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Mathematics of Uncertainty Modeling in the Analysis of Engineering and Science Problems

Book Description

For various scientific and engineering problems, how to deal with variables and parameters of uncertain value is an important issue. Full analysis of the specific errors in measurement, observations, experiments, and applications are vital in dealing with the parameters taken to simplify the problem. Mathematics of Uncertainty Modeling in the Analysis of Engineering and Science Problems aims to provide the reader with basic concepts for soft computing and other methods for various means of uncertainty in handling solutions, analysis, and applications. This book is an essential reference work for students, scholars, practitioners and researchers in the assorted fields of engineering and applied mathematics interested in a model for uncertain physical problems.

Table of Contents

  1. Cover
  2. Title Page
  3. Copyright Page
  4. Book Series
  5. Editorial Advisory Board and List of Reviewers
    1. Editorial Advisory Board
    2. List of Reviewers
  6. Preface
  7. Acknowledgment
  8. Chapter 1: Interval Mathematics as a Potential Weapon against Uncertainty
    1. ABSTRACT
    2. 1. INTRODUCTION: CAN WE KNOW ABOUT THE WORLD?
    3. 2. INTERVALS, ERRORS, AND UNCERTAINTIES
    4. 3. INTERVAL ARITHMETIC: A HISTORY AGAINST UNCERTAINTY
    5. 4. THE CLASSICAL THEORY OF INTERVAL ALGEBRA
    6. 5. MACHINE REALIZATION OF INTERVAL ARITHMETIC
    7. 6. INTERVAL DEPENDENCY AND ALTERNATE INTERVAL THEORIES
    8. 7. INTERVAL METHODS FOR UNCERTAINTY IN SCIENCE AND TECHNOLOGY
    9. 8. INTERVALS, AND MORE INTERVALS: A VIEW TO THE FUTURE
    10. ACKNOWLEDGMENT
    11. REFERENCES
    12. KEY TERMS AND DEFINITIONS
    13. ENDNOTES
  9. Chapter 2: Uncertainty Modeling Using Expert's Knowledge as Evidence
    1. ABSTRACT
    2. INTRODUCTION
    3. BACKGROUND
    4. REASONING WITH IGNORANCE
    5. UNCERTAINTY MEASURES OF EVIDENCE THEORY
    6. CONCLUSION
    7. REFERENCES
  10. Chapter 3: Evidence-Based Uncertainty Modeling
    1. ABSTRACT
    2. 1. INTRODUCTION
    3. 2. BASIC CONCEPTS OF DEMPSTER-SHAFER THEORY (DST)
    4. 3. COMBINING EVIDENCES
    5. 4. EVIDENCE THEORY IN RELATION TO POSSIBILITY AND PROBABILITY THEORY
    6. 5. DST WITH FUZZY FOCAL ELEMENTS
    7. 6. UNCERTAINTY MEASURES IN EVIDENCE THEORY
    8. CONCLUSION
    9. ACKNOWLEDGMENT
    10. REFERENCES
    11. KEY TERMS AND DEFINITIONS
  11. Chapter 4: Hybrid Set Structures for Soft Computing
    1. ABSTRACT
    2. 1. INTRODUCTION
    3. 2. PRELIMINARIES
    4. 3. GENERALIZED SETS
    5. 4. HYBRID SET STRUCTURES
    6. 5. HESITANT FUZZY SOFT SETS IN DECISION MAKING PROBLEMS
    7. 6. APPLICATIONS
    8. 7. FUTURE RESEARCH DIRECTIONS
    9. 8. CONCLUSION
    10. REFERENCES
  12. Chapter 5: Source and m-Source Distances of Fuzzy Numbers and their Properties
    1. ABSTRACT
    2. 1. INTRODUCTION
    3. 2. PRELIMINARIES
    4. 3. SOURCE DISTANCE AND NEAR APPROXIMATION
    5. 4. M-SOURCE DISTANCE AND THE NEAREST APPROXIMATION
    6. 5. PROPERTIES OF M-SOURCE DISTANCE
    7. 6. POWERS OF A TRAPEZOIDAL FUZZY NUMBER
    8. 7. THE NEAREST TRAPEZOIDAL FUZZY NUMBER TO MULTIPLICATION OF TWO TRAPEZOIDAL FUZZY NUMBERS
    9. CONCLUSION
    10. REFERENCES
  13. Chapter 6: Construction of Normal Fuzzy Numbers using the Mathematics of Partial Presence
    1. ABSTRACT
    2. 1. INTRODUCTION
    3. 2. THE MATHEMATICS OF PARTIAL PRESENCE
    4. 3. CASE STUDIES
    5. 4. DISCUSSION
    6. ACKNOWLEDGMENT
    7. REFERENCES
    8. KEY TERMS AND DEFINITIONS
    9. ENDNOTES
  14. Chapter 7: Numerical Solution of Fuzzy Differential Equations and its Applications
    1. ABSTRACT
    2. 1. INTRODUCTION
    3. 3. PROPOSED METHODS FOR SOLVING FUZZY DIFFERENTIAL EQUATIONS
    4. 4. FUZZY HPM FOR -TH ORDER FUZZY DIFFERENTIAL EQUATIONS
    5. 5. FUZZY REACTION DIFFUSION MODEL
    6. 6. FUZZY SOLUTION OF FIRE PROPAGATION
    7. CONCLUSION
    8. ACKNOWLEDGMENT
