Closer Look at the Diffusion Equation
Published on 20. October 2020
189 pages
978-1-5361-8488-4 (ISBN)
Description
Diffusion is a principle transport mechanism emerging widely at different scale, from nano to micro and macro levels. This is a contributed book of seventh chapters encompassing local and no-local diffusion phenomena modelled with integer-order (local) and non-local operators. This book collates research results developed by scientists from different countries but with common research interest in modelling of diffusion problems. The results reported encompass diffusion problems related to efficient numerical modelling, hypersonic flows, approximate analytical solutions of solvent diffusion in polymers and wetting of soils. Some chapters are devoted to fractional diffusion problem with operators with singular and non-singular memory kernels. The book content cannot present the entire rich area of problems related to modelling of diffusion phenomena but allow seeing some new trends and approaches in the modelling technologies. In this context, the fractional models with singular and non-singular kernels the numerical methods and the development of the integration techniques related to the integral-balance approach form fresh fluxes of ideas to this classical engineering area of research. The book is oriented to researchers; master and PhD students involved in diffusion problems with a variety of application and could serves as a rich reference source and a collection of texts provoking new ideas.
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Jordan Hristov
A Closer Look at the Diffusion Equation
Book
10/2020
Nova Science Publishers Inc
€126.26
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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
