Wavelet Methods for Solving Partial Differential Equations and Fractional Differential Equations

Wavelet Methods for Solving Partial Differential Equations and Fractional Differential Equations
Author: Santanu Saha Ray
Publisher: CRC Press
Total Pages: 251
Release: 2018-01-12
Genre: Mathematics
ISBN: 1351682210

The main focus of the book is to implement wavelet based transform methods for solving problems of fractional order partial differential equations arising in modelling real physical phenomena. It explores analytical and numerical approximate solution obtained by wavelet methods for both classical and fractional order partial differential equations.

Wavelet Methods for Solving Partial Differential Equations and Fractional Differential Equations

Wavelet Methods for Solving Partial Differential Equations and Fractional Differential Equations
Author: Santanu Saha Ray
Publisher: CRC Press
Total Pages: 273
Release: 2018-01-12
Genre: Mathematics
ISBN: 1351682229

The main focus of the book is to implement wavelet based transform methods for solving problems of fractional order partial differential equations arising in modelling real physical phenomena. It explores analytical and numerical approximate solution obtained by wavelet methods for both classical and fractional order partial differential equations.

Haar Wavelets

Haar Wavelets
Author: Ülo Lepik
Publisher: Springer Science & Business Media
Total Pages: 209
Release: 2014-01-09
Genre: Technology & Engineering
ISBN: 3319042955

This is the first book to present a systematic review of applications of the Haar wavelet method for solving Calculus and Structural Mechanics problems. Haar wavelet-based solutions for a wide range of problems, such as various differential and integral equations, fractional equations, optimal control theory, buckling, bending and vibrations of elastic beams are considered. Numerical examples demonstrating the efficiency and accuracy of the Haar method are provided for all solutions.

Optimal Mobile Sensing and Actuation Policies in Cyber-physical Systems

Optimal Mobile Sensing and Actuation Policies in Cyber-physical Systems
Author: Christophe Tricaud
Publisher: Springer Science & Business Media
Total Pages: 177
Release: 2011-10-14
Genre: Technology & Engineering
ISBN: 1447122623

A successful cyber-physical system, a complex interweaving of hardware and software with some part of the physical environment, depends on proper identification of the, often pre-existing, physical element. A bespoke “cyber” part of the system may then be designed from scratch. Optimal Mobile Sensing and Actuation Strategies in Cyber-physical Systems focuses on distributed-parameter systems the dynamics of which can be modelled with partial differential equations. These are very challenging to observe, their states and inputs being distributed throughout a spatial domain. Consequently, systematic approaches to the optimization of sensor location have to be devised for parameter estimation. The text begins by reviewing the field of cyber-physical systems and introducing background notions of distributed parameter systems and optimal observation theory. New research problems are then defined within this framework. Two important problems considered are optimal mobile sensor trajectory planning and the accuracy effects and allocation of remote sensors. These are followed up with a solution to the problem of optimal robust estimation. Actuation policies are then introduced into the framework with the purpose of improving estimation and optimizing the trajectories of both sensors and actuators simultaneously. The large number of illustrations within the text will assist the reader to visualize the application of the methods proposed. A group of similar examples are used throughout the book to help the reader assimilate the material more easily. The monograph concentrates on the use of methods for which a cyber-physical-systems infrastructure is required. The methods are computationally heavy and require mobile sensors and actuators with communications abilities. Application examples cover fields from environmental science to national security so that readers are encouraged to link the ideas of cyber-physical systems with their own research.

Fractional Order Analysis

Fractional Order Analysis
Author: Hemen Dutta
Publisher: John Wiley & Sons
Total Pages: 336
Release: 2020-08-06
Genre: Mathematics
ISBN: 1119654238

A guide to the new research in the field of fractional order analysis Fractional Order Analysis contains the most recent research findings in fractional order analysis and its applications. The authors—noted experts on the topic—offer an examination of the theory, methods, applications, and the modern tools and techniques in the field of fractional order analysis. The information, tools, and applications presented can help develop mathematical methods and models with better accuracy. Comprehensive in scope, the book covers a range of topics including: new fractional operators, fractional derivatives, fractional differential equations, inequalities for different fractional derivatives and fractional integrals, fractional modeling related to transmission of Malaria, and dynamics of Zika virus with various fractional derivatives, and more. Designed to be an accessible text, several useful, relevant and connected topics can be found in one place, which is crucial for an understanding of the research problems of an applied nature. This book: Contains recent development in fractional calculus Offers a balance of theory, methods, and applications Puts the focus on fractional analysis and its interdisciplinary applications, such as fractional models for biological models Helps make research more relevant to real-life applications Written for researchers, professionals and practitioners, Fractional Order Analysis offers a comprehensive resource to fractional analysis and its many applications as well as information on the newest research.

