Mathematical Methods in Interdisciplinary Sciences

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).

• Notes on Contributors xvPreface xxvAcknowledgments xxvii1 Connectionist Learning Models for Application Problems Involving Differential and Integral Equations 1Susmita Mall, Sumit Kumar Jeswal, and Snehashish Chakraverty1.1 Introduction 11.1.1 Artificial Neural Network 11.1.2 Types of Neural Networks 11.1.3 Learning in Neural Network 21.1.4 Activation Function 21.1.4.1 Sigmoidal Function 31.1.5 Advantages of Neural Network 31.1.6 Functional Link Artificial Neural Network (FLANN) 31.1.7 Differential Equations (DEs) 41.1.8 Integral Equation 51.1.8.1 Fredholm Integral Equation of First Kind 51.1.8.2 Fredholm Integral Equation of Second Kind 51.1.8.3 Volterra Integral Equation of First Kind 51.1.8.4 Volterra Integral Equation of Second Kind 51.1.8.5 Linear Fredholm Integral Equation System of Second Kind 61.2 Methodology for Differential Equations 61.2.1 FLANN-Based General Formulation of Differential Equations 61.2.1.1 Second-Order Initial Value Problem 61.2.1.2 Second-Order Boundary Value Problem 71.2.2 Proposed Laguerre Neural Network (LgNN) for Differential Equations 71.2.2.1 Architecture of Single-Layer LgNN Model 71.2.2.2 Training Algorithm of Laguerre Neural Network (LgNN) 81.2.2.3 Gradient Computation of LgNN 91.3 Methodology for Solving a System of Fredholm Integral Equations of Second Kind 91.3.1 Algorithm 101.4 Numerical Examples and Discussion 111.4.1 Differential Equations and Applications 111.4.2 Integral Equations 161.5 Conclusion 20References 202 Deep Learning in Population Genetics: Prediction and Explanation of Selection of a Population 23Romila Ghosh and Satyakama Paul2.1 Introduction 232.2 Literature Review 232.3 Dataset Description 252.3.1 Selection and Its Importance 252.4 Objective 262.5 Relevant Theory, Results, and Discussions 272.5.1 automl 272.5.2 Hypertuning the Best Model 282.6 Conclusion 30References 303 A Survey of Classification Techniques in Speech Emotion Recognition 33Tanmoy Roy, Tshilidzi Marwala, and Snehashish Chakraverty3.1 Introduction 333.2 Emotional Speech Databases 333.3 SER Features 343.4 Classification Techniques 353.4.1 Hidden Markov Model 363.4.1.1 Difficulties in Using HMM for SER 373.4.2 Gaussian Mixture Model 373.4.2.1 Difficulties in Using GMM for SER 383.4.3 Support Vector Machine 383.4.3.1 Difficulties with SVM 393.4.4 Deep Learning 393.4.4.1 Drawbacks of Using Deep Learning for SER 413.5 Difficulties in SER Studies 413.6 Conclusion 41References 424 Mathematical Methods in Deep Learning 49Srinivasa Manikant Upadhyayula and Kannan Venkataramanan4.1 Deep Learning Using Neural Networks 494.2 Introduction to Neural Networks 494.2.1 Artificial Neural Network (ANN) 504.2.1.1 Activation Function 524.2.1.2 Logistic Sigmoid Activation Function 524.2.1.3 tanh or Hyperbolic Tangent Activation Function 534.2.1.4 ReLU (Rectified Linear Unit) Activation Function 544.3 Other Activation Functions (Variant Forms of ReLU) 554.3.1 Smooth ReLU 554.3.2 Noisy ReLU 554.3.3 Leaky ReLU 554.3.4 Parametric ReLU 564.3.5 Training and Optimizing a Neural Network Model 564.4 Backpropagation Algorithm 564.5 Performance and Accuracy 594.6 Results and Observation 59References 615 Multimodal Data Representation and Processing Based on Algebraic System of Aggregates 63Yevgeniya Sulema and Etienne Kerre5.1 Introduction 635.2 Basic Statements of ASA 645.3 Operations on Aggregates and Multi-images 655.4 Relations and Digital Intervals 725.5 Data Synchronization 755.6 Fuzzy Synchronization 925.7 Conclusion 96References 966 Nonprobabilistic Analysis of Thermal and Chemical Diffusion Problems with Uncertain Bounded Parameters 99Sukanta Nayak, Tharasi Dilleswar Rao, and Snehashish Chakraverty6.1 Introduction 