Mathematics in Computational Science and Engineering

MATHEMATICS IN COMPUTATIONAL SCIENCE AND ENGINEERING This groundbreaking new volume, written by industry experts, is a must-have for engineers, scientists, and students across all engineering disciplines working in mathematics and computational science who want to stay abreast with the most current and provocative new trends in the industry. Applied science and engineering is the application of fundamental concepts and knowledge to design, build and maintain a product or a process, which provides a solution to a problem and fulfills a need. This book contains advanced topics in computational techniques across all the major engineering disciplines for undergraduate, postgraduate, doctoral and postdoctoral students. This will also be found useful for professionals in an industrial setting. It covers the most recent trends and issues in computational techniques and methodologies for applied sciences and engineering, production planning, and manufacturing systems. More importantly, it explores the application of computational techniques and simulations through mathematics in the field of engineering and the sciences. Whether for the veteran engineer, scientist, student, or other industry professional, this volume is a must-have for any library. Useful across all engineering disciplines, it is a multifactional tool that can be put to use immediately in practical applications. This groundbreaking new volume: Includes detailed theory with illustrations Uses an algorithmic approach for a unique learning experience Presents a brief summary consisting of concepts and formulae Is pedagogically designed to make learning highly effective and productive Is comprised of peer-reviewed articles written by leading scholars, researchers and professors AUDIENCE: Engineers, scientists, students, researchers, and other professionals working in the field of computational science and mathematics across multiple disciplines

Table of Contents:
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