V. Nataraj
  • Publisher :Unknown
  • Release Date :2018-02-03
  • Total pages :200
  • ISBN : 9789811075872
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Summary : This book addresses the topic of fractional-order modeling of nuclear reactors. Approaching neutron transport in the reactor core as anomalous diffusion, specifically subdiffusion, it starts with the development of fractional-order neutron telegraph equations. Using a systematic approach, the book then examines the development and analysis of various fractional-order models representing nuclear reactor dynamics, ultimately leading to the fractional-order linear and nonlinear control-oriented models. The book utilizes the mathematical tool of fractional calculus, the calculus of derivatives and integrals with arbitrary non-integer orders (real or complex), which has recently been found to provide a more compact and realistic representation to the dynamics of diverse physical systems. Including extensive simulation results and discussing important issues related to the fractional-order modeling of nuclear reactors, the book offers a valuable resource for students and researchers working in the areas of fractional-order modeling and control and nuclear reactor modeling.

Advances in Differential and Difference Equations with Applications 2020

Advances in Differential and Difference Equations with Applications 2020
  • Author : Dumitru Baleanu
  • Publisher :Unknown
  • Release Date :2021-01-20
  • Total pages :348
  • ISBN : 9783039368709
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Summary : It is very well known that differential equations are related with the rise of physical science in the last several decades and they are used successfully for models of real-world problems in a variety of fields from several disciplines. Additionally, difference equations represent the discrete analogues of differential equations. These types of equations started to be used intensively during the last several years for their multiple applications, particularly in complex chaotic behavior. A certain class of differential and related difference equations is represented by their respective fractional forms, which have been utilized to better describe non-local phenomena appearing in all branches of science and engineering. The purpose of this book is to present some common results given by mathematicians together with physicists, engineers, as well as other scientists, for whom differential and difference equations are valuable research tools. The reported results can be used by researchers and academics working in both pure and applied differential equations.

Basic Theory

Basic Theory
  • Author : Anatoly Kochubei,Yuri Luchko
  • Publisher :Unknown
  • Release Date :2019-02-19
  • Total pages :489
  • ISBN : 9783110571622
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Summary : This multi-volume handbook is the most up-to-date and comprehensive reference work in the field of fractional calculus and its numerous applications. This first volume collects authoritative chapters covering the mathematical theory of fractional calculus, including fractional-order operators, integral transforms and equations, special functions, calculus of variations, and probabilistic and other aspects.

Fractional Calculus with Applications in Mechanics

Fractional Calculus with Applications in Mechanics
  • Author : Teodor M. Atanackovic,Stevan Pilipovic,Bogoljub Stankovic,Dusan Zorica
  • Publisher :Unknown
  • Release Date :2014-02-19
  • Total pages :406
  • ISBN : 9781118909133
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Summary : The books Fractional Calculus with Applications in Mechanics: Vibrations and Diffusion Processes and Fractional Calculus with Applications in Mechanics: Wave Propagation, Impact and Variational Principles contain various applications of fractional calculus to the fields of classical mechanics. Namely, the books study problems in fields such as viscoelasticity of fractional order, lateral vibrations of a rod of fractional order type, lateral vibrations of a rod positioned on fractional order viscoelastic foundations, diffusion-wave phenomena, heat conduction, wave propagation, forced oscillations of a body attached to a rod, impact and variational principles of a Hamiltonian type. The books will be useful for graduate students in mechanics and applied mathematics, as well as for researchers in these fields. Part 1 of this book presents an introduction to fractional calculus. Chapter 1 briefly gives definitions and notions that are needed later in the book and Chapter 2 presents definitions and some of the properties of fractional integrals and derivatives. Part 2 is the central part of the book. Chapter 3 presents the analysis of waves in fractional viscoelastic materials in infinite and finite spatial domains. In Chapter 4, the problem of oscillations of a translatory moving rigid body, attached to a heavy, or light viscoelastic rod of fractional order type, is studied in detail. In Chapter 5, the authors analyze a specific engineering problem of the impact of a viscoelastic rod against a rigid wall. Finally, in Chapter 6, some results for the optimization of a functional containing fractional derivatives of constant and variable order are presented.

