John Wiley & Sons Mathematical Methods in Interdisciplinary Sciences Cover Brings mathematics to bear on your real-world, scientific problems Mathematical Methods in Interdis.. Product #: 978-1-119-58550-3 Regular price: $109.52 $109.52 Auf Lager

Mathematical Methods in Interdisciplinary Sciences

Chakraverty, Snehashish (Herausgeber)

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1. Auflage September 2020
464 Seiten, Hardcover
Wiley & Sons Ltd

ISBN: 978-1-119-58550-3
John Wiley & Sons

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Brings mathematics to bear on your real-world, scientific problems

Mathematical Methods in Interdisciplinary Sciences provides a practical and usable framework for bringing a mathematical approach to modelling real-life scientific and technological problems. The collection of chapters Dr. Snehashish Chakraverty has provided describe in detail how to bring mathematics, statistics, and computational methods to the fore to solve even the most stubborn problems involving the intersection of multiple fields of study. Graduate students, postgraduate students, researchers, and professors will all benefit significantly from the author's clear approach to applied mathematics.

The book covers a wide range of interdisciplinary topics in which mathematics can be brought to bear on challenging problems requiring creative solutions. Subjects include:
* Structural static and vibration problems
* Heat conduction and diffusion problems
* Fluid dynamics problems

The book also covers topics as diverse as soft computing and machine intelligence. It concludes with examinations of various fields of application, like infectious diseases, autonomous car and monotone inclusion problems.

Notes on Contributors xv

Preface xxv

Acknowledgments xxvii

1 Connectionist Learning Models for Application Problems Involving Differential and Integral Equations 1
Susmita Mall, Sumit Kumar Jeswal, and Snehashish Chakraverty

1.1 Introduction 1

1.1.1 Artificial Neural Network 1

1.1.2 Types of Neural Networks 1

1.1.3 Learning in Neural Network 2

1.1.4 Activation Function 2

1.1.4.1 Sigmoidal Function 3

1.1.5 Advantages of Neural Network 3

1.1.6 Functional Link Artificial Neural Network (FLANN) 3

1.1.7 Differential Equations (DEs) 4

1.1.8 Integral Equation 5

1.1.8.1 Fredholm Integral Equation of First Kind 5

1.1.8.2 Fredholm Integral Equation of Second Kind 5

1.1.8.3 Volterra Integral Equation of First Kind 5

1.1.8.4 Volterra Integral Equation of Second Kind 5

1.1.8.5 Linear Fredholm Integral Equation System of Second Kind 6

1.2 Methodology for Differential Equations 6

1.2.1 FLANN-Based General Formulation of Differential Equations 6

1.2.1.1 Second-Order Initial Value Problem 6

1.2.1.2 Second-Order Boundary Value Problem 7

1.2.2 Proposed Laguerre Neural Network (LgNN) for Differential Equations 7

1.2.2.1 Architecture of Single-Layer LgNN Model 7

1.2.2.2 Training Algorithm of Laguerre Neural Network (LgNN) 8

1.2.2.3 Gradient Computation of LgNN 9

1.3 Methodology for Solving a System of Fredholm Integral Equations of Second Kind 9

1.3.1 Algorithm 10

1.4 Numerical Examples and Discussion 11

1.4.1 Differential Equations and Applications 11

1.4.2 Integral Equations 16

1.5 Conclusion 20

References 20

2 Deep Learning in Population Genetics: Prediction and Explanation of Selection of a Population 23
Romila Ghosh and Satyakama Paul

2.1 Introduction 23

2.2 Literature Review 23

2.3 Dataset Description 25

2.3.1 Selection and Its Importance 25

2.4 Objective 26

2.5 Relevant Theory, Results, and Discussions 27

2.5.1 automl 27

2.5.2 Hypertuning the Best Model 28

2.6 Conclusion 30

References 30

3 A Survey of Classification Techniques in Speech Emotion Recognition 33
Tanmoy Roy, Tshilidzi Marwala, and Snehashish Chakraverty

