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Mathematical Modeling with Multidisciplinary Applications

Book Description

Features mathematical modeling techniques and real-world processes with applications in diverse fields

Mathematical Modeling with Multidisciplinary Applications details the interdisciplinary nature of mathematical modeling and numerical algorithms. The book combines a variety of applications from diverse fields to illustrate how the methods can be used to model physical processes, design new products, find solutions to challenging problems, and increase competitiveness in international markets.

Written by leading scholars and international experts in the field, the book presents new and emerging topics in areas including finance and economics, theoretical and applied mathematics, engineering and machine learning, physics, chemistry, ecology, and social science. In addition, the book thoroughly summarizes widely used mathematical and numerical methods in mathematical modeling and features:

  • Diverse topics such as partial differential equations (PDEs), fractional calculus, inverse problems by ordinary differential equations (ODEs), semigroups, decision theory, risk analysis, Bayesian estimation, nonlinear PDEs in financial engineering, perturbation analysis, and dynamic system modeling

  • Case studies and real-world applications that are widely used for current mathematical modeling courses, such as the green house effect and Stokes flow estimation

  • Comprehensive coverage of a wide range of contemporary topics, such as game theory, statistical models, and analytical solutions to numerical methods

  • Examples, exercises with select solutions, and detailed references to the latest literature to solidify comprehensive learning

  • New techniques and applications with balanced coverage of PDEs, discrete models, statistics, fractional calculus, and more

Mathematical Modeling with Multidisciplinary Applications is an excellent book for courses on mathematical modeling and applied mathematics at the upper-undergraduate and graduate levels. The book also serves as a valuable reference for research scientists, mathematicians, and engineers who would like to develop further insights into essential mathematical tools.

