
Closer Look at the Diffusion Equation
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Content
- Intro
- A CLOSER LOOK ATTHE DIFFUSION EQUATION
- A CLOSER LOOK ATTHE DIFFUSION EQUATION
- CONTENTS
- PREFACE
- Chapter 1A NUMERICAL APPROACH TO SOLVINGUNSTEADY ONE-DIMENSIONAL NONLINEARDIFFUSION EQUATIONS
- Abstract
- 1. INTRODUCTION
- 2. IMPLICIT TIME DISCRETIZATION
- 3. SOLVING THE NONLINEAR TPBVPS
- 3.1. Partial Derivatives for the Newton Method
- 3.2. Applying the Finite Difference Method
- 3.3. Solving the Nonlinear System by Newton Method
- 4. NUMERICAL COMPUTER EXPERIMENTS
- 5. MATLAB CODE
- APPENDIX
- CONCLUSION
- Acknowledgments
- REFERENCES
- Chapter 2DIFFUSION IN HYPERSONIC FLOWS
- Abstract
- 1. INTRODUCTION
- 2. PHYSICAL MODELING
- 2.1. Mixtures Properties
- 2.2. Conservation Equations
- 2.3. Thermodynamic and Chemical Models
- 2.4. Chemical Model
- 2.5. DiffusionModels
- 3. MATHEMATICAL MODELING
- 4. RESULTS AND DISCUSSION
- 4.1. Grid Independence Study and Validation
- 4.2. Results of Fick's Law of Diffusion
- 4.3. Results of Binary Collision Theory
- 4.4. Results of StefanMaxwell Equation
- CONCLUSION AND FUTURE WORK
- REFERENCES
- Chapter 3ON THE NONLINEAR DIFFUSIONWITH EXPONENTIALCONCENTRATION-DEPENDENTDIFFUSIVITY: INTEGRAL-BALANCESOLUTIONS AND ANALYZES
- Abstract
- 1. INTRODUCTION
- 1.1. Physical Origins of the Modelled Problem
- 1.2. Mathematical Problem Formulation andGoverningEquations
- 1.3. The Diffusivity Functional Relationship:Preliminary Comments
- 1.4. The Ranges of the Rate-Factor in Real Physical Situations
- 1.5. Existing Solution Approaches: A Short Overview
- 1.5.1. Boltzmann Similarity Transform Approach
- 1.5.2. Asymptotic Solutions
- Babu's Solution
- Budd and Stokie Solution
- Parlange's Solution
- 1.5.3. Moving Front Approaches
- 1.6. Motivation and Main Concept
- 1.7. Organization of the Following Part of the Chapter
- 2. APPROXIMATE INTEGRAL-BALANCE SOLUTION
- 2.1. Integral-Balance Method
- 2.1.1. HBIM Integration Technique
- 2.1.2. DIMIntegration Technique
- 2.2. Assumed Profile and Penetration Depths
- 2.2.1. DIM
- 2.2.2. HBIM
- 2.3. The Approximate Profile
- 2.3.1. DIMSolution
- 2.3.2. HBIM Solution
- 2.4. Dimensionless Penetration Depths
- HBIM solution
- DIM solution
- 2.5. Front Time Evolution Scaling
- 3. OPTIMIZATION OF THE APPROXIMATE SOLUTION
- 3.1. Residual Function
- 3.1.1. Residual Function in Terms of the Similarity Variable
- 3.1.2. Residual Function at the Boundaries of the Penetration Layer
- 3.1.3. Optimal Exponents of the Solutions
- 4. NUMERICAL EXPERIMENTS WITH THEAPPROXIMATE SOLUTION
- 4.1. Dimensionless Penetration Depth: Some Limits
- HBIM Solution
- DIM Solution
- 5. COMPARISON OF THE APPROXIMATE SOLUTIONTO OTHER SOLUTIONS OF THE PROBLEM
- 5.1. Solution of Riek et al.[38]
- 5.2. Solution of Lockington et al.[52]
- 5.3. Solution of Tzimopoulos et al. [55]
- 5.4. Briefs on the Comparisons of Solutions
- CONCLUSION
- REFERENCES
- Chapter 4SOLUTIONS FOR FRACTIONALREACTION-DIFFUSION EQUATIONS
- Abstract
- 1. INTRODUCTION
- 2. REACTION - DIFFUSION
- 2.1. Irreversible Reaction
- 2.2. Reversible Reaction
- 3. DISCUSSION AND CONCLUSION
- ACKNOWLEDGMENT
- REFERENCES
- Chapter 5SEMI-ANALYTICAL SOLUTION OF HRISTOVDIFFUSION EQUATION WITH SOURCE
- Abstract
- 1. INTRODUCTION
- 2. MATHEMATICAL BACKGROUND
- 3. SEPARABLE SOLUTIONS VIA FOURIER METHOD
- 4. NUMERICAL APPROXIMATION AND ILLUSTRA-TIVE EXAMPLES
- CONCLUSION
- REFERENCES
- Chapter 6NON-GAUSSIAN DIFFUSION EMERGENCEIN SUPERSTATISTICS
- Abstract
- 1. INTRODUCTION
- 2. NON-GAUSSIAN DIFFUSION ANDSUPERSTATISTICS
- 3. SUPER FOKKER-PLANCK EQUATION FOR THEHARMONIC POTENTIAL
- 47].
- REFERENCES
- Chapter 7MEAN SQUARE DISPLACEMENT OF THEFRACTIONAL DIFFUSION EQUATIONDESCRIBED BY CAPUTO GENERALIZEDFRACTIONAL DERIVATIVE
- Abstract
- 1. INTRODUCTION
- 2. FRACTIONAL DERIVATIVE OPERATORSAND THEIR PROPERTIES
- 3. QUALITATIVE PROPERTIES FOR THEFRACTIONALDIFFUSION EQUATION
- 4. ANALYTICAL SOLUTION OF THE FRACTIONALDIFFUSION EQUATION
- 5. MEAN SQUARE DISPLACEMENT FORFRACTIONAL DIFFUSION EQUATION
- 6. METHOD FOR GETTING THE MEAN SQUAREDISPLACEMENT
- 7. ANALYSIS AND INTERPRETATIONS OFTHE MAIN RESULTS
- CONCLUSION
- REFERENCES
- EDITOR CONTACT INFORMATION
- INDEX
- Blank Page
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