3.1 Introduction 33

3.2 Emotional Speech Databases 33

3.3 SER Features 34

3.4 Classification Techniques 35

3.4.1 Hidden Markov Model 36

3.4.1.1 Difficulties in Using HMM for SER 37

3.4.2 Gaussian Mixture Model 37

3.4.2.1 Difficulties in Using GMM for SER 38

3.4.3 Support Vector Machine 38

3.4.3.1 Difficulties with SVM 39

3.4.4 Deep Learning 39

3.4.4.1 Drawbacks of Using Deep Learning for SER 41

3.5 Difficulties in SER Studies 41

3.6 Conclusion 41

References 42

4 Mathematical Methods in Deep Learning 49
Srinivasa Manikant Upadhyayula and Kannan Venkataramanan

4.1 Deep Learning Using Neural Networks 49

4.2 Introduction to Neural Networks 49

4.2.1 Artificial Neural Network (ANN) 50

4.2.1.1 Activation Function 52

4.2.1.2 Logistic Sigmoid Activation Function 52

4.2.1.3 tanh or Hyperbolic Tangent Activation Function 53

4.2.1.4 ReLU (Rectified Linear Unit) Activation Function 54

4.3 Other Activation Functions (Variant Forms of ReLU) 55

4.3.1 Smooth ReLU 55

4.3.2 Noisy ReLU 55

4.3.3 Leaky ReLU 55

4.3.4 Parametric ReLU 56

4.3.5 Training and Optimizing a Neural Network Model 56

4.4 Backpropagation Algorithm 56

4.5 Performance and Accuracy 59

4.6 Results and Observation 59

References 61

5 Multimodal Data Representation and Processing Based on Algebraic System of Aggregates 63
Yevgeniya Sulema and Etienne Kerre

5.1 Introduction 63

5.2 Basic Statements of ASA 64

5.3 Operations on Aggregates and Multi-images 65

5.4 Relations and Digital Intervals 72

5.5 Data Synchronization 75

5.6 Fuzzy Synchronization 92

5.7 Conclusion 96

References 96

6 Nonprobabilistic Analysis of Thermal and Chemical Diffusion Problems with Uncertain Bounded Parameters 99
Sukanta Nayak, Tharasi Dilleswar Rao, and Snehashish Chakraverty

6.1 Introduction 99

6.2 Preliminaries 99

6.2.1 Interval Arithmetic 99

6.2.2 Fuzzy Number and Fuzzy Arithmetic 100

6.2.3 Parametric Representation of Fuzzy Number 101

6.2.4 Finite Difference Schemes for PDEs 102

6.3 Finite Element Formulation for Tapered Fin 102

6.4 Radon Diffusion and Its Mechanism 105

6.5 Radon Diffusion Mechanism with TFN Parameters 107

6.5.1 EFDM to Radon Diffusion Mechanism with TFN Parameters 108

6.6 Conclusion 112

References 112

7 Arbitrary Order Differential Equations with Fuzzy Parameters 115
Tofigh Allahviranloo and Soheil Salahshour

7.1 Introduction 115

7.2 Preliminaries 115

7.3 Arbitrary Order Integral and Derivative for Fuzzy-Valued Functions 116

7.4 Generalized Fuzzy Laplace Transform with Respect to Another Function 118

References 122

8 Fluid Dynamics Problems in Uncertain Environment 125
Perumandla Karunakar, Uddhaba Biswal, and Snehashish Chakraverty

8.1 Introduction 125

8.2 Preliminaries 126

8.2.1 Fuzzy Set 126

8.2.2 Fuzzy Number 126

8.2.3 delta-Cut 127

8.2.4 Parametric Approach 127

8.3 Problem Formulation 127

8.4 Methodology 129

8.4.1 Homotopy Perturbation Method 129

8.4.2 Homotopy Perturbation Transform Method 130

8.5 Application of HPM and HPTM 131

8.5.1 Application of HPM to Jeffery-Hamel Problem 131

8.5.2 Application of HPTM to Coupled Whitham-Broer-Kaup Equations 134

8.6 Results and Discussion 136

8.7 Conclusion 142

References 142

9 Fuzzy Rough Set Theory-Based Feature Selection: A Review 145
Tanmoy Som, Shivam Shreevastava, Anoop Kumar Tiwari, and Shivani Singh

