Chapter
1.3 Example of Some Aquifers in the World
1.4 Some Identified Aquifer Properties
2 Principle of Groundwater Flow
2.1 Groundwater Within Geological Formations
2.1.2 Groundwater Overdraft
2.1.3 Overview of Groundwater Within Subsurface
2.2 Concept of Groundwater Flow Motion
2.2.1 Darcy's Law and Its Application
2.2.2 Derivation of Darcy's Law
2.3 Theis Model of Groundwater Flow
2.3.1 Derivative of Theis Groundwater Flow Equation
2.3.2 Derivation of Exact Solution
2.3.3 Cooper and Jacob Approximation of Theis's Solution of Groundwater Flow Equation
2.4 Groundwater Flow Within an Unconfined Aquifer
2.5 Groundwater Flow in a Deformable Aquifer
2.5.1 Perturbation From Prolate Coordinates to Spherical Coordinates
2.5.2 Derivation of Solution via Asymptotic Method
2.7 Uncertainties Analysis of Aquifer Parameters for Groundwater Flow Model
2.7.2 Evaluation of Uncertainties by Mean of Statistic Formulas
Parameters uncertainties evaluation of groundwater flow equation
3.1 History of Groundwater Pollution: Love Canal Disaster
3.1.1 Love Canal Disaster
3.1.2 Construction of the 93rd Street School and the 99th Street School
3.1.3 Health Problems and Site Cleanup of Love Canal
3.4 Heath Problems Caused by Groundwater Pollution
3.5 Convection Dispersion Model
3.5.1 Derivation of the Mathematical Model
3.5.2 Derivation of Exact Solution
3.6 Groundwater Remediation: Techniques and Actions
3.6.1 Remediation Technique
Some techniques for groundwater remediation
3.7 Sensibility Analysis of Model Parameters
3.7.1 Some Commonly Used Methods for Sensitivity Analysis
3.7.2 Limitations of Sensibility Analysis Methods
3.8 Problems of Transboundary Aquifers
4 Limitations of Groundwater Models With Local Derivative
4.1 Limitations of Groundwater Flow Model
4.2 Limitations of Groundwater Convection Dispersion Model
5 Fractional Operators and Their Applications
5.1 Introduction to the Concept of Fractional Calculus
5.2 Riemann-Liouville Type
5.2.1 Some Useful Properties
5.7 Caputo-Fabrizio in Riemann-Liouville Sense
5.8 Atangana-Baleanu Derivatives With Fractional Order
5.8.1 Motivations and Definitions With New Kernel
5.8.2 Properties of Atangana-Baleanu Derivatives With Fractional Order
5.8.3 Relation With Integral Transforms
5.9 Physical Interpretation of Fractional Derivatives
5.10 Advantages and Limitations
5.10.1 Advantages of Fractional Derivatives
5.10.2 Disadvantages of Fractional Derivatives
6 Regularity of a General Parabolic Equation With Fractional Differentiation
6.1 Regularity With Caputo Fractional Differentiation
6.1.1 Some Useful Information About the Marchaud and Caputo Derivatives
6.1.2 The Main Focus of Investigation and Main Results
6.1.3 Investigation of the Weak Solutions of Equations in Non-divergence Form
6.1.4 Derivation of an Approximating Solutions
6.1.5 Hölder Continuity for the Time-Fractional Differentiation
6.1.6 Derivation of Sub-solution to an Ordinary Differential Equation
6.1.7 Analysis of the Hölder Continuity Within the Framework of Fractional Differentiation
6.2 Regularity of General Fractional Parabolic Equation With Atangana-Baleanu Derivative
6.2.1 Literature Review of Results Related to the Hölder Regularity
6.2.2 Preliminaries and Some Useful Denotations
6.2.3 The Fractional Time Derivative With Nonlocal and Nonsingular Mittag-Leffler Kernel
6.2.4 Detailed Proof of the Existence of Weak Solutions Using Approximating Solutions
6.2.5 Numerical Approximation of the Problem
