Fuzzy Arbitrary Order System

Snehashish Chakraverty, PhD, is Professor and Head of the Department of Mathematics at the National Institute of Technology, Rourkela in India. The author of five books and approximately 140 journal articles, his research interests include mathematical modeling, machine intelligence, uncertainty modeling, numerical analysis, and differential equations.

Presents a systematic treatment of fuzzy fractional differential equations as well as newly developed computational methods to model uncertain physical problemsComplete with comprehensive results and solutions, Fuzzy Arbitrary Order System: Fuzzy Fractional Differential Equations and Applications details newly developed methods of fuzzy computational techniquesneeded to model solve uncertainty. Fuzzy differential equations are solved via various analytical andnumerical methodologies, and this book presents their importance for problem solving, prototypeengineering design, and systems testing in uncertain environments.In recent years, modeling of differential equations for arbitrary and fractional order systems has been increasing in its applicability, and as such, the authors feature examples from a variety of disciplines to illustrate the practicality and importance of the methods within physics, applied mathematics, engineering, and chemistry, to name a few. The fundamentals of fractional differential equations and the basic preliminaries of fuzzy fractional differential equations are first introduced, followed by numerical solutions, comparisons of various methods, and simulated results. In addition, fuzzy ordinary, partial, linear, and nonlinear fractional differential equations are addressed to solve uncertainty in physical systems. In addition, this book features:* Basic preliminaries of fuzzy set theory, an introduction of fuzzy arbitrary order differential equations, and various analytical and numerical procedures for solving associated problems* Coverage on a variety of fuzzy fractional differential equations including structural, diffusion, and chemical problems as well as heat equations and biomathematical applications* Discussions on how to model physical problems in terms of nonprobabilistic methods and provides systematic coverage of fuzzy fractional differential equations and its applications* Uncertainties in systems and processes with a fuzzy conceptFuzzy Arbitrary Order System: Fuzzy Fractional Differential Equations and Applications is an ideal resource for practitioners, researchers, and academicians in applied mathematics, physics, biology, engineering, computer science, and chemistry who need to model uncertain physical phenomena and problems. The book is appropriate for graduate-level courses on fractional differential equations for students majoring in applied mathematics, engineering, physics, and computer science.

PREFACE ixACKNOWLEDGMENTS xiii1 Preliminaries of Fuzzy Set Theory 1Bibliography 72 Basics of Fractional and Fuzzy Fractional Differential Equations 9Bibliography 123 Analytical Methods for Fuzzy Fractional Differential Equations (FFDES) 153.1 n-Term Linear Fuzzy Fractional Linear Differential Equations 163.2 Proposed Methods 18Bibliography 284 Numerical Methods for Fuzzy Fractional Differential Equations 314.1 Homotopy Perturbation Method (HPM) 314.2 Adomian Decomposition Method (ADM) 354.3 Variational Iteration Method (VIM) 37Bibliography 395 Fuzzy Fractional Heat Equations 415.1 Arbitrary-Order Heat Equation 415.2 Solution of Fuzzy Arbitrary-Order Heat Equations by HPM 415.3 Numerical Examples 435.4 Numerical Results 45Bibliography 476 Fuzzy Fractional Biomathematical Applications 496.1 Fuzzy Arbitrary-Order Predator-Prey Equations 496.1.1 Particular Case 516.2 Numerical Results of Fuzzy Arbitrary-Order Predator-Prey Equations 54Bibliography 657 Fuzzy Fractional Chemical Problems 677.1 Arbitrary-Order Rossler's Systems 677.2 HPM Solution of Uncertain Arbitrary-Order Rossler's System 687.3 Particular Case 717.3.1 Special Case 737.4 Numerical Results 78Bibliography 838 Fuzzy Fractional Structural Problems 878.1 Fuzzy Fractionally Damped Discrete System 888.2 Uncertain Response Analysis 908.2.1 Uncertain Step Function Response 908.2.2 Uncertain Impulse Function Response 938.3 Numerical Results 968.3.1 Case Studies for Uncertain Step Function Response 978.3.2 Case Studies for Uncertain Impulse Function Response 1008.4 Fuzzy Fractionally Damped Continuous System 1018.5 Uncertain Response Analysis 1108.5.1 Unit step Function Response 1108.5.2 Unit Impulse Function Response 1118.6 Numerical Results 1128.6.1 Case Studies for Fuzzy Unit Step Response 1148.6.2 Case Studies for Fuzzy Unit Impulse Response 115Bibliography 1189 Fuzzy Fractional Diffusion Problems 1219.1 Fuzzy Fractional-Order Diffusion Equation 1219.1.1 Double-Parametric-Based Solution of UncertainFractional-Order Diffusion Equation 1239.1.2 Solution Bounds for Different External Forces 1259.2 Numerical Results of Fuzzy Fractional Diffusion Equation 130Bibliography 13910 Uncertain Fractional Fornberg-Whitham Equations 14110.1 Parametric-Based Interval Fractional Fornberg-WhithamEquation 14110.2 Solution by VIM 14310.3 Solution Bounds for Different Interval Initial Conditions 14510.4 Numerical Results 148Bibliography 15211 Fuzzy Fractional Vibration Equation of Large Membrane 15511.1 Double-Parametric-Based Solution of Uncertain Vibration Equation of Large Membrane 15611.2 Solutions of Fuzzy Vibration Equation of Large Membrane 15811.3 Case Studies (Solution Bounds for Particular Cases) 16011.4 Numerical Results for Fuzzy Fractional Vibration Equation for Large Membrane 172Bibliography 18812 Fuzzy Fractional Telegraph Equations 19112.1 Double-Parametric-Based Fuzzy Fractional Telegraph Equations 19112.2 Solutions of Fuzzy Telegraph Equations Using Homotopy Perturbation Method 19412.3 Solution Bounds for Particular Cases 19512.4 Numerical Results for Fuzzy Fractional Telegraph Equations 199Bibliography 20513 Fuzzy Fokker-Planck Equation with Space and Time Fractional Derivatives 20713.1 Fuzzy Fractional Fokker-Planck Equation with Space and Time Fractional Derivatives 20713.2 Double-Parametric-Based Solution of Uncertain Fractional Fokker-Planck Equation 20913.2.1 Solution by HPM 20913.2.2 Solution By ADM 21013.3 Case Studies Using HPM and ADM 21113.3.1 Using HPM 21113.3.2 Using ADM 21513.4 Numerical Results of Fuzzy Fractional Fokker-Planck Equation 218Bibliography 22014 Fuzzy Fractional Bagley-Torvik Equations 22314.1 Various Types of Fuzzy Fractional Bagley-Torvik Equations 22314.2 Results and Discussions 231Bibliography 241APPENDIX A 243A.1 Fractionally Damped Spring-Mass System (Problem 1) 243A.1.1 Response Analysis 246A.1.2 Analytical Solution Using Fractional Green's Function 247A.2 Fractionally Damped Beam (Problem 2) 248A.2.1 Response Analysis 250A.2.2 Numerical Results 251Bibliography 255INDEX 257