Mathematical Modelling, Optimization, Analytic and Numerical Solutions (Industrial and Applied Mathematics)

This book discusses a variety of topics related to industrial and applied mathematics, focusing on wavelet theory, sampling theorems, inverse problems and their applications, partial differential equations as a model of real-world problems, computational linguistics, mathematical models and methods for meteorology, earth systems, environmental and medical science, and the oil industry. It features papers presented at the International Conference in Conjunction with 14th Biennial Conference of ISIAM, held at Guru Nanak Dev University, Amritsar, India, on 2–4 February 2018. The conference has emerged as an influential forum, bringing together prominent academic scientists, experts from industry, and researchers. The topics discussed include Schrodinger operators, quantum kinetic equations and their application, extensions of fractional integral transforms, electrical impedance tomography, diffuse optical tomography, Galerkin method by using wavelets, a Cauchy problem associated with Korteweg–de Vries equation, and entropy solution for scalar conservation laws. This book motivates and inspires young researchers in the fields of industrial and applied mathematics. Preface Acknowledgements Contents About the Editors 1 Certain Areas of Industrial and Applied Mathematics 1.1 Introduction 1.2 Areas Pursued By Our Research Group 1.3 Ruchira Aneja—Variants of Wavelet in Medical Imaging 1.4 Wavelet Cross-Correlation 1.5 Fractal Dimension and Predictability References 2 Schrödinger Operators with a Switching Effect 2.1 Introduction 2.2 Smilansky–Solomyak Model 2.3 Variations on the Model 2.3.1 Regular Version of the Model 2.3.2 Magnetic Version of the Model 2.3.3 The δ' Version of the Model 2.4 Another Model 2.5 Resonances in Smilansky–Solomyak Model 2.6 Concluding Remarks References 3 Distribution Theory by Riemann Integrals 3.1 Overall Motivation 3.1.1 Psychological Aspects 3.1.2 The Search for a Banach Space of Test Functions 3.2 Notations and Preliminaries 3.3 Continuous Functions That Vanish at Infinity 3.4 The Wiener Algebra on mathbbRd 3.5 The Fourier Transform 3.6 Tensor Factorization 3.7 The Feichtinger Algebra 3.8 The Shortcut to Distribution Theory 3.9 The Kernel Theorem 3.10 Shannon's Sampling Theorem 3.11 Systems and Convolution Operators 3.12 Further References 3.13 The Relation to the Schwartz Theory References 4 Partial Differential Equations on Metric Graphs: A Survey of Results on Optimization, Control, and Stabilizability Problems with Special Focus on Shape and Topological Sensitivity Problems 4.1 Introduction 4.2 Planar Graphs 4.2.1 Steklov–Poincaré Map for Metric Graphs 4.2.2 Problems on General Metric Graphs 4.2.3 Shape Variations for Graph Problems 4.2.4 Shape Sensitivity for the First Eigenvalue 4.2.5 Transmission Problem 4.2.6 Topological Derivative with Respect to Material Properties 4.3 Stars with a Hole 4.3.1 Homogeneous Networks 4.4 The Topological Derivative 4.4.1 Homogeneous Graphs 4.4.2 Sensitivity with Respect to Edge Inclusion References 5 Topological Analysis of a Weighted Human Behaviour Model Coupled on a Street and Place Network in the Context of Urban Terrorist Attacks 5.1 Introduction 5.2 The PCR System: An A-Spatial Model for Analysing the Dynamics of Human Behaviour During a Disaster 5.2.1 Neuroscientific Background of the PCR System 5.2.2 Equations of the PCR System 5.2.3 Transitional Dynamics 5.3 Mathematical Weighted and Coupled PCR System 5.3.1 The Weighted PCR System 5.3.2 Flows and Bottleneck Coupling 5.3.3 Fixed Points of the Two-Node System 5.4 Influence of the Spatial Configuration on the Pace of Evacuation 5.4.1 Equation of the Three-Node Network 5.4.2 Scaling the Parameters 5.4.3 Speed of Convergence Towards the Fixed Points 5.4.4 Influence of the Intermediate Place Capacity on the Evacuation Dynamics 5.5 Conclusion References 6 A New Model for Transient Flow in Gas Transportation Networks 6.1 Introduction 6.2 The System 6.2.1 The Graph of the Network 6.2.2 The Partial Differential Equations 6.2.3 Node Conditions for the Pressure 6.2.4 Initial Conditions for the Flow Rate and the Pressure 6.3 Well-Posedness of the System 6.4 A Recursion for the System State 6.5 Conclusion References 7 Mixed-Integer Optimal Control for PDEs: Relaxation via Differential Inclusions and Applications to Gas Network Optimization 7.1 Introduction 7.2 Setting 7.3 Relaxation via Differential Inclusions 7.4 Application to Gas Network Optimization 7.4.1 Networks with Pipes, Valves, and Compressors 7.4.2 Abstract Problem Formulation 7.4.3 Numerical Example 7.5 Conclusion References 8 Application of Solution of the Quantum Kinetic Equations for Information Technology and Renewable Energy Problem 8.1 Introduction 8.2 Application of the Quantum Kinetic Equations for the Solution of Problems