6.2.6 The Derivation of the Point-Wise Estimation and Hölder Regularity
7 Applications of Fractional Operators to Groundwater Models
7.1 Theis Model With Fractional Derivatives
7.1.1 Theis Model With Caputo Derivative
7.1.2 Theis Model With Caputo-Fabrizio Derivative
7.2 Existence and Uniqueness for the New Model of Groundwater Flow in Confined Aquifer
7.2.1 Formulation of the Problem and Existence of Solutions
7.2.2 Uniqueness of the Exact Solution of the Problem
Derivation of an analytic solution
7.3 Numerical Analysis of the New Groundwater Model
7.3.1 Discretization of the Problem Using Crank-Nicholson Scheme
7.4 Application to Groundwater Flowing Within an Unconfined Aquifer
Groundwater Flowing Within an Unconfined Aquifer With Caputo Derivative
Groundwater Flowing Within an Unconfined Aquifer With Caputo-Fabrizio
7.5 Existence and Uniqueness of the Solution for the New Model of Groundwater Flow Within an Unconfined Aquifer
7.5.2 Uniqueness of the Solution
Groundwater flowing within an unconfined aquifer with Atangana-Baleanu fractional derivative
Existence and Uniqueness of the Solution
Numerical Analysis of the Modified Model
7.6 Hantush Model With Fractional Derivatives
7.6.1 Analytical Solution of the Flow Within the Leaky Aquifer With the Atangana-Baleanu Fractional Derivative
7.6.2 Existence and Uniqueness of the Solution of the Problem
7.6.3 Exact Solution of the Flow Within the Leaky Aquifer Based Upon Atangana-Baleanu Fractional Derivative
7.6.4 Numerical Analysis of the Groundwater Flowing Within the Leaky Aquifer Based Upon Atangana-Baleanu Fractional Derivative
7.7 Steady-State Model With Fractional Derivatives
7.7.1 Theim's Equilibrium Model Within an Unconfined Aquifer With Fractional Derivative
7.7.2 Theim's Equilibrium Model Within a Confined Aquifer With Fractional Derivative
7.8 Limitations of Fractional Derivative for Modeling Groundwater Problems
8 Models of Groundwater Pollution With Fractional Operators
8.1 Time Fractional Convection-Dispersion Equation With Caputo Type
8.1.1 Analytical Solution With the Riemann-Liouville Derivative
8.1.2 Analytical Solution With the Caputo Derivative
8.2 Time Fractional Convection-Dispersion Equation With Caputo-Fabrizio Type
8.3 Time Fractional Convection-Dispersion Equation With Atangana-Baleanu Derivative
8.3.1 Derivation of Numerical Solution
8.4 Limitation of Fractional Derivative for Groundwater Pollution Model
9 Fractional Variable Order Derivatives
9.1 Definition of Existing Variable Order Derivatives
9.2 Limitations and Advantages
9.3 Atangana-Koca Variable-Order Derivative
9.4 Relation With Some Integral Transform
9.5 Partial Derivative With Variable Order
9.6 Atangana-Koca Derivative With Variable Order in Caputo Sense
9.6.1 Relation With Integral Transforms
9.6.2 Numerical Approximation of Atangana-Koca Fractional Variable Order
9.6.3 Second Order Variable-Order Approximation of Atangana-Koca Derivative
10 Groundwater Flow Model in Self-similar Aquifer With Atangana-Baleanu Fractional Operators
10.1 Groundwater Flow Model in a Self-similar Aquifer
10.2 Positive Solution of the Modified Groundwater Flow in Self-similar Aquifer With Atangana-Baleanu Fractional Operators
10.3 Semi-Analytical and Numerical Solutions
10.3.1 Numerical Analysis With Fractional Integral
10.3.2 Numerical Approximation of Fractional Integral
10.3.3 Application to New Modified Model
10.3.4 Numerical Solution With Fractional Derivative
