Mathematics in Computational Science and Engineering

Ramakant Bhardwaj, PhD, is an associate professor of mathematics in Amity University, Kolkata, India with 15 years of teaching experience. He has published 135 research papers in reputed journals. He is also the co-author of six mathematics books, which are not only for mathematicians but written for practical applications for engineers and scientists. Satyendra Narayan, PhD, is a professor of applied computing at the Sheridan Institute of Technology and Advanced Learning in Oakville, Ontario, Canada. He has more than 35 years of teaching experience and has published several research papers in the field of computing in reputed journals. He is also the co-author of several books. Jyoti Mishra, PhD, is an associate professor in the Department of Mathematics, Gyan Ganga Institute of Technology, Jabalpur, India. She has more than ten years of teaching and research experience and has published close to 50 research papers in reputed journals. She is also the co-author of several books.

Preface xvii

1 Brownian Motion in EOQ 1
K. Suganthi and G. Jayalalitha

1.1 Introduction 2

1.2 Assumptions in EOQ 4

1.2.1 Model Formulation 4

1.2.1.1 Assumptions 4

1.2.1.2 Notations 4

1.2.1.3 Inventory Ordering Cost 4

1.2.1.4 Inventory Holding Cost 5

1.2.1.5 Inventory Total Cost in EOQ 5

1.2.2 Example 5

1.2.3 Inventory Control Commodities in Instantaneous Demand Method Under Development of the tock 7

1.2.3.1 Assumptions 8

1.2.3.2 Notations 8

1.2.3.3 Model Formulation 9

1.2.3.4 Numerical Examples 10

1.2.3.5 Sensitivity Analysis 11

1.2.4 Classic EOQ Method in Inventory 12

1.2.4.1 Assumptions 12

1.2.4.2 Notations 13

1.2.4.3 Mathematical Model 13

1.3 Methodology 15

1.3.1 Brownian Motion 16

1.4 Results 17

1.4.1 Numerical Examples 20

1.4.2 Sensitivity Analysis 20

1.4.3 Brownian Path in Hausdorff Dimension 21

1.4.4 The Hausdorff Measure 22

1.4.5 Levy Processes 22

1.5 Discussion 23

1.5.1 Future Research 23

1.6 Conclusions 24

References 24

2 Ill-Posed Resistivity Inverse Problems and its Application to Geoengineering Solutions 27
Satyendra Narayan

2.1 Introduction 28

2.2 Fundamentals of Ill-Posed Inverse Problems 29

2.3 Brief Historical Development of Resistivity Inversion 30

2.4 Overview of Inversion Schemes 31

2.5 Theoretical Basis for Multi-Dimensional Resistivity Inversion Technqiues 32

2.6 Mathematical Concept for Application to Geoengineering Problems 40

2.7 Mathematical Quantification of Resistivity Resolution and Detection 43

2.8 Scheme of Resistivity Data Presentation 45

2.9 Design Strategy for Monitoring Processes of IOR Projects, Geo-Engineering, and Geo-Environmental Problems 47

2.10 Final Remarks and Conclusions 49

References 51

3 Shadowed Set and Decision-Theoretic Three-Way Approximation of Fuzzy Sets 55
M. A. Ibrahim, T. O. William-West and D. Singh