Stochastic Calculus for Fractional Brownian Motion and Related Processes

Stochastic Calculus for Fractional Brownian Motion and Related Processes
  • Author : IUliia S. Mishura,I︠U︡lii︠a︡ S. Mishura,Yuliya Mishura,Julija S. Mišura,Ûliâ Stepanovna Mišura
  • Publisher :Unknown
  • Release Date :2008-01-02
  • Total pages :393
  • ISBN : 9783540758723
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Summary : This volume grew out of a series of preprints which were written and circulated - tween 1993 and 1994. Around the same time, related work was done independently by Harder 40] and Laumon 62]. In writing this text based on a revised version of these preprints that were widely distributed in summer 1995, I ?nally did not p- sue the original plan to completely reorganize the original preprints. After the long delay, one of the reasons was that an overview of the results is now available in 115]. Instead I tried to improve the presentation modestly, in particular by adding cross-references wherever I felt this was necessary. In addition, Chaps. 11 and 12 and Sects. 5. 1, 5. 4, and 5. 5 were added; these were written in 1998. I willgivea moredetailedoverviewofthecontentofthedifferentchaptersbelow. Before that I should mention that the two main results are the proof of Ramanujan's conjecture for Siegel modular forms of genus 2 for forms which are not cuspidal representations associated with parabolic subgroups(CAP representations), and the study of the endoscopic lift for the group GSp(4). Both topics are formulated and proved in the ?rst ?ve chapters assuming the stabilization of the trace formula. All the remaining technical results, which are necessary to obtain the stabilized trace formula, are presented in the remaining chapters. Chapter 1 gathers results on the cohomology of Siegel modular threefolds that are used in later chapters, notably in Chap. 3. At the beginning of Chap.

Stochastic Calculus for Fractional Brownian Motion and Applications

Stochastic Calculus for Fractional Brownian Motion and Applications
  • Author : Francesca Biagini,Yaozhong Hu,Bernt Øksendal,Tusheng Zhang
  • Publisher :Unknown
  • Release Date :2008-02-17
  • Total pages :330
  • ISBN : 9781846287978
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Summary : The purpose of this book is to present a comprehensive account of the different definitions of stochastic integration for fBm, and to give applications of the resulting theory. Particular emphasis is placed on studying the relations between the different approaches. Readers are assumed to be familiar with probability theory and stochastic analysis, although the mathematical techniques used in the book are thoroughly exposed and some of the necessary prerequisites, such as classical white noise theory and fractional calculus, are recalled in the appendices. This book will be a valuable reference for graduate students and researchers in mathematics, biology, meteorology, physics, engineering and finance.

Stochastics of Environmental and Financial Economics

Stochastics of Environmental and Financial Economics
  • Author : Fred Espen Benth,Giulia Di Nunno
  • Publisher :Unknown
  • Release Date :2015-10-23
  • Total pages :360
  • ISBN : 9783319234250
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Summary : These Proceedings offer a selection of peer-reviewed research and survey papers by some of the foremost international researchers in the fields of finance, energy, stochastics and risk, who present their latest findings on topical problems. The papers cover the areas of stochastic modeling in energy and financial markets; risk management with environmental factors from a stochastic control perspective; and valuation and hedging of derivatives in markets dominated by renewables, all of which further develop the theory of stochastic analysis and mathematical finance. The papers were presented at the first conference on “Stochastics of Environmental and Financial Economics (SEFE)”, being part of the activity in the SEFE research group of the Centre of Advanced Study (CAS) at the Academy of Sciences in Oslo, Norway during the 2014/2015 academic year.

Fractional Calculus

Fractional Calculus
  • Author : Dumitru Baleanu,Kai Diethelm,Enrico Scalas,Juan J Trujillo
  • Publisher :Unknown
  • Release Date :2016-09-15
  • Total pages :476
  • ISBN : 9789813140059
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Summary : This book will give readers the possibility of finding very important mathematical tools for working with fractional models and solving fractional differential equations, such as a generalization of Stirling numbers in the framework of fractional calculus and a set of efficient numerical methods. Moreover, we will introduce some applied topics, in particular fractional variational methods which are used in physics, engineering or economics. We will also discuss the relationship between semi-Markov continuous-time random walks and the space-time fractional diffusion equation, which generalizes the usual theory relating random walks to the diffusion equation. These methods can be applied in finance, to model tick-by-tick (log)-price fluctuations, in insurance theory, to study ruin, as well as in macroeconomics as prototypical growth models. All these topics are complementary to what is dealt with in existing books on fractional calculus and its applications. This book will keep in mind the trade-off between full mathematical rigor and the needs of readers coming from different applied areas of science and engineering. In particular, the numerical methods listed in the book are presented in a readily accessible way that immediately allows the readers to implement them on a computer in a programming language of their choice. The second edition of the book has been expanded and now includes a discussion of additional, newly developed numerical methods for fractional calculus and a chapter on the application of fractional calculus for modeling processes in the life sciences.