Table of Contents

  1. Cover
  2. Half Title page
  3. Title page
  4. Copyright page
  5. List of Figures
  6. Preface
  7. Acknowledgments
  8. Editor and Contributors
  9. Part I: Introduction and Foundations
    1. Chapter 1: Differential Equations
      1. 1.1 Ordinary Differential Equations
      2. 1.2 Partial Differential Equations
      3. 1.3 Classic Mathematical Models
      4. 1.4 Other Mathematical Models
      5. 1.5 Solution Techniques
      6. Exercises
      7. References
    2. Chapter 2: Mathematical Modeling
      1. 2.1 Mathematical Modeling
      2. 2.2 Model Formulation
      3. 2.3 Parameter Estimation
      4. 2.4 Mathematical Models
      5. 2.5 Numerical Methods
      6. Exercises
      7. References
    3. Chapter 3: Numerical Methods: An Introduction
      1. 3.1 Direct Integration
      2. 3.2 Finite Difference Methods
      3. Exercises
      4. References
    4. Chapter 4: Teaching Mathematical Modeling in Teacher Education: Efforts and Results
      1. 4.1 Introduction
      2. 4.2 Theoretical Frameworks Connected to Mathematical Modeling
      3. 4.3 Mathematical Modeling Tasks
      4. 4.4 Conclusions
      5. Exercises
      6. References
  10. Part II: Mathematical Modeling with Multidisciplinary Applications
    1. Chapter 5: Industrial Mathematics with Applications
      1. 5.1 Industrial Mathematics
      2. 5.2 Numerical Simulation of Metallurgical Electrodes
      3. 5.3 Numerical Simulation of Pit Lake Water Quality
      4. Exercises
      5. References
    2. Chapter 6: Binary and Ordinal Data Analysis in Economics: Modeling and Estimation
      1. 6.1 Introduction
      2. 6.2 Theoretical Foundations
      3. 6.3 Estimation
      4. 6.4 Applications
      5. 6.5 Conclusions
      6. Exercises
      7. References
    3. Chapter 7: Inverse Problems in ODEs
      1. 7.1 Banach’s Fixed Point Theorem & The Collage Theorem
      2. 7.2 Existence-Uniqueness of Solutions to Initial Value Problems
      3. 7.3 Solving Inverse Problems for ODEs
      4. Exercises
      5. References
    4. Chapter 8: Estimation of Model Parameters
      1. 8.1 Estimation is an Inverse Problem
      2. 8.2 The Multivariate Normal Distribution
      3. 8.3 Model of Observations
      4. 8.4 Estimation
      5. 8.5 Conclusion
      6. Exercises
      7. References
    5. Chapter 9: Linear and Nonlinear Parabolic Partial Differential Equations in Financial Engineering
      1. 9.1 Financial Derivatives
      2. 9.2 Motivation for a Model for the Price of Stocks
      3. 9.3 Stock Prices Involving the Wiener Process
      4. 9.4 Connection Between the Wiener Process and PDEs
      5. 9.5 The Black-Scholes-Merton Equation
      6. 9.6 Solution of the Black-Scholes-Merton Equation
      7. 9.7 Free Boundary-Value Problems
      8. 9.8 The Hamilton-Jacobi-Bellman Equation
      9. 9.9 Numerical Methods
      10. 9.10 Conclusion
      11. Exercises
      12. References
    6. Chapter 10: Decision Modeling in Supply Chain Management
      1. 10.1 Introduction to Decision Modeling
      2. 10.2 Mathematical Programming Models
      3. 10.3 Introduction of Supply Chain Management
      4. 10.4 Applications in Supply Chain Management
      5. 10.5 Summary
      6. Exercises
      7. References
    7. Chapter 11: Modeling Temperature for Pricing Weather Derivatives
      1. 11.1 Introduction
      2. 11.2 Stochastic Temperature Modeling
      3. 11.3 Continuous-Time Autoregressive Processes
      4. 11.4 Pricing of Temperature Futures Contracts
      5. Exercises
      6. References
    8. Chapter 12: Decision Theory under Risk and Applications in Social Sciences: I. Individual Decision Making
      1. 12.1 Introduction
      2. 12.2 The Fundamental Framework
      3. 12.3 A Brief Introduction to Theory of Choice
      4. 12.4 Collective Choice
      5. 12.5 Preferences Under Uncertainty
      6. 12.6 Decisions Over Time
      7. 12.7 The Problem of Aggregation
      8. 12.8 Conclusion
      9. Exercises
      10. References
    9. Chapter 13: Fractals, with Applications to Signal and Image Modeling 307
      1. 13.1 Iterated Function Systems
      2. 13.2 Fractal Dimension
      3. 13.3 More on the Definition of Iterated Function System
      4. 13.4 The Chaos Game
      5. 13.5 An Application to Image Analysis
      6. References
    10. Chapter 14: Efficient Numerical Methods for Singularly Perturbed Differential Equations
      1. 14.1 Introduction
      2. 14.2 Characterization of SPPs
      3. 14.3 Numerical Approximate Solution
      4. 14.4 SPPs Arising in Chemical Reactor Theory
      5. 14.5 Layer-Adapted Nonuniform Meshes
      6. References
  11. Part III: Advanced Modeling Topics
    1. Chapter 15: Fractional Calculus and its Applications
      1. 15.1 Introduction
      2. 15.2 Fractional Calculus Fundamentals
      3. 15.3 Fractional-Order Systems and Controllers
      4. 15.4 Stability of Fractional-Order Systems
      5. 15.5 Applications of Fractional Calculus
      6. Exercises
      7. References
    2. Chapter 16: The Goal Programming Model: Theory and Applications
      1. 16.1 Multi-Criteria Decision Aid
      2. 16.2 The Goal Programming Model
      3. 16.3 Scenario-based Goal Programming
      4. 16.4 Applications
      5. Exercises
      6. References
    3. Chapter 17: Decision Theory under Risk and Applications in Social Sciences: II. Game Theory
      1. 17.1 Introduction
      2. 17.2 Best Replies and Nash Equilibria
      3. 17.3 Mixed Strategies and Minimax
      4. 17.4 Nash Equilibria and Conservative Strategies
      5. 17.5 Zero-Sum Games and the Minimax Theorem
      6. 17.6 Nash Equilibria for Mixed Strategies
      7. 17.7 Cooperative Games
      8. 17.8 Conclusion
      9. Exercises
      10. References
    4. Chapter 18: Control Problems on Differential Equations
      1. 18.1 Introduction
      2. 18.2 Ordinary Differential Equations
      3. 18.3 Partial Differential Equations
      4. Exercises
      5. References
    5. Chapter 19: Markov-Jump Stochastic Models for Tropical Convection
      1. 19.1 Introduction
      2. 19.2 Random Numbers: Theory and Simulations
      3. 19.3 Markov Chains and Birth-Death Processes
      4. 19.4 A Birth-Death Process for Convective Inhibition
      5. 19.5 A Birth-Death Process for Cloud-Cloud Interactions
      6. 19.6 Further Reading
      7. Exercises
      8. References
  12. Problem Solutions
  13. Index