9.1 Introduction 145

9.2 Preliminaries 146

9.2.1 Rough Set Theory 146

9.2.1.1 Rough Set 146

9.2.1.2 Rough Set-Based Feature Selection 147

9.2.2 Fuzzy Set Theory 147

9.2.2.1 Fuzzy Tolerance Relation 148

9.2.2.2 Fuzzy Rough Set Theory 149

9.2.2.3 Degree of Dependency-Based Fuzzy Rough Attribute Reduction 149

9.2.2.4 Discernibility Matrix-Based Fuzzy Rough Attribute Reduction 149

9.3 Fuzzy Rough Set-Based Attribute Reduction 149

9.3.1 Degree of Dependency-Based Approaches 150

9.3.2 Discernibility Matrix-Based Approaches 154

9.4 Approaches for Semisupervised and Unsupervised Decision Systems 154

9.5 Decision Systems with Missing Values 158

9.6 Applications in Classification, Rule Extraction, and Other Application Areas 158

9.7 Limitations of Fuzzy Rough Set Theory 159

9.8 Conclusion 160

References 160

10 Universal Intervals: Towards a Dependency-Aware Interval Algebra 167
Hend Dawood and Yasser Dawood

10.1 Introduction 167

10.2 The Need for Interval Computations 169

10.3 On Some Algebraic and Logical Fundamentals 170

10.4 Classical Intervals and the Dependency Problem 174

10.5 Interval Dependency: A Logical Treatment 176

10.5.1 Quantification Dependence and Skolemization 177

10.5.2 A Formalization of the Notion of Interval Dependency 179

10.6 Interval Enclosures Under Functional Dependence 184

10.7 Parametric Intervals: How Far They Can Go 186

10.7.1 Parametric Interval Operations: From Endpoints to Convex Subsets 186

10.7.2 On the Structure of Parametric Intervals: Are They Properly Founded? 188

10.8 Universal Intervals: An Interval Algebra with a Dependency Predicate 192

10.8.1 Universal Intervals, Rational Functions, and Predicates 193

10.8.2 The Arithmetic of Universal Intervals 196

10.9 The S-Field Algebra of Universal Intervals 201

10.10 Guaranteed Bounds or Best Approximation or Both? 209

Supplementary Materials 210

Acknowledgments 211

References 211

11 Affine-Contractor Approach to Handle Nonlinear Dynamical Problems in Uncertain Environment 215
Nisha Rani Mahato, Saudamini Rout, and Snehashish Chakraverty

11.1 Introduction 215

11.2 Classical Interval Arithmetic 217

11.2.1 Intervals 217

11.2.2 Set Operations of Interval System 217

11.2.3 Standard Interval Computations 218

11.2.4 Algebraic Properties of Interval 219

11.3 Interval Dependency Problem 219

11.4 Affine Arithmetic 220

11.4.1 Conversion Between Interval and Affine Arithmetic 220

11.4.2 Affine Operations 221

11.5 Contractor 223

11.5.1 SIVIA 223

11.6 Proposed Methodology 225

11.7 Numerical Examples 230

11.7.1 Nonlinear Oscillators 230

11.7.1.1 Unforced Nonlinear Differential Equation 230

11.7.1.2 Forced Nonlinear Differential Equation 232

11.7.2 Other Dynamic Problem 233

11.7.2.1 Nonhomogeneous Lane-Emden Equation 233

11.8 Conclusion 236

References 236

12 Dynamic Behavior of Nanobeam Using Strain Gradient Model 239
Subrat Kumar Jena, Rajarama Mohan Jena, and Snehashish Chakraverty

12.1 Introduction 239

12.2 Mathematical Formulation of the Proposed Model 240

12.3 Review of the Differential Transform Method (DTM) 241

12.4 Application of DTM on Dynamic Behavior Analysis 242

12.5 Numerical Results and Discussion 244

12.5.1 Validation and Convergence 244

12.5.2 Effect of the Small-Scale Parameter 245

12.5.3 Effect of Length-Scale Parameter 247

12.6 Conclusion 248

Acknowledgment 249

References 250

13 Structural Static and Vibration Problems 253
M. Amin Changizi and Ion Stiharu

13.1 Introduction 253

13.2 One-parameter Groups 254

13.3 Infinitesimal Transformation 254

13.4 Canonical Coordinates 254

13.5 Algorithm for Lie Symmetry Point 255

13.6 Reduction of the Order of the ODE 255

13.7 Solution of First-Order ODE with Lie Symmetry 255

13.8 Identification 256

13.9 Vibration of a Microcantilever Beam Subjected to Uniform Electrostatic Field 258

13.10 Contact Form for the Equation 259

13.11 Reducing in the Order of the Nonlinear ODE Representing the Vibration of a Microcantilever Beam Under Electrostatic Field 260

