Methods of Mathematical Modelling and Computation for Complex Systems

Preface
Contents
About the Editors
On the Diffusion with Decaying Time-Dependent Diffusivity: Formulations and Approximate Solutions Pertinent to Diffusion in Concretes
1 Introduction
1.1 General Assumptions
1.2 Common Approaches in Modelling and Solutions
2 Motivation of This Study
2.1 Aim and Chapter Organization
3 Existing Functional Relationships of D(t): A Brief Overview
3.1 Singular (Power-Law) Time Dependence of D(t)
3.2 Non-singular Forms of D(t)
4 New Forms of D(t) and Method of Approximate Solutions
4.1 New Non-singular Forms of D(t)
4.2 Integral-Balance Method
5 Approximate Solutions of (4) with Different D(t)
5.1 Solutions with D1(t)
5.2 Solutions with D2(t)
5.3 Solutions with D3(t)
5.4 Numerical Experiments with the Approximate Solutions
5.5 Comparative Study with Non-singular D(t) From Literature Sources
5.6 Solution Optimization
6 Outcomes of the Approximate Solutions
6.1 Causality of Dynamic Models: Basic Principles
6.2 Subdiffusion Front Propagation: Two Formal Explanations
6.3 Finite Penetration Depth and Finite Speed of the Front
7 Final Comments
References
Laminar Convection of Power-Law Fluids in Differentially Heated Closed Region: CFD Analysis
1 Introduction
2 Numerical Method
2.1 Governing Equations
2.2 Boundary Conditions
2.3 Grid Dependency
2.4 Benchmark Comparison
3 Discussion and Computational Results
3.1 The Effects of Rayleigh Number
3.2 Bingham Fluids
3.3 Pseudo Plastic Fluid
3.4 Dilatant Fluids
4 Conclusions
References
Mathematical Perspective of Hodgkin-Huxley Model and Bifurcation Analysis
1 Neurological Backgrounds
1.1 How Does Neuron Work?
1.2 How Does a Neuron Fire?
1.3 How Neurons Communicate?
2 Hodgkin and Huxley Model (HH Model)
3 Review of HH Model:
4 Modified HH Models
5 Comparative Studies of Modified HH Models
5.1 Stability Analysis
5.2 Comparative Study of Time Series Analysis
5.3 Comparative Study of One-Parameter Bifurcation
5.4 Comparative Study of Two-Parameter Bifurcation
6 Results
7 Discussions
References
Mathematical Analysis of Two Unequal Collinear Cracks in a Piezo-Electro-Magnetic Media
1 Introduction
2 Basic Equations for Piezoelectromagnetic Media
2.1 Crack Face Boundary Conditions
3 Fundamental Formulation and Solution Methodology
4 Statement of the Problem
5 Solution of the Problem
6 Applications
6.1 Crack Opening Displacement (COD)
6.2 Crack Opening Potential Drop (COPD)
6.3 Crack Opening Induction Drop (COID)
6.4 Stress Intensity Factor (SIF)
6.5 Electric Displacement Intensity Factor (EDIF)
6.6 Magnetic Induction Intensity Factor (MIIF)
7 Case Study
7.1 Effect of Inter-Crack Distance
7.2 Effect of Crack Length
8 Conclusions
References
Advanced Analysis of Local Fractional Calculus Applied to the Rice Theory in Fractal Fracture Mechanics
1 Introduction
2 The LFD and LFI in the Fractional Real and Complex Sets
2.1 The LFD and LFI in the Fractional Real Set
2.2 The LFD and LFI in the Fractional Complex Set
3 Analysis of the Local Fractional Differential and Integral Equations
3.1 The Uniqueness of the Solutions of the Local Fractional Differential Equations
3.2 The Uniqueness of the Solutions of the Local Fractional Integral Equations
4 Local Fractional Inequalities
5 The Series and Transforms Involving the Mittag-Leffler Function Defined on Cantor Sets
5.1 The Fourier Type Series Via the Mittag-Leffler Function Defined On Cantor Sets
5.2 The Fourier Type Transform Via Mittag-Leffler Function Defined on Cantor Sets
6 Local Fractional Vector Calculus with an Application in Fractal Fracture Mechanics
6.1 Local Fractional Vector Calculus
6.2 An Application to Rice Theory in Fractal Mechanics
7 Conclusion
References
General Fractional Calculus with Nonsingular Kernels: New Prospective on Viscoelasticity
1 Introduction
2 Mathematical Tools
2.1 The Special Functions with Power Law
2.2 GFC in a Kernel Function
2.3 FC Within the Singular Power-Law Kernel
2.4 GFC with the Nonsingular Power-Law Kernel
2.5 GFC with the Nonsingular Mittag–Leffler Function Kernel
2.6 GFC with the Nonsingular Wiman Kernel
2.7 GFC with the Nonsingular Prabhakar Kernel
3 The Rheological Models with GFCs Involving the Nonsingular Kernels
3.1 Complex Phenomena in Viscoelasticity
3.2 The Viscoelastic Elements with GFDs
3.3 The Maxwell Models with GFDs
3.4 The Voigt Models with GFDs
