Fuzzy Differential Equations and Applications for Engineers and Scientists  book cover
1st Edition

Fuzzy Differential Equations and Applications for Engineers and Scientists

ISBN 9781482244731
Published November 23, 2016 by CRC Press
240 Pages

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Book Description

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.

Table of Contents



Preliminaries of Fuzzy Set Theory


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


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)


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)


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)


Fuzzy Structural Problems

Double Parametric Based Solution of Uncertain Beam

Solution of Uncertain Beam

Uncertain Response Analysis

Numerical Results


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


Non Probabilistic Uncertainty Analysis of Forest Fire Model

Fuzzy Modelling of Forest Fire

Fuzzy Solution of Fire Propagation

Numerical Results


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


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


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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.