3.1 Introduction 55

3.2 Preliminaries on Three-Way Approximation of Fuzzy Sets 57

3.2.1 Shadowed Set Approximation 57

3.2.2 Decision-Theoretic Three-Way Approximation 58

3.3 Theoretical Foundations of Shadowed Sets 60

3.3.1 Uncertainty Balance Models 61

3.3.1.1 Pedrycz’s (Pd) Model 61

3.3.1.2 Tahayori-Sadeghian-Pedrycz (TSP) Model 61

3.3.1.3 Ibrahim-William-West-Kana-Singh (IWKS) Model 62

3.3.2 Minimum Error or Deng-Yao (DY) Model 63

3.3.3 Average Uncertainty or Ibrahim-West (IW) Model 64

3.3.4 Nearest Quota of Uncertainty (WIK) Model 65

3.3.5 Algorithm for Constructing Shadowed Sets 65

3.3.6 Examples on Shadowed Set Approximation 66

3.4 Principles for Constructing Decision-Theoretic Approximation 73

3.4.1 Deng and Yao Special Decision-Theoretic (DYSD) Model 74

3.4.2 Zhang, Xia, Liu and Wang (ZXLW) Generalized Decision-Theoretic Model 77

3.4.3 A General Perspective to Decision-Theoretic Three-Way Approximation 78

3.4.3.1 Determination of n, m and p for Decision- Theoretic Three-Way Approximation 79

3.4.3.2 A General Decision-Theoretic Three-Way Approximation Partition Thresholds 81

3.4.4 Example on Decision-Theoretic Three-Way Approximation 83

3.5 Concluding Remarks and Future Directions 87

References 88

4 Intuitionistic Fuzzy Rough Sets: Theory to Practice 91
Shivani Singh and Tanmoy Som

4.1 Introduction 92

4.2 Preliminaries 93

4.2.1 Rough Set Theory 94

4.2.2 Intuitionistic Fuzzy Set Theory 95

4.2.3 Intuitionistic Fuzzy-Rough Set Theory 96

4.3 Intuitionistic Fuzzy Rough Sets 97

4.4 Extension and Hybridization of Intuitionistic Fuzzy Rough Sets 110

4.4.1 Extension 110

4.4.1.1 Dominance-Based Intuitionistic Fuzzy Rough Sets 111

4.4.1.2 Covering-Based Intuitionistic Fuzzy Rough Sets 111

4.4.1.3 Kernel Intuitionistic Fuzzy Rough Sets 112

4.4.1.4 Tolerance-Based Intuitionistic Fuzzy Rough Sets 112

4.4.1.5 Interval-Valued Intuitionistic Fuzzy Rough Sets 112

4.4.2 Hybridization 113

4.4.2.1 Variable Precision Intuitionistic Fuzzy Rough Sets 113

4.4.2.2 Intuitionistic Fuzzy Neighbourhood Rough Sets 114

4.4.2.3 Intuitionistic Fuzzy Multigranulation Rough Sets 114

4.4.2.4 Intuitionistic Fuzzy Decision-Theoretic Rough Sets 114

4.4.2.5 Intuitionistic Fuzzy Rough Sets and Soft Intuitionistic Fuzzy Rough Sets 115

4.4.2.6 Multi-Adjoint Intuitionistic Fuzzy Rough Sets 115

4.4.2.7 Intuitionistic Fuzzy Quantified Rough Sets 116

4.4.2.8 Genetic Algorithm and IF Rough Sets 116

4.5 Applications of Intuitionistic Fuzzy Rough Sets 116

4.5.1 Attribute Reduction 116

4.5.2 Decision Making 118

4.5.3 Other Applications 119

4.6 Work Distribution of IFRS Country-Wise and Year-Wise 123

4.6.1 Country-Wise Work Distribution 123

4.6.2 Year-Wise Work Distribution 124

4.6.3 Limitations of Intuitionistic Fuzzy Rough Set Theory 124

4.7 Conclusion 125

Acknowledgement 125

References 125

5 Satellite-Based Estimation of Ambient Particulate Matters (pm 2.5) Over a Metropolitan City in Eastern India 135
Tamanna Nasrin, Sharadia Dey and Sabyasachi Mondal

5.1 Introduction 136

5.2 Methodology 137

5.3 Result and Discussions 138

5.4 Conclusion 143

References 144

6 Computational Simulation Techniques in Inventory Management 147
Dr. Abhijit Pandit and Dr. Pulak Konar

6.1 Introduction 147

6.1.1 Inventory Management 147

6.1.2 Simulation 148

6.2 Conclusion 164

References 165

7 Workability of Cement Mortar Using Nano Materials and PVA 167
Dr. Mohan Kantharia and Dr. Pankaj Mishra

7.1 Introduction 167

7.2 Literature Survey 168

7.3 Materials and Methods 171

7.4 Results and Discussion 171

7.5 Conclusion 177

References 178

8 Distinctive Features of Semiconducting and Brittle Half-Heusler Alloys; LiXP (X=Zn, Cd) 181
Madhu Sarwan, Abdul Shukoor V. and Sadhna Singh