996.2 Preliminaries 996.2.1 Interval Arithmetic 996.2.2 Fuzzy Number and Fuzzy Arithmetic 1006.2.3 Parametric Representation of Fuzzy Number 1016.2.4 Finite Difference Schemes for PDEs 1026.3 Finite Element Formulation for Tapered Fin 1026.4 Radon Diffusion and Its Mechanism 1056.5 Radon Diffusion Mechanism with TFN Parameters 1076.5.1 EFDM to Radon Diffusion Mechanism with TFN Parameters 1086.6 Conclusion 112References 1127 Arbitrary Order Differential Equations with Fuzzy Parameters 115Tofigh Allahviranloo and Soheil Salahshour7.1 Introduction 1157.2 Preliminaries 1157.3 Arbitrary Order Integral and Derivative for Fuzzy-Valued Functions 1167.4 Generalized Fuzzy Laplace Transform with Respect to Another Function 118References 1228 Fluid Dynamics Problems in Uncertain Environment 125Perumandla Karunakar, Uddhaba Biswal, and Snehashish Chakraverty8.1 Introduction 1258.2 Preliminaries 1268.2.1 Fuzzy Set 1268.2.2 Fuzzy Number 1268.2.3 𝛿-Cut 1278.2.4 Parametric Approach 1278.3 Problem Formulation 1278.4 Methodology 1298.4.1 Homotopy Perturbation Method 1298.4.2 Homotopy Perturbation Transform Method 1308.5 Application of HPM and HPTM 1318.5.1 Application of HPM to Jeffery–Hamel Problem 1318.5.2 Application of HPTM to Coupled Whitham–Broer–Kaup Equations 1348.6 Results and Discussion 1368.7 Conclusion 142References 1429 Fuzzy Rough Set Theory-Based Feature Selection: A Review 145Tanmoy Som, Shivam Shreevastava, Anoop Kumar Tiwari, and Shivani Singh9.1 Introduction 1459.2 Preliminaries 1469.2.1 Rough Set Theory 1469.2.1.1 Rough Set 1469.2.1.2 Rough Set-Based Feature Selection 1479.2.2 Fuzzy Set Theory 1479.2.2.1 Fuzzy Tolerance Relation 1489.2.2.2 Fuzzy Rough Set Theory 1499.2.2.3 Degree of Dependency-Based Fuzzy Rough Attribute Reduction 1499.2.2.4 Discernibility Matrix-Based Fuzzy Rough Attribute Reduction 1499.3 Fuzzy Rough Set-Based Attribute Reduction 1499.3.1 Degree of Dependency-Based Approaches 1509.3.2 Discernibility Matrix-Based Approaches 1549.4 Approaches for Semisupervised and Unsupervised Decision Systems 1549.5 Decision Systems with Missing Values 1589.6 Applications in Classification, Rule Extraction, and Other Application Areas 1589.7 Limitations of Fuzzy Rough Set Theory 1599.8 Conclusion 160References 16010 Universal Intervals: Towards a Dependency-Aware Interval Algebra 167Hend Dawood and Yasser Dawood10.1 Introduction 16710.2 The Need for Interval Computations 16910.3 On Some Algebraic and Logical Fundamentals 17010.4 Classical Intervals and the Dependency Problem 17410.5 Interval Dependency: A Logical Treatment 17610.5.1 Quantification Dependence and Skolemization 17710.5.2 A Formalization of the Notion of Interval Dependency 17910.6 Interval Enclosures Under Functional Dependence 18410.7 Parametric Intervals: How Far They Can Go 18610.7.1 Parametric Interval Operations: From Endpoints to Convex Subsets 18610.7.2 On the Structure of Parametric Intervals: Are They Properly Founded? 18810.8 Universal Intervals: An Interval Algebra with a Dependency Predicate 19210.8.1 Universal Intervals, Rational Functions, and Predicates 19310.8.2 The Arithmetic of Universal Intervals 19610.9 The S-Field Algebra of Universal Intervals 20110.10 Guaranteed Bounds or Best Approximation or Both? 209Supplementary Materials 210Acknowledgments 211References 21111 Affine-Contractor Approach to Handle Nonlinear Dynamical Problems in Uncertain Environment 215Nisha Rani Mahato, Saudamini Rout, and Snehashish Chakraverty11.1 Introduction 21511.2 Classical Interval Arithmetic 21711.2.1 Intervals 21711.2.2 Set Operations of Interval System 21711.2.3 Standard Interval Computations 21811.2.4 Algebraic Properties of Interval 21911.3 Interval Dependency Problem 21911.4 Affine Arithmetic 22011.4.1 Conversion Between Interval and Affine