1st Edition
Fuzzy Differential Equations and Applications for Engineers and Scientists
Differential equations play a vital role in the modeling of physical and engineering problems, such as those in solid and fluid mechanics, viscoelasticity, biology, physics, and many other areas. In general, the parameters, variables and initial conditions within a model are considered as being defined exactly. In reality there may be only vague, imprecise or incomplete information about the variables and parameters available. This can result from errors in measurement, observation, or experimental data; application of different operating conditions; or maintenance induced errors. To overcome uncertainties or lack of precision, one can use a fuzzy environment in parameters, variables and initial conditions in place of exact (fixed) ones, by turning general differential equations into Fuzzy Differential Equations ("FDEs"). In real applications it can be complicated to obtain exact solution of fuzzy differential equations due to complexities in fuzzy arithmetic, creating the need for use of reliable and efficient numerical techniques in the solution of fuzzy differential equations. These include fuzzy ordinary and partial, fuzzy linear and nonlinear, and fuzzy arbitrary order differential equations.
This unique work provides a new direction for the reader in the use of basic concepts of fuzzy differential equations, solutions and its applications. It can serve as an essential reference work for students, scholars, practitioners, researchers and academicians in engineering and science who need to model uncertain physical problems.
Preface
Acknowledgements
Preliminaries of Fuzzy Set Theory
Interval
Fuzzy Number
Triangular Fuzzy Number (TFN)
Trapezodial Fuzzy Number (TrFN)
Gaussian Fuzzy Number (GFN)
Double Parametric Form of Fuzzy Number
Fuzzy Centre
Fuzzy Radius
Fuzzy Width
Fuzzy Arithmetic
Legendre Polynominals
Chebyshev Polynominals
Hermite Polynominals
Fibonacci Polynominals
References
Basics Concepts of Fuzzy and Fuzzy Fractional Differential Equations
n -th Order Fuzzy Differential Equations
Fractional Initial Value Problem (FIVP)
Fuzzy Fractional Initial Value Problem (FFIVP)
Analytical Methods of Fuzzy Differential Equations
Recent Proposed Methods
Method 1: Fuzzy Centre Based Method (FCM)
Method 2: Method Based on Addition and Subtraction of Fuzzy Numbers (ASFM)
Method 3: Fuzzy Centre and Fuzzy Radius Based Method (FCFRM)
Method 4: Double Parametric Based Method (DPM)
References
Numerical Methods for Fuzzy Ordinary and Partial Differential Equations
Euler Type Methods
Method 1: Max-Min Euler Method (MMEM)
Method 2: Average Euler Method (AEM)
Improved Euler Type Methods
Method 3: Max-Min Improved Euler Method (MMIEM)
Method 4: Average Improved Euler Method (AIEM)
Weighted Residual Methods (WRMs)
Method 5: Collocation Type Method (CM)
Method 6: Galerkin Type Method (GM)
Homotopy Perturbation Method (HPM)
Adomian Decomposition Method (ADM)
Variational Iteration Method (VIM)
References
Application of Numerical Methods to Fuzzy Ordinary Differential Equations
Implementation of Methods 1 and 2 (MMEM and AEM)
Implementation of Methods 3 and 4 (MMIEM and AIEM)
Implementation of Method 5 (CM)
Implementation of Method 6 (GM)
Implementation of Homotopy Perturbation Method (HPM)
References
Fuzzy Structural Problems
Double Parametric Based Solution of Uncertain Beam
Solution of Uncertain Beam
Uncertain Response Analysis
Numerical Results
References
Fuzzy Vibration Equation of Large Membrane
Double Parametric Based Solution of Uncertain Vibration Equation of Large Membrane
Solutions of fuzzy vibration equation of large membrane
Solution Bounds for Particular Cases
Numerical Results
References
Non Probabilistic Uncertainty Analysis of Forest Fire Model
Fuzzy Modelling of Forest Fire
Fuzzy Solution of Fire Propagation
Numerical Results
References
Fuzzy Inverse Heat Conduction Problems
Formulation of Uncertain Inverse Heat Conduction Problem
Double Parametric Based Uncertain Inverse Heat Conduction Problem
Solution of the Fuzzy Inverse Heat Conduction Problem
Solution Bounds for Different Fuzzy Initial Conditions
Numerical Results and Discussions
References
Fuzzy Fractional Klein-Gordon Equation
Double Parametric Based Fuzzy Fractional Klein-Gordan Equation
Solutions of Fuzzy Fractional Klein-Gordan Equation Using Homotopy Perturbation Method
Solution Bounds for Particular Cases
Numerical Results
References
Biography
Snehashish Chakraverty, Ph.D. is Professor of Mathematics at the National Institute of Technology, Rourkela in India, Ph. D. from IIT Roorkee and post-doctoral research from ISVR, University of Southampton, UK, and Concordia University, Canada. He was visiting professor at Concordia, McGill and Johannesburg universities. He published five books, 239 research papers, reviewer of many international journals, recipient of CSIR Young Scientist, BOYSCAST, UCOST, Golden Jubilee CBRI, INSA International Bilateral Exchange, Platinum Jubilee ISCA Lecture and Roorkee University gold medal awards. Dr. Chakraverty is the Chief Editor of International Journal of Fuzzy Computation and Modelling (IJFCM), Inderscience Publisher, Switzerland (http://www.inderscience.com/ijfcm) and happens to be the Guest Editor for other few journals. He was the President of the Section of Mathematical sciences (including Statistics) of Indian Science Congress (2015-2016) and was the Vice President – Orissa Mathematical Society (2011-2013). He has already guided eleven (11) Ph. D. students and seven are ongoing. Dr. Chakraverty has undertaken around 16 research projects as Principle Investigator funded by international and national agencies totaling about Rs.1.5 crores. His research area includes Differential Equations, Numerical Analysis, Soft Computing,Vibration and Inverse Vibration problems.
Smita Tapaswini, Ph.D. 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. She has received her Ph.D. degree in Mathematics from National Institute of Technology Rourkela, Odisha, 769 008, India on January 2015. She has been awarded Rajiv Gandhi National Fellowship (RGNF), under University Grant Commission (UGC), Government of India and also qualified Graduate Aptitude Test in Engineering (GATE) in the year 2011. Her research interests include fuzzy differential equations, fuzzy fractional differential equations and numerical analysis.
Diptiranjan Behera, Ph.D. is working as a Post-Doctoral Fellow at the Sichuan Provincial Key Laboratory of Reliability Engineering, School of Mechatronics Engineering, University of Electronic Science and Technology of China (UESTC), China. After completing B. Sc. (Bachelor of Science) degree in 2008 with Mathematics honours from Banki College (Utkal University, Odisha, India), his career started from National Institute of Technology (NIT) Rourkela, Odisha 769008, India and did M. Sc. (Master of Science) and Ph. D. degree in Mathematics from there. He has completed his M. Sc. in the year 2010 and received his Ph. D. degree in January 2015. During Ph. D. he had been doing research as a Junior and Senior Research Fellow on a research project funded by Board of Research in Nuclear Sciences, Department of Atomic Energy, Government of India. His current research interest includes in the areas of interval and fuzzy mathematics, fuzzy finite element methods, fuzzy structural analysis, fuzzy differential equations, fuzzy fractional differential equations, fuzzy system of linear equations, fuzzy eigenvalue problem and fuzzy linear programming problem.