8.1 Introduction 182

8.2 Computation Method 183

8.3 Result and Discussion 183

8.3.1 Structural Properties 183

8.3.2 Elastic Properties 185

8.3.3 Electronic Properties 187

8.3.4 Thermodynamic Properties 190

8.4 Conclusions 195

Acknowledgement 196

References 196

9 Fixed Point Results with Fuzzy Sets 199
Qazi Aftab Kabir, Sanath Kumar H.G. and Ramakant Bhardwaj

9.1 Introduction 199

9.2 Definitions and Preliminaries 200

9.3 Main Results 201

References 208

10 Role of Mathematics in Novel Artificial Intelligence Realm 211
Kavita Rawat and Manas Kumar Mishra

10.1 Introduction 212

10.2 Mathematical Concepts Applied in Artificial Intelligence 212

10.2.1 Linear Algebra 213

10.2.1.1 Matrix and Vectors 213

10.2.1.2 Eigen Value and Eigen Vector 214

10.2.1.3 Matrix Operations 217

10.2.1.4 Artificial Intelligence Algorithms That Use Linear Algebra 217

10.2.2 Calculus 218

10.2.2.1 Objective Function 219

10.2.2.2 Loss Function & Cost Function 219

10.2.2.3 Artificial Intelligence Algorithms That Use Calculus 222

10.2.3 Probability and Statistics 222

10.2.3.1 Population Versus Sample 224

10.2.3.2 Descriptive Statistics 224

10.2.3.3 Distributions 225

10.2.3.4 Probability 225

10.2.3.5 Correlation 226

10.2.3.6 Data Visualization Using Statistics 226

10.2.3.7 Artificial Intelligence Algorithms That Use Probability and Statistics 227

10.3 Work Flow of Artificial Intelligence & Application Areas 227

10.3.1 Application Areas 229

10.3.2 Trending Areas 229

10.4 Conclusion 230

References 231

11 Study of Corona Epidemic: Predictive Mathematical Model 233
K. Sruthila Gopala Krishnan, Ramakant Bhardwaj, Amit Kumar Mishra and Rakesh Mohan Shrraf

11.1 Mathematical Modelling 234

11.2 Need of Mathematical Modelling 235

11.3 Methods of Construction of Mathematical Models 236

11.3.1 Mathematical Modelling with the Help of Geometry 236

11.3.2 Mathematical Modelling with the Help of Algebra 237

11.3.3 Mathematical Modelling Using Trigonometry 239

11.3.4 Mathematical Modelling with the Help of Ordinary Differential Equation (ODE) 239

11.3.5 Mathematical Modelling Using Partial Differential Equation (PDE) 240

11.3.6 Mathematical Modelling Using Difference Equation 240

11.4 Comparative Study of Mathematical Model in the Time of Covid-19 – A Review 241

11.4.1 Review 241

11.4.2 Case Study 246

11.5 Corona Epidemic in the Context of West Bengal: Predictive Mathematical Model 247

11.5.1 Overview 247

11.5.2 Case Study 248

11.5.3 Methodology 250

11.5.3.1 Exponential Model 250

11.5.3.2 Model Based on Geometric Progression (g.p.) 252

11.5.3.3 Model for Stay At Home 253

11.5.4 Discussion 255

References 255

12 Application of Mathematical Modeling in Various Fields in Light of Fuzzy Logic 257
Dr. Dhirendra Kumar Shukla

12.1 Introduction 257

12.1.1 Mathematical Modeling 257

12.1.2 Principles of Mathematical Models 259

12.2 Fuzzy Logic 261

12.2.1 Fuzzy Cognitive Maps & Induced Fuzzy Cognitive Maps 262

12.2.2 Fuzzy Cluster Means 263

12.3 Literature Review 264

12.4 Applications of Fuzzy Logic 268

12.4.1 Controller of Temperature 269

12.4.2 Usage of Fuzzy Logic in a Washing Machine 270

12.4.3 Air Conditioner 271

12.4.4 Aeronautics 272

12.4.5 Automotive Field 272

12.4.6 Business 274

12.4.7 Finance 275

12.4.8 Chemical Engineering 276

12.4.9 Defence 278

12.4.10 Electronics 279

12.4.11 Medical Science and Bioinformatics 280

12.4.12 Robotics 282

12.4.13 Signal Processing and Wireless Communication 283

12.4.14 Transportation Problems 283

12.5 Conclusion 285

References 285

13 A Mathematical Approach Using Set & Sequence Similarity Measure for Item Recommendation Using Sequential Web Data 287
Vishal Paranjape, Dr. Neelu Nihalani, Dr. Nishchol Mishra and Dr. Jyoti Mishra