Arithmetic 22011.4.2 Affine Operations 22111.5 Contractor 22311.5.1 SIVIA 22311.6 Proposed Methodology 22511.7 Numerical Examples 23011.7.1 Nonlinear Oscillators 23011.7.1.1 Unforced Nonlinear Differential Equation 23011.7.1.2 Forced Nonlinear Differential Equation 23211.7.2 Other Dynamic Problem 23311.7.2.1 Nonhomogeneous Lane–Emden Equation 23311.8 Conclusion 236References 23612 Dynamic Behavior of Nanobeam Using Strain Gradient Model 239Subrat Kumar Jena, Rajarama Mohan Jena, and Snehashish Chakraverty12.1 Introduction 23912.2 Mathematical Formulation of the Proposed Model 24012.3 Review of the Differential Transform Method (DTM) 24112.4 Application of DTM on Dynamic Behavior Analysis 24212.5 Numerical Results and Discussion 24412.5.1 Validation and Convergence 24412.5.2 Effect of the Small-Scale Parameter 24512.5.3 Effect of Length-Scale Parameter 24712.6 Conclusion 248Acknowledgment 249References 25013 Structural Static and Vibration Problems 253M. Amin Changizi and Ion Stiharu13.1 Introduction 25313.2 One-parameter Groups 25413.3 Infinitesimal Transformation 25413.4 Canonical Coordinates 25413.5 Algorithm for Lie Symmetry Point 25513.6 Reduction of the Order of the ODE 25513.7 Solution of First-Order ODE with Lie Symmetry 25513.8 Identification 25613.9 Vibration of a Microcantilever Beam Subjected to Uniform Electrostatic Field 25813.10 Contact Form for the Equation 25913.11 Reducing in the Order of the Nonlinear ODE Representing the Vibration of a Microcantilever Beam Under Electrostatic Field 26013.12 Nonlinear Pull-in Voltage 26113.13 Nonlinear Analysis of Pull-in Voltage of Twin Microcantilever Beams 26613.14 Nonlinear Analysis of Pull-in Voltage of Twin Microcantilever Beams of Different Thicknesses 268References 27214 Generalized Differential and Integral Quadrature: Theory and Applications 273Francesco Tornabene and Rossana Dimitri14.1 Introduction 27314.2 Differential Quadrature 27414.2.1 Genesis of the Differential Quadrature Method 27414.2.2 Differential Quadrature Law 27514.3 General View on Differential Quadrature 27714.3.1 Basis Functions 27814.3.1.1 Lagrange Polynomials 28114.3.1.2 Trigonometric Lagrange Polynomials 28214.3.1.3 Classic Orthogonal Polynomials 28214.3.1.4 Monomial Functions 29114.3.1.5 Exponential Functions 29114.3.1.6 Bernstein Polynomials 29114.3.1.7 Fourier Functions 29214.3.1.8 Bessel Polynomials 29214.3.1.9 Boubaker Polynomials 29214.3.2 Grid Distributions 29314.3.2.1 Coordinate Transformation 29314.3.2.2 𝛿-Point Distribution 29314.3.2.3 Stretching Formulation 29314.3.2.4 Several Types of Discretization 29314.3.3 Numerical Applications: Differential Quadrature 29714.4 Generalized Integral Quadrature 31014.4.1 Generalized Taylor-Based Integral Quadrature 31214.4.2 Classic Integral Quadrature Methods 31414.4.2.1 Trapezoidal Rule with Uniform Discretization 31414.4.2.2 Simpson’s Method (One-third Rule) with Uniform Discretization 31414.4.2.3 Chebyshev–Gauss Method (Chebyshev of the First Kind) 31414.4.2.4 Chebyshev–Gauss Method (Chebyshev of the Second Kind) 31414.4.2.5 Chebyshev–Gauss Method (Chebyshev of the Third Kind) 31514.4.2.6 Chebyshev–Gauss Method (Chebyshev of the Fourth Kind) 31514.4.2.7 Chebyshev–Gauss–Radau Method (Chebyshev of the First Kind) 31514.4.2.8 Chebyshev–Gauss–Lobatto Method (Chebyshev of the First Kind) 31514.4.2.9 Gauss–Legendre or Legendre–Gauss Method 31514.4.2.10 Gauss–Legendre–Radau or Legendre–Gauss–Radau Method 31514.4.2.11 Gauss–Legendre–Lobatto or Legendre–Gauss–Lobatto Method 31614.4.3 Numerical Applications: Integral Quadrature 31614.4.4 Numerical Applications: Taylor-Based Integral Quadrature 32014.5 General View: The Two-Dimensional Case 324References 34015 Brain Activity Reconstruction by Finding a Source Parameter in an Inverse Problem 