    9. REFERENCES
  15. Chapter 8: Modeling with Stochastic Fuzzy Differential Equations
    1. ABSTRACT
    2. 1. INTRODUCTION
    3. 2. PRELIMINARIES
    4. 3. STOCHASTIC FUZZY DIFFERENTIAL EQUATIONS
    5. 4. APPLICATIONS OF STOCHASTIC FUZZY DIFFERENTIAL EQUATIONS
    6. 5. FUTURE RESEARCH DIRECTIONS
    7. 6. CONCLUSION
    8. REFERENCES
    9. KEY TERMS AND DEFINITIONS
  16. Chapter 9: Mathematics of Probabilistic Uncertainty Modeling
    1. ABSTRACT
    2. INTRODUCTION
    3. BACKGROUND
    4. MATHEMATICS OF MONTE CARLO SIMULATION
    5. SOLVED PROBLEMS USING MONTE CARLO METHOD
    6. UNCERTAINTY MODELING USING POLYNOMIAL CHAOS THEORY
    7. CONCLUSION
    8. REFERENCES
  17. Chapter 10: Reaction-Diffusion Problems with Stochastic Parameters Using the Generalized Stochastic Finite Difference Method
    1. ABSTRACT
    2. 1. INTRODUCTION
    3. 2. DETERMINISTIC MODEL
    4. 3. A STOCHASTIC PROCESS APPROACH
    5. 4. REACTION-DIFFUSION WITH UNCERTAINTY
    6. 5. NUMERICAL ILLUSTRATION
    7. 6. CONCLUSION
    8. REFERENCES
  18. Chapter 11: MV-Partitions and MV-Powers
    1. ABSTRACT
    2. 1. INTRODUCTION
    3. 2. MV - PARTITIONS
    4. 3. MV - POWERS
    5. 4. MV-ELEMENTS
    6. REFERENCES
  19. Chapter 12: An Algebraic Study of the Notion of Independence of Frames
    1. ABSTRACT
    2. INTRODUCTION
    3. BACKGROUND
    4. TOWARDS AN ALGEBRAIC APPROACH TO CONFLICT
    5. AN ALGEBRAIC STUDY OF INDEPENDENCE OF FRAMES
    6. INDEPENDENCE ON LATTICES VS. INDEPENDENCE OF FRAMES
    7. FUTURE RESEARCH DIRECTIONS
    8. CONCLUSION
    9. REFERENCES
    10. KEY TERMS AND DEFINITIONS
    11. APPENDIX
  20. Chapter 13: Pricing and Lot-Sizing Decisions in Retail Industry
    1. ABSTRACT
    2. 1. INTRODUCTION
    3. 2. LITERATURE REVIEW
    4. 3. MAIN FOCUS OF THE CHAPTER
    5. 4. PROBLEM DEFINITION AND FORMULATION
    6. 5. THE SOLUTION METHOD
    7. 6. SOLVING THE CRISP MULTI-OBJECTIVE PROGRAMMING MODEL
    8. 7. META-HEURISTIC ALGORITHM
    9. 8. AN ILLUSTRATIVE EXAMPLE
    10. 9. CONCLUDING REMARKS AND FUTURE STUDIES
    11. REFERENCES
    12. ADDITIONAL READING
    13. KEY TERMS AND DEFINITIONS
  21. Chapter 14: A New Approach for Suggesting Takeover Targets Based on Computational Intelligence and Information Retrieval Methods
    1. ABSTRACT
    2. 1. INTRODUCTION
    3. 2. REVIEW OF LITERATURE
    4. 3. PROBLEM DESCRIPTION
    5. 4. METHODOLOGY
    6. 5. ANALYSIS AND EXPERIMENTAL EVALUATION
    7. 6. DISCUSSION
    8. 7. CONCLUSION
    9. REFERENCES
    10. KEY TERMS AND DEFINITIONS:
  22. Chapter 15: Fuzzy Finite Element Method in Diffusion Problems
    1. ABSTRACT
    2. 1. INTRODUCTION
    3. 2. INTERVAL AND FUZZY ARITHMETIC
    4. 3. A SIMPLE EXAMPLE OF FULLY FUZZY SYSTEM USING THE PROPOSED METHOD
    5. 4. TRADITIONAL FINITE ELEMENT FORMULATION FOR HEAT EQUATION
    6. 5. FUZZY FINITE ELEMENT METHOD IN HEAT EQUATION
    7. 6. TRADITIONAL FINITE ELEMENT FORMULATION FOR NEUTRON DIFFUSION EQUATION
    8. 7. FUZZY FINITE ELEMENT METHOD IN NEUTRON DIFFUSION EQUATION
    9. 8. CONCLUSION
    10. ACKNOWLEDGMENT
    11. REFERENCES
  23. Chapter 16: Uncertainty Quantification of Aeroelastic Stability
    1. ABSTRACT
    2. INTRODUCTION
    3. BACKGROUND TO AEROELASTICITY
    4. BACKGROUND TO UNCERTAINTY QUANTIFICATION
    5. SURROGATE MODELS
    6. PROBLEM DESCRIPTION
    7. FUTURE RESEARCH DIRECTIONS
    8. CONCLUSION
    9. ACKNOWLEDGMENT
    10. REFERENCES
    11. ADDITIONAL READING
    12. KEY TERMS AND DEFINITIONS
  24. Chapter 17: Uncertain Static and Dynamic Analysis of Imprecisely Defined Structural Systems
    1. ABSTRACT
    2. INTRODUCTION
    3. PRELIMINARIES
    4. FUZZY SYSTEM OF LINEAR EQUATIONS
    5. FULLY FUZZY SYSTEM OF LINEAR EQUATIONS
    6. GENERALISED FUZZY EIGENVALUE PROBLEM
    7. NUMERICAL EXAMPLES
    8. FUZZY FRACTIONAL SPRING-MASS SYSTEMS
    9. FUTURE RESEARCH DIRECTIONS
    10. CONCLUSION
    11. ACKNOWLEDGMENT
    12. REFERENCES
  25. Compilation of References
  26. About the Contributors