Theory and Applications of Fractional Differential Equations

Theory and Applications of Fractional Differential Equations
  • Author : A.A. Kilbas,H. M. Srivastava,J.J. Trujillo
  • Publisher :Unknown
  • Release Date :2006-03-02
  • Total pages :523
  • ISBN : 0444518320
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Summary : This work aims to present, in a systematic manner, results including the existence and uniqueness of solutions for the Cauchy Type and Cauchy problems involving nonlinear ordinary fractional differential equations.

Advanced Mathematical Methods for Finance

Advanced Mathematical Methods for Finance
  • Author : Julia Di Nunno,Bernt Øksendal
  • Publisher :Unknown
  • Release Date :2011-03-29
  • Total pages :536
  • ISBN : 364218412X
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Summary : This book presents innovations in the mathematical foundations of financial analysis and numerical methods for finance and applications to the modeling of risk. The topics selected include measures of risk, credit contagion, insider trading, information in finance, stochastic control and its applications to portfolio choices and liquidation, models of liquidity, pricing, and hedging. The models presented are based on the use of Brownian motion, Lévy processes and jump diffusions. Moreover, fractional Brownian motion and ambit processes are also introduced at various levels. The chosen blend of topics gives an overview of the frontiers of mathematics for finance. New results, new methods and new models are all introduced in different forms according to the subject. Additionally, the existing literature on the topic is reviewed. The diversity of the topics makes the book suitable for graduate students, researchers and practitioners in the areas of financial modeling and quantitative finance. The chapters will also be of interest to experts in the financial market interested in new methods and products. This volume presents the results of the European ESF research networking program Advanced Mathematical Methods for Finance.

Applications of Fractional Calculus in Physics

Applications of Fractional Calculus in Physics
  • Author : R Hilfer
  • Publisher :Unknown
  • Release Date :2000-03-02
  • Total pages :472
  • ISBN : 9789814496209
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Summary : Fractional calculus is a collection of relatively little-known mathematical results concerning generalizations of differentiation and integration to noninteger orders. While these results have been accumulated over centuries in various branches of mathematics, they have until recently found little appreciation or application in physics and other mathematically oriented sciences. This situation is beginning to change, and there are now a growing number of research areas in physics which employ fractional calculus. This volume provides an introduction to fractional calculus for physicists, and collects easily accessible review articles surveying those areas of physics in which applications of fractional calculus have recently become prominent. Contents:An Introduction to Fractional Calculus (P L Butzer & U Westphal)Fractional Time Evolution (R Hilfer)Fractional Powers of Infinitesimal Generators of Semigroups (U Westphal)Fractional Differences, Derivatives and Fractal Time Series (B J West & P Grigolini)Fractional Kinetics of Hamiltonian Chaotic Systems (G M Zaslavsky)Polymer Science Applications of Path-Integration, Integral Equations, and Fractional Calculus (J F Douglas)Applications to Problems in Polymer Physics and Rheology (H Schiessel et al.)Applications of Fractional Calculus Techniques to Problems in Biophysics (T F Nonnenmacher & R Metzler)Fractional Calculus and Regular Variation in Thermodynamics (R Hilfer) Readership: Statistical, theoretical and mathematical physicists. Keywords:Fractional Calculus in PhysicsReviews: “This monograph provides a systematic treatment of the theory and applications of fractional calculus for physicists. It contains nine review articles surveying those areas in which fractional calculus has become important. All the chapters are self-contained.” Mathematics Abstracts

Fractional Calculus

Fractional Calculus
  • Author : Dumitru Baleanu,Juan J. Trujillo
  • Publisher :Unknown
  • Release Date :2016-09-19
  • Total pages :448
  • ISBN : 9813140038
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Summary : This book will give readers the possibility of finding very important mathematical tools for working with fractional models and solving fractional differential equations, such as a generalization of Stirling numbers in the framework of fractional calculus and a set of efficient numerical methods. Moreover, we will introduce some applied topics, in particular fractional variational methods which are used in physics, engineering or economics. We will also discuss the relationship between semi-Markov continuous-time random walks and the space-time fractional diffusion equation, which generalizes the usual theory relating random walks to the diffusion equation. These methods can be applied in finance, to model tick-by-tick (log)-price fluctuations, in insurance theory, to study ruin, as well as in macroeconomics as prototypical growth models. All these topics are complementary to what is dealt with in existing books on fractional calculus and its applications. This book will keep in mind the trade-off between full mathematical rigor and the needs of readers coming from different applied areas of science and engineering. In particular, the numerical methods listed in the book are presented in a readily accessible way that immediately allows the readers to implement them on a computer in a programming language of their choice. The second edition of the book has been expanded and now includes a discussion of additional, newly developed numerical methods for fractional calculus and a chapter on the application of fractional calculus for modeling processes in the life sciences.

