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.

Table of Contents:
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 𝛿-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 𝛿-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 Arıtürk, Sırma 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