4 Conclusion
References
Group Dynamical Systems on C*-Algebras Generated by Countable Infinitely Many Semicircular Elements
1 Introduction
1.1 Background
1.2 Motivation
1.3 Overview
2 Preliminaries
2.1 Free Probability
2.2 Semicircularity
3 Catalan Numbers
4 Free Distributions of Multi Semicircular Elements
4.1 Free-Distributional Data (13) of ρ
4.2 Free-Distributional Data (14) of ρ
4.3 The Joint Free Distribution ρx1,…,xN
5 A C*-Probability Space ( mathfrakX, ) Generated by | mathbbN| -Many Semicircular Elements
5.1 Free-Homomorphisms
5.2 A C*-Probability Space ( mathfrakX, midmathfrakX)
5.3 Free-Distributional Data on mathfrakX
6 Free-Isomorphisms on mathfrakX
6.1 Shifts on mathbbZ
6.2 Integer Shifts on mathfrakX
6.3 Free-Isomorphic Relations on mathfrakX
7 A Group Dynamical System ( λ, ΛmathfrakX, α)
7.1 A Group C*-Dynamical System Γ=( λ, ΛmathfrakX, α)
7.2 The Γ-Dynamical Algebra mathbbXΓ of λ
8 Free Probability on mathbbXΓ
8.1 Free Probability on ΛmathfrakX = ( ΛmathfrakX, τ)
8.2 Free-Distributional Data on mathbbXΓ
References
Lie Group Theory for Nonlinear Fractional K(m, n) Type Equation with Variable Coefficients
1 Introduction
2 Lie Symmetry Analysis Method for FPDEs
3 Fractional Lie Symmetries for Time-Fractional K(m, n)
4 Fractional Lie Symmetries for Time-Fractional K(2, 3)
5 Reduced Equations and Invariant Solution of Eq. (20)
6 Conclusion
References
Generalized Rayleigh Wave Propagation in Heterogeneous Substratum Over Homogeneous Half-Space Under Gravity
1 Introduction
2 Formulation of the Problem
2.1 Equations of Motion and Solution for Upper Layer
2.2 Equations of Motion and Solution for Half Space
3 Boundary Conditions
3.1 Particular Cases
4 Results and Discussions
5 Conclusion
References
On Defining Trigonometric Box Spline-Like Surface on Type-I Triangulation
1 Introduction
2 Trigonometric Spline Surface on Arbitrary Topology
3 Construction of the New Non-stationary Subdivision Scheme
3.1 Construction of the Scheme
3.2 Regular Subdivision Rules
3.3 Subdivision Surface
4 Convergence Analysis
5 Application of the Proposed Scheme
6 Conclusion
References
Mathematical Modelling for Perishable Product Supply Chain Under Inflation and Variable Lead Time
1 Introduction
2 Literature Review
3 Research Gap
4 The Model Framework
5 Result and Discussion
5.1 Future Scope and Limitations of the Research Work
6 Conclusion
References
Mexican Hat Wavelet Transform and Its Applications
1 Introduction
2 Necessary Conditions for Mexican Hat Wavelet and MHWST
3 Sufficient Conditions for Mexican Hat Wavelet and MHWST
References
Fractal Fractional Derivative Operator Method on MCF-7 Cell Line Dynamics
1 Introduction
2 Mathematical Analysis of the Models
2.1 Preliminaries
2.2 Uniqueness and Continuous Dependence of the Solution
2.3 Physiological Level for the Dynamics
2.4 Asymptotic Stability Conditions for the Dynamics
3 Construction of the Fractal Fractional Operator Numerical Method
4 Numerical Results and Discussions
5 Conclusion
References
The Exponentiated Half Logistic-Topp-Leone-G Power Series Class of Distributions: Model, Properties and Applications
1 Introduction
2 The Model, Sub-Classes and Properties
2.1 The Model
2.2 Quantile Function
2.3 Expansion of Density
2.4 Moments and Generating Function
2.5 Conditional Moments
2.6 Mean Deviation, Lorenz and Bonferroni Curves
2.7 Order Statistics and Rényi Entropy
2.8 Maximum Likelihood Estimation
3 Some Special Classes of the EHL-TL-GPS Class of Distributions
3.1 Exponentiated Half Logistic Topp-Leone-Log-Logistic Power Series (EHL-TL-LLoGPS) Class of Distributions
3.2 Exponentiated Half Logistic-Topp-Leone-Weibull Power Series (EHL-TL-WPS) Class of Distributions
4 Simulation Study
5 Applications
5.1 Guinea Pigs Survival Times Data
5.2 Skin Folds Data
6 Concluding Remarks
References
Fixed Points of Multivalued (αast-φ)-Contractions and Metric Transforms
1 Introduction and Preliminaries
2 Extensions by Using Multivalued ( αast-φ) Contractions
3 Generalized (αast-φ) Multivalued Contractions
References
Approximation of Signals Via Different Summability Means with Effects of Gibbs Phenomenon
1 Introduction and Preliminaries
2 Definitions and Motivation
3 Classical Approximation Theorems
4 Statistical Korovkin-Type Approximation Results
5 Effects of Gibbs Phenomenon
6 Concluding Remarks and Observations
References