Stochastic Processes and Calculus

Stochastic Processes and Calculus
  • Author : Uwe Hassler
  • Publisher :Unknown
  • Release Date :2015-12-12
  • Total pages :391
  • ISBN : 9783319234281
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Summary : This textbook gives a comprehensive introduction to stochastic processes and calculus in the fields of finance and economics, more specifically mathematical finance and time series econometrics. Over the past decades stochastic calculus and processes have gained great importance, because they play a decisive role in the modeling of financial markets and as a basis for modern time series econometrics. Mathematical theory is applied to solve stochastic differential equations and to derive limiting results for statistical inference on nonstationary processes. This introduction is elementary and rigorous at the same time. On the one hand it gives a basic and illustrative presentation of the relevant topics without using many technical derivations. On the other hand many of the procedures are presented at a technically advanced level: for a thorough understanding, they are to be proven. In order to meet both requirements jointly, the present book is equipped with a lot of challenging problems at the end of each chapter as well as with the corresponding detailed solutions. Thus the virtual text - augmented with more than 60 basic examples and 40 illustrative figures - is rather easy to read while a part of the technical arguments is transferred to the exercise problems and their solutions.

Statistical Inference for Fractional Diffusion Processes

Statistical Inference for Fractional Diffusion Processes
  • Author : B. L. S. Prakasa Rao
  • Publisher :Unknown
  • Release Date :2011-07-05
  • Total pages :280
  • ISBN : 9780470975763
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Summary : Stochastic processes are widely used for model building in the social, physical, engineering and life sciences as well as in financial economics. In model building, statistical inference for stochastic processes is of great importance from both a theoretical and an applications point of view. This book deals with Fractional Diffusion Processes and statistical inference for such stochastic processes. The main focus of the book is to consider parametric and nonparametric inference problems for fractional diffusion processes when a complete path of the process over a finite interval is observable. Key features: Introduces self-similar processes, fractional Brownian motion and stochastic integration with respect to fractional Brownian motion. Provides a comprehensive review of statistical inference for processes driven by fractional Brownian motion for modelling long range dependence. Presents a study of parametric and nonparametric inference problems for the fractional diffusion process. Discusses the fractional Brownian sheet and infinite dimensional fractional Brownian motion. Includes recent results and developments in the area of statistical inference of fractional diffusion processes. Researchers and students working on the statistics of fractional diffusion processes and applied mathematicians and statisticians involved in stochastic process modelling will benefit from this book.

Future Perspectives in Risk Models and Finance

Future Perspectives in Risk Models and Finance
  • Author : Alain Bensoussan,Dominique Guegan,Charles S. Tapiero
  • Publisher :Unknown
  • Release Date :2014-11-20
  • Total pages :315
  • ISBN : 9783319075242
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Summary : This book provides a perspective on a number of approaches to financial modelling and risk management. It examines both theoretical and practical issues. Theoretically, financial risks models are models of a real and a financial “uncertainty”, based on both common and private information and economic theories defining the rules that financial markets comply to. Financial models are thus challenged by their definitions and by a changing financial system fueled by globalization, technology growth, complexity, regulation and the many factors that contribute to rendering financial processes to be continuously questioned and re-assessed. The underlying mathematical foundations of financial risks models provide future guidelines for risk modeling. The book’s chapters provide selective insights and developments that can contribute to better understand the complexity of financial modelling and its ability to bridge financial theories and their practice. Future Perspectives in Risk Models and Finance begins with an extensive outline by Alain Bensoussan et al. of GLM estimation techniques combined with proofs of fundamental results. Applications to static and dynamic models provide a unified approach to the estimation of nonlinear risk models. A second section is concerned with the definition of risks and their management. In particular, Guegan and Hassani review a number of risk models definition emphasizing the importance of bi-modal distributions for financial regulation. An additional chapter provides a review of stress testing and their implications. Nassim Taleb and Sandis provide an anti-fragility approach based on “skin in the game”. To conclude, Raphael Douady discusses the noncyclical CAR (Capital Adequacy Rule) and their effects of aversion of systemic risks. A third section emphasizes analytic financial modelling approaches and techniques. Tapiero and Vallois provide an overview of mathematical systems and their use in financial modeling. These systems span the fundamental Arrow-Debreu framework underlying financial models of complete markets and subsequently, mathematical systems departing from this framework but yet generalizing their approach to dynamic financial models. Explicitly, models based on fractional calculus, on persistence (short memory) and on entropy-based non-extensiveness. Applications of these models are used to define a modeling approach to incomplete financial models and their potential use as a “measure of incompleteness”. Subsequently Bianchi and Pianese provide an extensive overview of multi-fractional models and their important applications to Asset price modeling. Finally, Tapiero and Jinquyi consider the binomial pricing model by discussing the effects of memory on the pricing of asset prices.