of Information Technology 8.3 Application of the Quantum Kinetic Equations for the Solution of Problems of Renewable Energy References 9 Inverse Problems Involving PDEs with Applications to Imaging 9.1 Introduction 9.2 Electrical Impedance Tomography 9.2.1 Analytical Setting for the EIT Model 9.2.2 Inverse Problem in EIT 9.3 Diffuse Optical Tomography 9.3.1 Radiative Transport Equation 9.3.2 DOT Model 9.4 Computational Aspect and the Regularization of the Ill-Posed Problem 9.4.1 Iteratively Regularized Gauss–Newton Method 9.4.2 The Statistical Inverse Problem 9.4.3 Regularization Functions 9.4.4 The Markov Chain Monte Carlo Method 9.5 Conclusion References 10 Critical Growth Elliptic Problems with Choquard Type Nonlinearity: A Survey 10.1 A Brief Survey 10.1.1 Existence and Multiplicity Results 10.1.2 Radial Symmetry and Regularity of Solutions 10.1.3 Choquard Equations Involving the p(.)-Laplacian 10.2 Choquard Equations Involving the Fractional Laplacian 10.2.1 Brezis–Nirenberg Type Existence Results 10.2.2 Magnetic Choquard Equations 10.2.3 Singular Problems Involving Choquard Nonlinearity 10.3 System of Equations with Choquard Type Nonlinearity 10.3.1 Doubly Nonlocal p-Fractional Coupled Elliptic System 10.3.2 Doubly Nonlocal System with Critical Nonlinearity 10.4 Some Open Questions References 11 Wavelet Galerkin Methods for Higher Order Partial Differential Equations 11.1 Introduction 11.2 Basic Background 11.2.1 Periodized Wavelet 11.3 Wavelet Galerkin Methods for Higher Order PDEs 11.4 Numerical Results 11.5 Conclusion References 12 Resilience and Dynamics of Coral Reefs Impacted by Chemically Rich Seaweeds and Unsustainable Fishing 12.1 Introduction 12.2 The Basic Model 12.3 Equilibria and Their Stability 12.4 Seasonally Forced System 12.5 Discussion References 13 Multigrid Methods for the Simulations of Surfactant Spreading on a Thin Liquid Film 13.1 Introduction 13.2 Model Description 13.3 Discretization 13.4 Numerical Approach—Nonlinear Multigrid Method (FAS) 13.5 Result and Discussions 13.5.1 Validation 13.5.2 Mesh and Time Convergence Study 13.5.3 Comparison with Matlab Solver 13.5.4 Mesh Refinement Analysis 13.6 Validation Case with Sharp Changes in Gradient in the Free Surface Profile and Surfactant Concentration 13.7 Concluding Remarks References 14 Hopf Bifurcation in a Mathematical Model of Tuberculosis with Delay 14.1 Introduction 14.2 Mathematical Model Formulation 14.2.1 Existence of Steady States 14.3 Stability of Equilibria 14.3.1 Characteristic Equation 14.3.2 Length of Delay and Hopf Bifurcation 14.4 Numerical Simulations 14.5 Discussion and Conclusion References 15 Treatment of Psoriasis by Interleukin-10 Through Impulsive Control Strategy: A Mathematical Study 15.1 Introduction 15.2 The Model 15.2.1 Model Formulation with Suitable Assumptions 15.2.2 Model Properties 15.3 Equilibrium Analysis 15.3.1 Existence Condition 15.3.2 Stability Criteria 15.4 Impulse Therapeutic Approaches 15.4.1 System Under Fixed IL-10 Injecting Process 15.5 Numerical Simulations 15.6 Discussion 15.7 Conclusion References 16 On Fractional Partial Differential Equations of Diffusion Type with Integral Kernel 16.1 Introduction 16.2 Basic Concepts 16.3 Fractional Partial Differential Equations with Kernel 16.3.1 Dirichlet Boundary Condition 16.3.2 Neumann Boundary Condition References 17 Mathematical Study on Human Cells Interaction Dynamics for HIV-TB Co-infection 17.1 Introduction 17.2 The Model 17.2.1 The Deterministic Model 17.2.2 Boundedness 17.3 Equilibrium Analysis 17.3.1 Stability of the Endemic Equilibrium 17.3.2 Reproduction Number 17.3.3 Stability of Disease-Free Equilibrium 17.3.4 Sensitivity Analysis 17.4 Numerical Simulation 17.5 Discussion 17.6 Appendix References 18 Relative Controllability of Nonlinear Fractional Damped Delay Systems with Multiple Delays in Control 18.1 Introduction 18.2 Preliminaries 18.3 Linear System with Multiple Delays in Control 18.4 Nonlinear Systems with Multiple Delays in Control 18.5 Example 18.6 Conclusion References 19 A Graphical User Interface-Based Fingerprint Recognition 19.1 Introduction 19.2 System Development 19.3 Proposed System Functions 19.3.1 Panel-1: Basic Operation Panel 19.3.2 Panel-2: Histogram 19.3.3 Panel-3: Image Enhancements 19.3.4 Panel-4: Feature Extraction 19.3.5 Panel-5: Noises and Filters 19.3.6 Panel-6: Similarity Measure 19.3.7 Panel-7: Performance Parameters 19.3.8 Panel-8: Results 19.4 Results and Discussion 19.5 Conclusion References 20 Existence and Stability Results for Stochastic Fractional Delay Differential Equations with Gaussian Noise 20.1 Introduction 20.2 Preliminaries 20.3 Existence and Uniqueness 20.4 Stability Analysis 20.5 Examples References 21 Asymptotic Stability of Implicit Fractional Volterra Integrodifferential Equations 21.1 Introduction 21.2 Preliminaries 21.3 Main Results 21.4 Examples 21.5 Conclusion References