Fuzzy Arbitrary Order System: Fuzzy Fractional Differential Equations and Applications

Presents a systematic treatment of fuzzy fractional differential equations as well as newly developed computational methods to model uncertain physical problems

Complete with comprehensive results and solutions, Fuzzy Arbitrary Order System: Fuzzy Fractional Differential Equations and Applications details newly developed methods of fuzzy computational techniques needed to model solve uncertainty.  Fuzzy differential equations are solved via various analytical and numerical methodologies, and this book presents their importance for problem solving and prototype engineering 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, inverse, and chemical problems as well as heat equations and bio–mathematical applications Discussions on how to model physical problems in terms of non–probabilistic methods and provides systematic coverage of fuzzy fractional differential equations and its applications Uncertainties in systems and processes with a fuzzy concept

Fuzzy 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.

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, and differential equations. 

SmitaTapaswini, PhD, is Assistant Professor in the Department of Mathematics at the Kalinga Institute of Industrial Technology University in India and is also Post–Doctoral Fellow at the College of Mathematics and Statistics at Chongqing University in China. Her research interests include fuzzy differential equations, fuzzy fractional differential equations, and numerical analysis. 

Diptiranjan Behera, PhD, is Post–Doctoral Fellow at the Institute of Reliability Engineering in the School of Mechatronics Engineering at the University of Electronic Science and Technology in China. His current research interests include interval and fuzzy mathematics, fuzzy finite element methods, and fuzzy structural analysis.

 

Preface

Chapter 1: Preliminaries of Fuzzy Set Theory

1.1. Interval

1.2. Fuzzy Number

1.3. Triangular Fuzzy Number (TFN)

1.4. Trapezoidal Fuzzy Number (TrFN)

1.5. Gaussian Fuzzy Number (GFN)

1.6. Double Parametric Form of Fuzzy Number

1.7. Fuzzy Centre

1.8. Fuzzy Radius

1.9. Fuzzy Width

1.10. Fuzzy Arithmetic

References

Chapter 2: Basics of fractional and fuzzy fractional differential equations

2.1. Fuzzy Initial Value Problem (FIVP)

2.2. Fuzzy Boundary Value Problem (FBVP)

2.3. Riemann–Liouville Fractional integral

2.4. Fuzzy Riemann–Liouville Fractional integral

2.5. Caputo Derivative

2.6. Caputo–type fuzzy fractional derivatives

2.7. Fractional Initial Value Problem

2.8. Fuzzy Fractional Initial Value Problem

References

Chapter 3: Analytical Methods for Fuzzy Fractional Differential Equations (FFDEs)

3.1. n–term Linear Fuzzy Fractional Linear Differential Equations

3.2. Proposed Methods

References

Chapter 4: Numerical Methods for Fuzzy Fractional Differential Equations.

4.1. Homotopy Perturbation Method (HPM)

4.2. Adomian Decomposition Method (ADM)

4.3. Variational Iteration Method (VIM)

References

Chapter 5: Fuzzy Fractional Heat Equations

5.1. Arbitrary–Order Heat Equation

5.2. Solution of Fuzzy Arbitrary Order Heat Equations by HPM

5.3. Numerical Examples

5.4. Numerical Results

References

Chapter 6: Fuzzy Fractional Bio–Mathematical Applications

6.1. Fuzzy Arbitrary Order Predator–Prey Equations

6.2. Numerical Results of Fuzzy Arbitrary Order Predator–Prey Equations

References

Chapter 7: Fuzzy Fractional Chemical Problems

7.1. Arbitrary–order Rossler s systems

7.2. HPM Solution of Uncertain Arbitrary Order Rossler s System

7.3. Particular case

7.4. Numerical Results

References

Chapter 8: Fuzzy Fractional Structural Problems

8.1. Fuzzy Fractionally Damped Discrete System

8.2. Uncertain Response Analysis

8.3. Numerical Results

8.4. Fuzzy Fractionally Damped Continuous System

8.5. Uncertain Response Analysis

8.6. Numerical Results

References

Chapter 9: Fuzzy Fractional Diffusion Problems

9.1. Fuzzy Fractional Order Diffusion Equation

9.2. Numerical Results of Fuzzy Fractional Diffusion Equation

References

Chapter 10: Uncertain Fractional Fornberg–Whitham Equations

10.1. Parametric based interval fractional Fornberg–Whitham equation

10.2. Solution by VIM using proposed methodology

10.3. Solution bounds for different interval initial conditions

10.4. Numerical results

References

Chapter 11: Fuzzy Fractional Vibration Equation of Large Membrane

11.1. Double Parametric Based Solution of Uncertain Vibration Equation of Large Membrane

11.2. Solutions of fuzzy vibration equation of large membrane

11.3. Case Studies (Solution bounds for particular cases)

11.4. Numerical results

References

Chapter 12: Fuzzy Fractional Telegraph equations

12.1. Double Parametric Based Fuzzy Fractional Telegraph Equations

12.2. Solutions of Fuzzy Telegraph Equations Using Homotopy Perturbation Method

12.3. Solution Bounds for Particular Cases

12.4. Numerical Results

References

Chapter 13: Fuzzy Fokker–Planck equation with space and time fractional derivatives

13.1. Fuzzy fractional Fokker–Planck equation with space and time fractional derivatives

13.2. Double parametric based solution of uncertain fractional Fokker–Planck equation

13.3. Case studies using HPM and ADM.

13.4. Numerical Results

References

Chapter 14: Fuzzy fractional Bagley–Torvik equations

14.1. Various types of Fuzzy fractional Bagley–Torvik equations

14.2. Results and Discussions

References

Appendix

Index