Theory and Applications of Long-Range Dependence

Theory and Applications of Long-Range Dependence
  • Author : Paul Doukhan,George Oppenheim,Murad Taqqu
  • Publisher :Unknown
  • Release Date :2002-12-13
  • Total pages :720
  • ISBN : 0817641688
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Summary : The area of data analysis has been greatly affected by our computer age. For example, the issue of collecting and storing huge data sets has become quite simplified and has greatly affected such areas as finance and telecommunications. Even non-specialists try to analyze data sets and ask basic questions about their structure. One such question is whether one observes some type of invariance with respect to scale, a question that is closely related to the existence of long-range dependence in the data. This important topic of long-range dependence is the focus of this unique work, written by a number of specialists on the subject. The topics selected should give a good overview from the probabilistic and statistical perspective. Included will be articles on fractional Brownian motion, models, inequalities and limit theorems, periodic long-range dependence, parametric, semiparametric, and non-parametric estimation, long-memory stochastic volatility models, robust estimation, and prediction for long-range dependence sequences. For those graduate students and researchers who want to use the methodology and need to know the "tricks of the trade," there will be a special section called "Mathematical Techniques." Topics in the first part of the book are covered from probabilistic and statistical perspectives and include fractional Brownian motion, models, inequalities and limit theorems, periodic long-range dependence, parametric, semiparametric, and non-parametric estimation, long-memory stochastic volatility models, robust estimation, prediction for long-range dependence sequences. The reader is referred to more detailed proofs if already found in the literature. The last part of the book is devoted to applications in the areas of simulation, estimation and wavelet techniques, traffic in computer networks, econometry and finance, multifractal models, and hydrology. Diagrams and illustrations enhance the presentation. Each article begins with introductory background material and is accessible to mathematicians, a variety of practitioners, and graduate students. The work serves as a state-of-the art reference or graduate seminar text.

Proceedings

Proceedings
  • Author : Anonim
  • Publisher :Unknown
  • Release Date :2004
  • Total pages :229
  • ISBN : UCSD:31822022870554
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Summary :

Handbook of Mellin Transforms

Handbook of Mellin Transforms
  • Author : Yu. A. Brychkov,O. I. Marichev,N. V. Savischenko
  • Publisher :Unknown
  • Release Date :2018-10-10
  • Total pages :587
  • ISBN : 9780429784446
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Summary : The Mellin transformation is widely used in various problems of pure and applied mathematics, in particular, in the theory of differential and integral equations and the theory of Dirichlet series. It is found in extensive applications in mathematical physics, number theory, mathematical statistics, theory of asymptotic expansions, and especially, in the theory of special functions and integral transformations. It is essentially used in algorithms of integration in computer algebra systems. Since the majority of integrals encountered in applications can be reduced to the form of the corresponding Mellin transforms with specific parameters, this handbook can also be used for definite and indefinite integrals. By changes in variables, the Mellin transform can be turned into the Fourier and Laplace transforms. The appendices contain formulas of connection with other integral transformations, and an algorithm for determining regions of convergence of integrals. The Handbook of Mellin Transforms will be of interest and useful to all researchers and engineers who use mathematical methods. It will become the main source of formulas of Mellin transforms, as well as indefinite and definite integrals.

Fractional Calculus

Fractional Calculus
  • Author : Praveen Agarwal,Dumitru Baleanu,YangQuan Chen,Shaher Momani,José António Tenreiro Machado
  • Publisher :Unknown
  • Release Date :2019-11-23
  • Total pages :249
  • ISBN : 9789811504303
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Summary : This book collects papers presented at the International Conference on Fractional Differentiation and its Applications (ICFDA), held at the University of Jordan, Amman, Jordan, on 16–18 July 2018. Organized into 13 chapters, the book discusses the latest trends in various fields of theoretical and applied fractional calculus. Besides an essential mathematical interest, its overall goal is a general improvement of the physical world models for the purpose of computer simulation, analysis, design and control in practical applications. It showcases the development of fractional calculus as an acceptable tool for a large number of diverse scientific communities due to more adequate modeling in various fields of mechanics, electricity, chemistry, biology, medicine, economics, control theory, as well as signal and image processing. The book will be a valuable resource for graduate students and researchers of mathematics and engineering.