11 Groundwater Flow Within a Fracture, Matrix Rock and Leaky Aquifers: Fractal Geometry
11.1 Model of Groundwater Flow in a Fractal Dual Media Accounting for Elasticity
11.1.1 Existence of System Solutions by Means of Fixed-Point Theorem Method
11.2 Derivation of the Numerical Solution
11.3 Model of Groundwater Flow in Fractal Dual Media With Viscoelasticity Effect
11.3.1 Existence of System Solutions
11.3.2 Derivation of the Numerical Solution With Mittag-Leffler Law
11.4 Model of Groundwater Flow in a Fractal Dual Media With Heterogeneity and Viscoelasticity Properties
11.4.1 Numerical Simulation for Different Values of Fractional Order
11.5 Modeling Groundwater Flow With Variable Order Derivatives in a Leaky Aquifer
11.5.1 Definition and Problem Modification
11.5.2 Problem Formulation
11.5.3 Numerical Solution of the Modified Model by Means of Crank-Nicolson Scheme for the Modified Equation 178
11.5.4 Stability Analysis of the Crank-Nicolson Scheme by Means of Fourier Method
11.5.5 Convergence Analysis of the Crank-Nicolson Scheme
12 Modeling Groundwater Pollution With Variable Order Derivatives
12.1 Properties of Soils and Validity of Variable Order Derivatives
12.2 Modeling With Caputo Variable Order
12.3.1 Crank-Nicolson Scheme
12.3.2 Stability Analysis of the Crank-Nicolson Scheme
12.3.3 Convergence Analysis of the Crank-Nicolson Scheme
12.4 Groundwater Flow Model With Space Time Riemann-Liouville Fractional Derivatives Approximation
12.5 Modified Groundwater Flow Model by Means of Riemann-Liouville Approximate Fractional Differentiation
12.6 Derivation of Solutions of the Modified Groundwater Flow
12.6.1 Variational Iteration Method
12.6.2 Solution by Means of Green Function Methods
12.7 Application of the Modified Riemann-Liouville Variable Order on the Advection Dispersion Equation
12.7.1 Analysis and Possible Solutions
12.7.3 Possible Analytical Solution
Variational iteration method
12.8 A Model of the Groundwater Flowing Within a Leaky Aquifer Using the Concept of Local Variable Order Derivative
12.8.1 Groundwater Flow Equation Using the Local Variable Order Derivative
12.8.2 Construction of a Possible Special Solution
12.8.3 Uniqueness of the Solution
12.8.4 Stability Analysis of the Method
13 Groundwater Recharge Model With Fractional Differentiation
13.1 Motivation for the New Development
13.2 Analytical Solution Using the Green Function Approach
13.3 Analysis of Uncertainties Within the Scope of Statistics
13.4 Groundwater Recharge Model With Power Law
13.5 Groundwater Recharge Model With Caputo-Fabrizio Fractional Differentiation
13.6 Modeling Groundwater Recharge With Atangana-Baleanu Fractional Differentiation
13.7 The Model With Eton Approach
13.7.1 Application of the Atangana Derivative With Memory
13.7.2 Properties of the Atangana Derivative With Memory
13.7.3 Application to the Selected Groundwater Recharge Equation
13.7.4 Iterative Method for the New Model
13.7.5 Numerical Solution Using Crank-Nicolson
14 Atangana Derivative With Memory and Application
14.1 Definition and Properties
14.1.1 Properties of Atangana Derivative With Memory
14.2 Application to the Flow Model in a Confined Aquifer
14.2.1 Existence and Uniqueness of Exact Solution
14.3 Modeling the Movement of Pollution With Uncertain Derivative
14.4.1 Stability Analysis of the Numerical Scheme
14.5 Numerical Simulations
Fractional Operators with Constant and Variable Order with Application to Geo-hydrology
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