Wen Yu, PhD, is Professor at CINVESTAV-IPN (National Polytechnic Institute), Mexico City, Mexico. He is Associate Editor of IEEE Transactions on Cybernetics, Neurocomputing, and Journal of Intelligent and Fuzzy Systems. Dr. Yu is a member of the Mexican Academy of Sciences. Raheleh Jafari is a postdoctoral research fellow at Centre for Artificial Intelligence Research (CAIR), University of Agder, Grimstad, Norway. She is on the editorial board of the Journal of Intelligent and Fuzzy Systems, and served as a reviewer in various journals and conferences. Her research interest is in the field of artificial intelligence, fuzzy control, machine learning, nonlinear systems, neural networks, and fuzzy engineering.

• List of Figures xiList of Tables xiiiPreface xv1 Fuzzy Equations 11.1 Introduction 11.2 Fuzzy Equations 11.3 Algebraic Fuzzy Equations 31.4 Numerical Methods for Solving Fuzzy Equations 51.4.1 Newton Method 51.4.2 Steepest Descent Method 71.4.3 Adomian Decomposition Method 81.4.4 Ranking Method 91.4.5 Intelligent Methods 101.4.5.1 Genetic Algorithm Method 101.4.5.2 Neural Network Method 111.4.5.3 Fuzzy Linear Regression Model 141.5 Summary 202 Fuzzy Differential Equations 212.1 Introduction 212.2 Predictor–Corrector Method 212.3 Adomian Decomposition Method 232.4 Euler Method 232.5 Taylor Method 252.6 Runge–Kutta Method 252.7 Finite Difference Method 262.8 Differential Transform Method 282.9 Neural Network Method 292.10 Summary 363 Modeling and Control Using Fuzzy Equations 393.1 Fuzzy Modeling with Fuzzy Equations 393.1.1 Fuzzy Parameter Estimation with Neural Networks 453.1.2 Upper Bounds of the Modeling Errors 483.2 Control with Fuzzy Equations 523.3 Simulations 593.4 Summary 674 Modeling and Control Using Fuzzy Differential Equations 694.1 Introduction 694.2 Fuzzy Modeling with Fuzzy Differential Equations 694.3 Existence of a Solution 724.4 Solution Approximation using Bernstein Neural Networks 794.5 Solutions Approximation using the Fuzzy Sumudu Transform 834.6 Simulations 854.7 Summary 995 System Modeling with Partial Differential Equations 1015.1 Introduction 1015.2 Solutions using Burgers–Fisher Equations 1015.3 Solution using Wave Equations 1065.4 Simulations 1095.5 Summary 1176 System Control using Z-numbers 1196.1 Introduction 1196.2 Modeling using Dual Fuzzy Equations and Z-numbers 1196.3 Controllability using Dual Fuzzy Equations 1246.4 Fuzzy Controller 1286.5 Nonlinear System Modeling 1316.6 Controllability using Fuzzy Differential Equations 1316.7 Fuzzy Controller Design using Fuzzy Differential Equations and Z-number 1356.8 Approximation using a Fuzzy Sumudu Transform and Z-numbers 1386.9 Simulations 1396.10 Summary 151References 153Index 167

Wen Yu, PhD, is Professor at CINVESTAV-IPN (National Polytechnic Institute), Mexico City, Mexico. He is Associate Editor of IEEE Transactions on Cybernetics, Neurocomputing, and Journal of Intelligent and Fuzzy Systems. Dr. Yu is a member of the Mexican Academy of Sciences. Raheleh Jafari is a postdoctoral research fellow at Centre for Artificial Intelligence Research (CAIR), University of Agder, Grimstad, Norway. She is on the editorial board of the Journal of Intelligent and Fuzzy Systems, and served as a reviewer in various journals and conferences. Her research interest is in the field of artificial intelligence, fuzzy control, machine learning, nonlinear systems, neural networks, and fuzzy engineering.