An Introduction to Wavelets

An Introduction to Wavelets
Author: Charles K. Chui
Publisher: Elsevier
Total Pages: 281
Release: 2016-06-03
Genre: Science
ISBN: 1483282864

Wavelet Analysis and its Applications, Volume 1: An Introduction to Wavelets provides an introductory treatise on wavelet analysis with an emphasis on spline-wavelets and time-frequency analysis. This book is divided into seven chapters. Chapter 1 presents a brief overview of the subject, including classification of wavelets, integral wavelet transform for time-frequency analysis, multi-resolution analysis highlighting the important properties of splines, and wavelet algorithms for decomposition and reconstruction of functions. The preliminary material on Fourier analysis and signal theory is covered in Chapters 2 and 3. Chapter 4 covers the introductory study of cardinal splines, while Chapter 5 describes a general approach to the analysis and construction of scaling functions and wavelets. Spline-wavelets are deliberated in Chapter 6. The last chapter is devoted to an investigation of orthogonal wavelets and wavelet packets. This volume serves as a textbook for an introductory one-semester course on "wavelet analysis for upper-division undergraduate or beginning graduate mathematics and engineering students.

Nonlinear Differential Equations in Physics

Nonlinear Differential Equations in Physics
Author: Santanu Saha Ray
Publisher: Springer Nature
Total Pages: 409
Release: 2019-12-28
Genre: Mathematics
ISBN: 9811516561

This book discusses various novel analytical and numerical methods for solving partial and fractional differential equations. Moreover, it presents selected numerical methods for solving stochastic point kinetic equations in nuclear reactor dynamics by using Euler–Maruyama and strong-order Taylor numerical methods. The book also shows how to arrive at new, exact solutions to various fractional differential equations, such as the time-fractional Burgers–Hopf equation, the (3+1)-dimensional time-fractional Khokhlov–Zabolotskaya–Kuznetsov equation, (3+1)-dimensional time-fractional KdV–Khokhlov–Zabolotskaya–Kuznetsov equation, fractional (2+1)-dimensional Davey–Stewartson equation, and integrable Davey–Stewartson-type equation. Many of the methods discussed are analytical–numerical, namely the modified decomposition method, a new two-step Adomian decomposition method, new approach to the Adomian decomposition method, modified homotopy analysis method with Fourier transform, modified fractional reduced differential transform method (MFRDTM), coupled fractional reduced differential transform method (CFRDTM), optimal homotopy asymptotic method, first integral method, and a solution procedure based on Haar wavelets and the operational matrices with function approximation. The book proposes for the first time a generalized order operational matrix of Haar wavelets, as well as new techniques (MFRDTM and CFRDTM) for solving fractional differential equations. Numerical methods used to solve stochastic point kinetic equations, like the Wiener process, Euler–Maruyama, and order 1.5 strong Taylor methods, are also discussed.

Generalized Fractional Order Differential Equations Arising in Physical Models

Generalized Fractional Order Differential Equations Arising in Physical Models
Author: Santanu Saha Ray
Publisher: Chapman & Hall/CRC
Total Pages: 0
Release: 2018-11-13
Genre: Mathematics
ISBN: 9780429771774

This book analyzes the various semi-analytical and analytical methods for finding approximate and exact solutions of fractional order partial differential equations. It explores approximate and exact solutions obtained by various analytical methods for fractional order partial differential equations arising in physical models.

Methods of Mathematical Modelling

Methods of Mathematical Modelling
Author: Harendra Singh
Publisher: CRC Press
Total Pages: 255
Release: 2019-09-17
Genre: Mathematics
ISBN: 1000596788

This book features original research articles on the topic of mathematical modelling and fractional differential equations. The contributions, written by leading researchers in the field, consist of chapters on classical and modern dynamical systems modelled by fractional differential equations in physics, engineering, signal processing, fluid mechanics, and bioengineering, manufacturing, systems engineering, and project management. The book offers theory and practical applications for the solutions of real-life problems and will be of interest to graduate level students, educators, researchers, and scientists interested in mathematical modelling and its diverse applications. Features Presents several recent developments in the theory and applications of fractional calculus Includes chapters on different analytical and numerical methods dedicated to several mathematical equations Develops methods for the mathematical models which are governed by fractional differential equations Provides methods for models in physics, engineering, signal processing, fluid mechanics, and bioengineering Discusses real-world problems, theory, and applications