13.12 Nonlinear Pull-in Voltage 261

13.13 Nonlinear Analysis of Pull-in Voltage of Twin Microcantilever Beams 266

13.14 Nonlinear Analysis of Pull-in Voltage of Twin Microcantilever Beams of Different Thicknesses 268

References 272

14 Generalized Differential and Integral Quadrature: Theory and Applications 273
Francesco Tornabene and Rossana Dimitri

14.1 Introduction 273

14.2 Differential Quadrature 274

14.2.1 Genesis of the Differential Quadrature Method 274

14.2.2 Differential Quadrature Law 275

14.3 General View on Differential Quadrature 277

14.3.1 Basis Functions 278

14.3.1.1 Lagrange Polynomials 281

14.3.1.2 Trigonometric Lagrange Polynomials 282

14.3.1.3 Classic Orthogonal Polynomials 282

14.3.1.4 Monomial Functions 291

14.3.1.5 Exponential Functions 291

14.3.1.6 Bernstein Polynomials 291

14.3.1.7 Fourier Functions 292

14.3.1.8 Bessel Polynomials 292

14.3.1.9 Boubaker Polynomials 292

14.3.2 Grid Distributions 293

14.3.2.1 Coordinate Transformation 293

14.3.2.2 delta-Point Distribution 293

14.3.2.3 Stretching Formulation 293

14.3.2.4 Several Types of Discretization 293

14.3.3 Numerical Applications: Differential Quadrature 297

14.4 Generalized Integral Quadrature 310

14.4.1 Generalized Taylor-Based Integral Quadrature 312

14.4.2 Classic Integral Quadrature Methods 314

14.4.2.1 Trapezoidal Rule with Uniform Discretization 314

14.4.2.2 Simpson's Method (One-third Rule) with Uniform Discretization 314

14.4.2.3 Chebyshev-Gauss Method (Chebyshev of the First Kind) 314

14.4.2.4 Chebyshev-Gauss Method (Chebyshev of the Second Kind) 314

14.4.2.5 Chebyshev-Gauss Method (Chebyshev of the Third Kind) 315

14.4.2.6 Chebyshev-Gauss Method (Chebyshev of the Fourth Kind) 315

14.4.2.7 Chebyshev-Gauss-Radau Method (Chebyshev of the First Kind) 315

14.4.2.8 Chebyshev-Gauss-Lobatto Method (Chebyshev of the First Kind) 315

14.4.2.9 Gauss-Legendre or Legendre-Gauss Method 315

14.4.2.10 Gauss-Legendre-Radau or Legendre-Gauss-Radau Method 315

14.4.2.11 Gauss-Legendre-Lobatto or Legendre-Gauss-Lobatto Method 316

14.4.3 Numerical Applications: Integral Quadrature 316

14.4.4 Numerical Applications: Taylor-Based Integral Quadrature 320

14.5 General View: The Two-Dimensional Case 324

References 340

15 Brain Activity Reconstruction by Finding a Source Parameter in an Inverse Problem 343
Amir H. Hadian-Rasanan and Jamal Amani Rad

15.1 Introduction 343

15.1.1 Statement of the Problem 344

15.1.2 Brief Review of Other Methods Existing in the Literature 345

15.2 Methodology 346

15.2.1 Weighted Residual Methods and Collocation Algorithm 346

15.2.2 Function Approximation Using Chebyshev Polynomials 349

15.3 Implementation 353

15.4 Numerical Results and Discussion 354

15.4.1 Test Problem 1 355

15.4.2 Test Problem 2 357

15.4.3 Test Problem 3 358

15.4.4 Test Problem 4 359

15.4.5 Test Problem 5 362

15.5 Conclusion 365

References 365

16 Optimal Resource Allocation in Controlling Infectious Diseases 369
A.C. Mahasinghe, S.S.N. Perera, and K.K.W.H. Erandi