• List of Figures xiList of Tables xiiiPreface xv1 Fuzzy Equations 11.1 Introduction 11.2 Fuzzy Equations 11.3 Algebraic Fuzzy Equations 31.4 Numerical Methods for Solving Fuzzy Equations 51.4.1 Newton Method 51.4.2 Steepest Descent Method 71.4.3 Adomian Decomposition Method 81.4.4 Ranking Method 91.4.5 Intelligent Methods 101.4.5.1 Genetic Algorithm Method 101.4.5.2 Neural Network Method 111.4.5.3 Fuzzy Linear Regression Model 141.5 Summary 202 Fuzzy Differential Equations 212.1 Introduction 212.2 Predictor–Corrector Method 212.3 Adomian Decomposition Method 232.4 Euler Method 232.5 Taylor Method 252.6 Runge–Kutta Method 252.7 Finite Difference Method 262.8 Differential Transform Method 282.9 Neural Network Method 292.10 Summary 363 Modeling and Control Using Fuzzy Equations 393.1 Fuzzy Modeling with Fuzzy Equations 393.1.1 Fuzzy Parameter Estimation with Neural Networks 453.1.2 Upper Bounds of the Modeling Errors 483.2 Control with Fuzzy Equations 523.3 Simulations 593.4 Summary 674 Modeling and Control Using Fuzzy Differential Equations 694.1 Introduction 694.2 Fuzzy Modeling with Fuzzy Differential Equations 694.3 Existence of a Solution 724.4 Solution Approximation using Bernstein Neural Networks 794.5 Solutions Approximation using the Fuzzy Sumudu Transform 834.6 Simulations 854.7 Summary 995 System Modeling with Partial Differential Equations 1015.1 Introduction 1015.2 Solutions using Burgers–Fisher Equations 1015.3 Solution using Wave Equations 1065.4 Simulations 1095.5 Summary 1176 System Control using Z-numbers 1196.1 Introduction 1196.2 Modeling using Dual Fuzzy Equations and Z-numbers 1196.3 Controllability using Dual Fuzzy Equations 1246.4 Fuzzy Controller 1286.5 Nonlinear System Modeling 1316.6 Controllability using Fuzzy Differential Equations 1316.7 Fuzzy Controller Design using Fuzzy Differential Equations and Z-number 1356.8 Approximation using a Fuzzy Sumudu Transform and Z-numbers 1386.9 Simulations 1396.10 Summary 151References 153Index 167

Wen Yu, PhD, is Professor at CINVESTAV-IPN (National Polytechnic Institute), Mexico City, Mexico. He is Associate Editor of IEEE Transactions on Cybernetics, Neurocomputing, and Journal of Intelligent and Fuzzy Systems. Dr. Yu is a member of the Mexican Academy of Sciences. Raheleh Jafari is a postdoctoral research fellow at Centre for Artificial Intelligence Research (CAIR), University of Agder, Grimstad, Norway. She is on the editorial board of the Journal of Intelligent and Fuzzy Systems, and served as a reviewer in various journals and conferences. Her research interest is in the field of artificial intelligence, fuzzy control, machine learning, nonlinear systems, neural networks, and fuzzy engineering.

• List of Figures xiList of Tables xiiiPreface xv1 Fuzzy Equations 11.1 Introduction 11.2 Fuzzy Equations 11.3 Algebraic Fuzzy Equations 31.4 Numerical Methods for Solving Fuzzy Equations 51.4.1 Newton Method 51.4.2 Steepest Descent Method 71.4.3 Adomian Decomposition Method 81.4.4 Ranking Method 91.4.5 Intelligent Methods 101.4.5.1 Genetic Algorithm Method 101.4.5.2 Neural Network Method 111.4.5.3 Fuzzy Linear Regression Model 141.5 Summary 202 Fuzzy Differential Equations 212.1 Introduction 212.2 Predictor–Corrector Method 212.3 Adomian Decomposition Method 232.4 Euler Method 232.5 Taylor Method 252.6 Runge–Kutta Method 252.7 Finite Difference Method 262.8 Differential Transform Method 282.9 Neural Network Method 292.10 Summary 363 Modeling and Control Using Fuzzy Equations 393.1 Fuzzy Modeling with Fuzzy Equations 393.1.1 Fuzzy Parameter Estimation with Neural Networks 453.1.2 Upper Bounds of the Modeling Errors 483.2 Control with Fuzzy Equations 523.3 Simulations 593.4 Summary 674 Modeling and Control Using Fuzzy Differential Equations 694.1 Introduction 694.2 Fuzzy Modeling with Fuzzy Differential Equations 694.3 Existence of a Solution 724.4 Solution Approximation using Bernstein Neural Networks 794.5 Solutions Approximation using the Fuzzy Sumudu Transform 834.6 Simulations 854.7 Summary 995 System Modeling with Partial Differential Equations 1015.1 Introduction 1015.2 Solutions using Burgers–Fisher Equations 1015.3 Solution using Wave Equations 1065.4 Simulations 1095.5 Summary 1176 System Control using Z-numbers 1196.1 Introduction 1196.2 Modeling using Dual Fuzzy Equations and Z-numbers 1196.3 Controllability using Dual Fuzzy Equations 1246.4 Fuzzy Controller 1286.5 Nonlinear System Modeling 1316.6 Controllability using Fuzzy Differential Equations 1316.7 Fuzzy Controller Design using Fuzzy Differential Equations and Z-number 1356.8 Approximation using a Fuzzy Sumudu Transform and Z-numbers 1386.9 Simulations 1396.10 Summary 151References 153Index 167