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E-raamat: Artificial Neural Networks for Engineers and Scientists: Solving Ordinary Differential Equations

(National Institute of Technology, Rourkela, India), (National Institute of Technology Rourkela, India)
  • Formaat: 168 pages
  • Ilmumisaeg: 20-Jul-2017
  • Kirjastus: CRC Press Inc
  • Keel: eng
  • ISBN-13: 9781351651318
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Artificial Neural Networks for Engineers and Scientists: Solving Ordinary Differential EquationsSuurem pilt
  • Formaat: 168 pages
  • Ilmumisaeg: 20-Jul-2017
  • Kirjastus: CRC Press Inc
  • Keel: eng
  • ISBN-13: 9781351651318
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Differential equations play a vital role in the fields of engineering and science. Problems in engineering and science can be modeled using ordinary or partial differential equations. Analytical solutions of differential equations may not be obtained easily, so numerical methods have been developed to handle them. Machine intelligence methods, such as Artificial Neural Networks (ANN), are being used to solve differential equations, and these methods are presented in Artificial Neural Networks for Engineers and Scientists: Solving Ordinary Differential Equations. This book shows how computation of differential equation becomes faster once the ANN model is properly developed and applied.
Preface ix
Acknowledgments xiii
Authors xv
Reviewers xvii
1 Preliminaries of Artificial Neural Network
1(10)
1.1 Introduction
1(1)
1.2 Architecture of ANN
2(2)
1.2.1 Feed-Forward Neural Network
3(1)
1.2.2 Feedback Neural Network
3(1)
1.3 Paradigms of Learning
4(1)
1.3.1 Supervised Learning or Associative Learning
4(1)
1.3.2 Unsupervised or Self-Organization Learning
4(1)
1.4 Learning Rules or Learning Processes
5(3)
1.4.1 Error Back-Propagation Learning Algorithm or Delta Learning Rule
5(3)
1.5 Activation Functions
8(3)
1.5.1 Sigmoid Function
8(1)
1.5.1.1 Unipolar Sigmoid Function
8(1)
1.5.1.2 Bipolar Sigmoid Function
9(1)
1.5.2 Tangent Hyperbolic Function
9(1)
References
9(2)
2 Preliminaries of Ordinary Differential Equations
11(6)
2.1 Definitions
12(5)
2.1.1 Order and Degree of DEs
12(1)
2.1.2 Ordinary Differential Equation
12(1)
2.1.3 Partial Differential Equation
12(1)
2.1.4 Linear and Nonlinear Differential Equations
13(1)
2.1.5 Initial Value Problem
13(1)
2.1.6 Boundary Value Problem
14(1)
References
15(2)
3 Multilayer Artificial Neural Network
17(20)
3.1 Structure of Multilayer ANN Model
18(1)
3.2 Formulations and Learning Algorithm of Multilayer ANN Model
18(9)
3.2.1 General Formulation of ODEs Based on ANN Model
18(2)
3.2.2 Formulation of nth-Order IVPs
20(1)
3.2.2.1 Formulation of First-Order IVPs
21(1)
3.2.2.2 Formulation of Second-Order IVPs
21(1)
3.2.3 Formulation of BVPs
22(1)
3.2.3.1 Formulation of Second-Order BVPs
22(1)
3.2.3.2 Formulation of Fourth-Order BVPs
23(1)
3.2.4 Formulation of a System of First-Order ODEs
24(1)
3.2.5 Computation of Gradient of ODEs for Multilayer ANN Model
25(2)
3.3 First-Order Linear ODEs
27(5)
3.4 Higher-Order ODEs
32(2)
3.5 System of ODEs
34(3)
References
36(1)
4 Regression-Based ANN
37(20)
4.1 Algorithm of RBNN Model
37(2)
4.2 Structure of RBNN Model
39(1)
4.3 Formulation and Learning Algorithm of RBNN Model
39(1)
4.4 Computation of Gradient for RBNN Model
40(1)
4.5 First-Order Linear ODEs
40(10)
4.6 Higher-Order Linear ODEs
50(7)
References
56(1)
5 Single-Layer Functional Link Artificial Neural Network
57(20)
5.1 Single-Layer FLANN Models
58(10)
5.1.1 ChNN Model
58(1)
5.1.1.1 Structure of the ChNN Model
58(1)