16.1 Introduction 369

16.2 Mobility-Based Resource Distribution 370

16.2.1 Distribution of National Resources 370

16.2.2 Transmission Dynamics 371

16.2.2.1 Compartment Models 371

16.2.2.2 SI Model 371

16.2.2.3 Exact Solution 371

16.2.2.4 Transmission Rate and Potential 372

16.2.3 Nonlinear Problem Formulation 373

16.2.3.1 Piecewise Linear Reformulation 374

16.2.3.2 Computational Experience 374

16.3 Connection-Strength Minimization 376

16.3.1 Network Model 376

16.3.1.1 Disease Transmission Potential 376

16.3.1.2 An Example 376

16.3.2 Nonlinear Problem Formulation 377

16.3.2.1 Connection Strength Measure 377

16.3.2.2 Piecewise Linear Approximation 378

16.3.2.3 Computational Experience 379

16.4 Risk Minimization 379

16.4.1 Novel Strategies for Individuals 379

16.4.1.1 Epidemiological Isolation 380

16.4.1.2 Identifying Objectives 380

16.4.2 Minimizing the High-Risk Population 381

16.4.2.1 An Example 381

16.4.2.2 Model Formulation 382

16.4.2.3 Linear Integer Program 383

16.4.2.4 Computational Experience 383

16.4.3 Minimizing the Total Risk 384

16.4.4 Goal Programming Approach 384

16.5 Conclusion 386

References 387

17 Artificial Intelligence and Autonomous Car 391
Merve Ar1türk, S1rma Yavuz, and Tofigh Allahviranloo

17.1 Introduction 391

17.2 What is Artificial Intelligence? 391

17.3 Natural Language Processing 391

17.4 Robotics 393

17.4.1 Classification by Axes 393

17.4.1.1 Axis Concept in Robot Manipulators 393

17.4.2 Classification of Robots by Coordinate Systems 394

17.4.3 Other Robotic Classifications 394

17.5 Image Processing 395

17.5.1 Artificial Intelligence in Image Processing 395

17.5.2 Image Processing Techniques 395

17.5.2.1 Image Preprocessing and Enhancement 396

17.5.2.2 Image Segmentation 396

17.5.2.3 Feature Extraction 396

17.5.2.4 Image Classification 396

17.5.3 Artificial Intelligence Support in Digital Image Processing 397

17.5.3.1 Creating a Cancer Treatment Plan 397

17.5.3.2 Skin Cancer Diagnosis 397

17.6 Problem Solving 397

17.6.1 Problem-solving Process 397

17.7 Optimization 399

17.7.1 Optimization Techniques in Artificial Intelligence 399

17.8 Autonomous Systems 400

17.8.1 History of Autonomous System 400

17.8.2 What is an Autonomous Car? 401

17.8.3 Literature of Autonomous Car 402

17.8.4 How Does an Autonomous Car Work? 405

17.8.5 Concept of Self-driving Car 406

17.8.5.1 Image Classification 407

17.8.5.2 Object Tracking 407

17.8.5.3 Lane Detection 408

17.8.5.4 Introduction to Deep Learning 408

17.8.6 Evaluation 409

17.9 Conclusion 410

References 410

18 Different Techniques to Solve Monotone Inclusion Problems 413
Tanmoy Som, Pankaj Gautam, Avinash Dixit, and D. R. Sahu

18.1 Introduction 413

18.2 Preliminaries 414

18.3 Proximal Point Algorithm 415

18.4 Splitting Algorithms 415

18.4.1 Douglas-Rachford Splitting Algorithm 416

18.4.2 Forward-Backward Algorithm 416

18.5 Inertial Methods 418

18.5.1 Inertial Proximal Point Algorithm 419

18.5.2 Splitting Inertial Proximal Point Algorithm 421

18.5.3 Inertial Douglas-Rachford Splitting Algorithm 421

18.5.4 Pock and Lorenz's Variable Metric Forward-Backward Algorithm 422

18.5.5 Numerical Example 428

18.6 Numerical Experiments 429

References 430

Index 433
PROFESSOR SNEHASHISH CHAKRAVERTY, is working in the Department of Mathematics (Applied Mathematics Group), National Institute of Technology Rourkela, Odisha, India as a Senior (Higher Administrative Grade) Professor. Prior to this he was with CSIR-Central Building Research Institute, Roorkee, India. Prof. Chakraverty received his Ph. D. from University of Roorkee (now IIT Roorkee). There after 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 has authored/co-authored 20 books, published 356 research papers in journals and conferences. Prof. Chakraverty is the Chief Editor of "International Journal of Fuzzy Computation and Modelling" (IJFCM), Inderscience Publisher, Switzerland (http://www.inderscience.com/ijfcm) and Associate Editor of "Computational Methods in Structural Engineering, Frontiers in Built Environment". He has been 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). Prof. Chakraverty is recipient of prestigious awards viz. Indian National Science Academy (INSA) nomination under International Collaboration/Bilateral Exchange Program, Platinum Jubilee ISCA Lecture Award (2014), CSIR Young Scientist (1997), BOYSCAST (DST), UCOST Young Scientist (2007, 2008), Golden Jubilee Director's (CBRI) Award (2001), Roorkee University Gold Medals (1987, 1988) etc. His present research area includes Differential Equations (Ordinary, Partial and Fractional), Numerical Analysis and Computational Methods, Structural Dynamics (FGM, Nano) and Fluid Dynamics, Mathematical Modeling and Uncertainty Modeling, Soft Computing and Machine Intelligence (Artificial Neural Network, Fuzzy, Interval and Affine Computations).