343Amir H. Hadian-Rasanan and Jamal Amani Rad15.1 Introduction 34315.1.1 Statement of the Problem 34415.1.2 Brief Review of Other Methods Existing in the Literature 34515.2 Methodology 34615.2.1 Weighted Residual Methods and Collocation Algorithm 34615.2.2 Function Approximation Using Chebyshev Polynomials 34915.3 Implementation 35315.4 Numerical Results and Discussion 35415.4.1 Test Problem 1 35515.4.2 Test Problem 2 35715.4.3 Test Problem 3 35815.4.4 Test Problem 4 35915.4.5 Test Problem 5 36215.5 Conclusion 365References 36516 Optimal Resource Allocation in Controlling Infectious Diseases 369A.C. Mahasinghe, S.S.N. Perera, and K.K.W.H. Erandi16.1 Introduction 36916.2 Mobility-Based Resource Distribution 37016.2.1 Distribution of National Resources 37016.2.2 Transmission Dynamics 37116.2.2.1 Compartment Models 37116.2.2.2 SI Model 37116.2.2.3 Exact Solution 37116.2.2.4 Transmission Rate and Potential 37216.2.3 Nonlinear Problem Formulation 37316.2.3.1 Piecewise Linear Reformulation 37416.2.3.2 Computational Experience 37416.3 Connection–Strength Minimization 37616.3.1 Network Model 37616.3.1.1 Disease Transmission Potential 37616.3.1.2 An Example 37616.3.2 Nonlinear Problem Formulation 37716.3.2.1 Connection Strength Measure 37716.3.2.2 Piecewise Linear Approximation 37816.3.2.3 Computational Experience 37916.4 Risk Minimization 37916.4.1 Novel Strategies for Individuals 37916.4.1.1 Epidemiological Isolation 38016.4.1.2 Identifying Objectives 38016.4.2 Minimizing the High-Risk Population 38116.4.2.1 An Example 38116.4.2.2 Model Formulation 38216.4.2.3 Linear Integer Program 38316.4.2.4 Computational Experience 38316.4.3 Minimizing the Total Risk 38416.4.4 Goal Programming Approach 38416.5 Conclusion 386References 38717 Artificial Intelligence and Autonomous Car 391Merve Arıtürk, Sırma Yavuz, and Tofigh Allahviranloo17.1 Introduction 39117.2 What is Artificial Intelligence? 39117.3 Natural Language Processing 39117.4 Robotics 39317.4.1 Classification by Axes 39317.4.1.1 Axis Concept in Robot Manipulators 39317.4.2 Classification of Robots by Coordinate Systems 39417.4.3 Other Robotic Classifications 39417.5 Image Processing 39517.5.1 Artificial Intelligence in Image Processing 39517.5.2 Image Processing Techniques 39517.5.2.1 Image Preprocessing and Enhancement 39617.5.2.2 Image Segmentation 39617.5.2.3 Feature Extraction 39617.5.2.4 Image Classification 39617.5.3 Artificial Intelligence Support in Digital Image Processing 39717.5.3.1 Creating a Cancer Treatment Plan 39717.5.3.2 Skin Cancer Diagnosis 39717.6 Problem Solving 39717.6.1 Problem-solving Process 39717.7 Optimization 39917.7.1 Optimization Techniques in Artificial Intelligence 39917.8 Autonomous Systems 40017.8.1 History of Autonomous System 40017.8.2 What is an Autonomous Car? 40117.8.3 Literature of Autonomous Car 40217.8.4 How Does an Autonomous Car Work? 40517.8.5 Concept of Self-driving Car 40617.8.5.1 Image Classification 40717.8.5.2 Object Tracking 40717.8.5.3 Lane Detection 40817.8.5.4 Introduction to Deep Learning 40817.8.6 Evaluation 40917.9 Conclusion 410References 41018 Different Techniques to Solve Monotone Inclusion Problems 413Tanmoy Som, Pankaj Gautam, Avinash Dixit, and D. R. Sahu18.1 Introduction 41318.2 Preliminaries 41418.3 Proximal Point Algorithm 41518.4 Splitting Algorithms 41518.4.1 Douglas–Rachford Splitting Algorithm 41618.4.2 Forward–Backward Algorithm 41618.5 Inertial Methods 41818.5.1 Inertial Proximal Point Algorithm 41918.5.2 Splitting Inertial Proximal Point Algorithm 42118.5.3 Inertial Douglas–Rachford Splitting Algorithm 42118.5.4 Pock and Lorenz’s Variable Metric Forward–Backward Algorithm 42218.5.5 Numerical Example 42818.6 Numerical Experiments 429References 430Index 433

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