5.1.1.2 Formulation of the ChNN Model
59(1)
5.1.1.3 Gradient Computation of the ChNN Model
60(2)
5.1.2 LeNN Model
62(1)
5.1.2.1 Structure of the LeNN Model
62(1)
5.1.2.2 Formulation of the LeNN Model
63(1)
5.1.2.3 Gradient Computation of the LeNN Model
63(1)
5.1.3 HeNN Model
64(1)
5.1.3.1 Architecture of the HeNN Model
64(1)
5.1.3.2 Formulation of the HeNN Model
65(1)
5.1.4 Simple Orthogonal Polynomial-Based Neural Network (SOPNN) Model
66(1)
5.1.4.1 Structure of the SOPNN Model
66(1)
5.1.4.2 Formulation of the SOPNN Model
67(1)
5.1.4.3 Gradient Computation of the SOPNN Model
68(1)
5.2 First-Order Linear ODEs
68(1)
5.3 Higher-Order ODEs
69(2)
5.4 System of ODEs
71(6)
References
74(3)
6 Single-Layer Functional Link Artificial Neural Network with Regression-Based Weights
77(10)
6.1 ChNN Model with Regression-Based Weights
78(1)
6.1.1 Structure of the ChNN Model
78(1)
6.1.2 Formulation and Gradient Computation of the ChNN Model
79(1)
6.2 First-Order Linear ODEs
79(4)
6.3 Higher-Order ODEs
83(4)
References
85(2)
7 Lane-Emden Equations
87(18)
7.1 Multilayer ANN-Based Solution of Lane-Emden Equations
89(4)
7.2 FLANN-Based Solution of Lane-Emden Equations
93(12)
7.2.1 Homogeneous Lane-Emden Equations
94(7)
7.2.2 Nonhomogeneous Lane-Emden Equation
101(1)
References
102(3)
8 Emden--Fowler Equations
105(12)
8.1 Multilayer ANN-Based Solution of Emden-Fowler Equations
106(4)
8.2 FLANN-Based Solution of Emden-Fowler Equations
110(7)
References
113(4)
9 Duffing Oscillator Equations
117(16)
9.1 Governing Equation
117(1)
9.2 Unforced Duffing Oscillator Equations
118(5)
9.3 Forced Duffing Oscillator Equations
123(10)
References
131(2)
10 Van der Pol-Duffing Oscillator Equation
133(14)
10.1 Model Equation
134(1)
10.2 Unforced Van der Pol-Duffing Oscillator Equation
135(1)
10.3 Forced Van der Pol-Duffing Oscillator Equation
135(12)
References
144(3)
Index 147
Dr. S. Chakraverty has over 25 years of experience as a researcher and teacher. Currently, he is working at the National Institute of Technology, Rourkela, Odisha as a full Professor and Head of the Department of Mathematics. Prior to this, he was with CSIRCentral Building Research Institute, Roorkee, India. After graduating from St. Columbas College (Ranchi University), he obtained his M. Sc in Mathematics and M. Phil in Computer Applications from the University of Roorkee (now the Indian Institute of Technology Roorkee), earning First Position in the University honors. Dr. Chakraverty received his Ph. D. from IIT Roorkee in 1992. Afterwards, he did his post-doctoral research at Institute of Sound and Vibration Research (ISVR), University of Southampton, U.K. and at the Faculty of Engineering and Computer Science, Concordia University, Canada. He was also a visiting professor at Concordia and McGill Universities, Canada, during 1997-1999 and visiting professor of University of Johannesburg, South Africa during 2011-2014.



Mrs. Susmita Mall received her M. Sc. degree in Mathematics from Ravenshaw University, Cuttack, Odisha, India in 2003. Currently she is a Senior Research Fellow in National Institute of Technology, Rourkela - 769 008, Odisha, India. She has been awarded Women Scientist Scheme-A (WOS-A) fellowship, under Department of Science and Technology (DST), Government of India to undertake her Ph. D. studies. Her current research interest includes Mathematical Modeling, Artificial Neural Network, Differential equations and Numerical analysis. To date, she has published seven research papers in international refereed journals and five in conferences.
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