13.1 Introduction 288

13.2 Measures of Assessment for Recommendation Engines 294

13.3 Related Work 295

13.4 Methodology/Research Design 296

13.4.1 Web Data Collection Through Web Logs 296

13.4.2 Web User Sessions Classification 300

13.5 Finding or Result 305

13.6 Conclusion and Future Work 306

References 307

14 Neural Network and Genetic Programming Based Explicit Formulations for Shear Capacity Estimation of Adhesive Anchors 311
Tawfik Kettanah and Satyendra Narayan

14.1 General Introduction 312

14.2 Research Significance 313

14.3 Biological Nervous System 314

14.4 Constructing Artificial Neural Network Model 317

14.5 Genetic Programming (GP) 320

14.6 Administering Genetic Programming Scheme 320

14.7 Genetic Programming In Details 320

14.8 Genetic Expression Programming 322

14.9 Developing Model With Genexpo Software 322

14.10 Comparing NN and GEP Results 325

14.11 Conclusions 326

References 327

15 Adaptive Heuristic - Genetic Algorithms 329
R. Anandan

15.1 Introduction 329

15.2 Genetic Algorithm 330

15.3 The Genetic Algorithm 331

15.4 Evaluation Module 331

15.5 Populace Module 331

15.5.1 Introduction 331

15.5.2 Initialisation Technique 331

15.5.3 Deletion Technique 332

15.5.4 Parent Selection Procedure 332

15.5.5 Fitness Technique 333

15.5.6 Populace Size 333

15.5.7 Elitism 334

15.6 Reproduction Module 334

15.6.1 Introduction 334

15.6.2 Operators 334

15.6.3 Mutation 338

15.6.4 Mutation Rate 338

15.6.5 Crossover Rate 338

15.6.6 Dynamic Mutation and Crossover Rates 338

15.7 Example 339

15.8 Schema Theorem 341

15.8.1 Introduction 341

15.9 Conclusion 342

15.10 Future Scope 342

References 342

16 Mathematically Enhanced Corrosion Detection 343
SeyedBijan Mahbaz, Giovanni Cascante, Satyendra Narayan, Maurice B. Dusseault and Philippe Vanheeghe

16.1 Introduction 344

16.1.1 Mathematics in NDT 346

16.1.2 Principal Component Analysis (PCA) 347

16.2 Case Study: PCA Applied to PMI Data for Defect Detection 347

16.3 PCA Feature Extraction for PMI Method 349

16.4 Experimental Setup and Test 351

16.5 Results 352

16.6 Conclusions 355

References 355

17 Dynamics of Malaria Parasite with Effective Control Analysis 359
Nagadevi Bala Nagaram and Suresh Rasappan

17.1 Introduction 359

17.2 The Mathematical Structure of EGPLC 361

17.3 The Modified EGPLC Model 363

17.4 Equilibria and Local Stability Analysis 364

17.5 Analysis of Global Stability 365

17.6 Global Stability Analysis with Back Propagation 367

17.7 Stability Analysis of Non-Deterministic EGPLC Model 373

17.8 Discussion on Numerical Simulation 378

17.9 Conclusion 381

17.10 Future Scope of the Work 381

References 381

18 Dynamics, Control, Stability, Diffusion and Synchronization of Modified Chaotic Colpitts Oscillator with Triangular Wave Non-Linearity Depending on the States 383
Suresh Rasappan and Niranjan Kumar K.A.

18.1 Introduction 384

18.2 The Mathematical Model of Chaotic Colpitts Oscillator 385

18.3 Adaptive Backstepping Control of the Modified Colpitts Oscillator with Unknown Parameters 395

18.3.1 Proposed System 395

18.3.2 Numerical Simulation 400

18.4 Synchronization of Modified Chaotic Colpitts Oscillator 400

18.4.1 Synchronization of Modified Chaotic Colpitts Oscillator using Non-Linear Feedback Method 402

18.4.2 Numerical Simulation 404

18.5 The Synchronization of Colpitts Oscillator via Backstepping Control 405

18.5.1 Analysis of the Error Dynamics 405

18.5.2 Numerical Simulation 408

18.6 Circuit Implementation 409

18.7 Conclusion 412

References 412

Index 415