Most complex physical phenomena can be described by nonlinear equations, specifically, differential equations. In water engineering, nonlinear differential equations play a vital role in modeling physical processes. Analytical solutions to strong nonlinear problems are not easily tractable, and existing techniques are problem-specific and applicable for specific types of equations. Exploring the concept of homotopy from topology, different kinds of homotopy-based methods have been proposed for analytically solving nonlinear differential equations, given by approximate series solutions. Homotopy-Based Methods in Water Engineering attempts to present the wide applicability of these methods to water engineering problems. It solves all kinds of nonlinear equations, namely algebraic/transcendental equations, ordinary differential equations (ODEs), systems of ODEs, partial differential equations (PDEs), systems of PDEs, and integro-differential equations using the homotopy-based methods. The content of the book deals with some selected problems of hydraulics of open-channel flow (with or without sediment transport), groundwater hydrology, surface-water hydrology, general Burger’s equation, and water quality.
Features:
- Provides analytical treatments to some key problems in water engineering
- Describes the applicability of homotopy-based methods for solving nonlinear equations, particularly differential equations
- Compares different approaches in dealing with issues of nonlinearity
Author(s): Manotosh Kumbhakar, Vijay P. Singh
Publisher: CRC Press
Year: 2023
Language: English
Pages: 450
City: Boca Raton
Cover
Half Title
Title Page
Copyright Page
Contents
Preface
About the Authors
PART I: Introduction
Chapter 1: Introduction
References
Chapter 2: Basic Concepts
2.1. Definition of Homotopy
2.2. Homotopy Perturbation Method
2.3. Homotopy Analysis Method
2.4. Optimal Homotopy Asymptotic Method
2.5. An Illustrative Example
2.5.1. Solution Using Various Analytical Methods
2.5.1.1. Exact Solution
2.5.1.2. Perturbation Solution
2.5.1.3. Lyapunov’s Artificial Small Parameter Method–Based Solution
2.5.1.4. Adomian Decomposition Method–Based Solution
2.5.1.5. Homotopy Perturbation Method–Based Solution
2.6. Homotopy Analysis Method–Based Solution
2.6.1. Solution in Terms of a Polynomial
2.6.2. Solution in Terms of Exponential Functions
2.6.3. Optimal Homotopy Asymptotic Method–Based Solution
2.7. Homotopy Derivative and Its Properties
2.8. Convergence Theorem of a HAM-Based Solution
2.9. Convergence Theorem of an OHAM-Based Solution
2.10. Padé Approximant
2.11. Some Remarks
Supplement to Chapter 2
References
Further Reading
PART II: Algebraic/Transcendental Equations
Chapter 3: Numerical Solutions for the Colebrook Equation
3.1. Introduction
3.2. Newton-Like Methods for Nonlinear Equations Using HPM and HAM
3.2.1. Newton-Raphson Method
3.2.2. HPM-Based Method
3.2.3. HAM-Based Method
3.3. Convergence Theorem of the HAM-Based Solution
3.4. Examples
3.5. Application to the Colebrook Equation
3.6. Concluding Remarks
Supplement to Chapter 3
References
Further Reading
PART III: Ordinary Differential Equations (Single and System)
Chapter 4: Velocity Distribution in Smooth Uniform Open-Channel Flow
4.1. Introduction
4.2. Velocity Model
4.3. HAM-Based Solution
4.4. HPM-Based Solution
4.5. OHAM-Based Solution
4.6. Convergence Theorems
4.6.1. Convergence Theorem of the HAM-Based Solution
4.6.2. Convergence Theorem of the OHAM-Based Solution
4.7. Results and Discussion
4.7.1. Numerical Convergence and Validation of the HAM-Based Solution
4.7.2. Validation of the HPM-Based Solution
4.7.3. Validation of the OHAM-Based Solution
4.8. Concluding Remarks
References
Further Reading
Chapter 5: Sediment Concentration Distribution in Open-Channel Flow
5.1. Introduction
5.2. Sediment Concentration Models
5.2.1. Rouse Equation
5.2.2. Chiu et al. (2000) Models
5.2.2.1. Sediment Concentration Model I
5.2.2.2. Sediment Concentration Model II
5.3. HAM-Based Analytical Solutions
5.3.1. HAM Solution for Sediment Concentration Model I
5.3.2. HAM Solution for Sediment Concentration Model II
5.4. HPM-Based Analytical Solutions
5.4.1. HPM Solution for Sediment Concentration Model I
5.5. OHAM-Based Analytical Solutions
5.5.1. OHAM Solution for Sediment Concentration Model I
5.6. Convergence Theorems
5.6.1. Convergence Theorem of HAM-Based Solution
5.6.2. Convergence Theorem of the OHAM-Based Solution
5.7. Results and Discussion
5.7.1. Numerical Convergence and Validation of the HAM-Based Solution
5.7.2. Validation of the HPM-Based Solution
5.7.3. Validation of the OHAM-Based Solution
5.8. Concluding Remarks
References
Further Reading
Chapter 6: Richards Equation under Gravity-Driven Infiltration and Constant Rainfall Intensity
6.1. Introduction
6.2. Governing Equation and HAM-Based Solution
6.2.1. Torricelli’s Law
6.2.2. Brooks and Corey’s Hydraulic Conductivity Function
6.3. HPM-Based Solution
6.4. OHAM-Based Solution
6.5. Convergence Theorems
6.5.1. Convergence Theorem of HAM-Based Solutions
6.5.2. Convergence Theorem of OHAM-Based Solution
6.6. Results and Discussion
6.6.1. Numerical Convergence and Validation of the HAM-Based Solution
6.6.2. Validation of HPM-Based Solution
6.6.3. Validation of OHAM-Based Solution
6.6.4. Behavior of the Solution
6.7. Concluding Remarks
References
Further Reading
Chapter 7: Error Equation for Unsteady Uniform Flow
7.1. Introduction
7.2. Governing Equation
7.3. Standard HAM-Based Solution
7.4. Modified HAM-Based Solution
7.5. HPM-Based Solution
7.6. OHAM-Based Solution
7.7. Convergence Theorems
7.7.1. Convergence Theorem of HAM-Based Solution
7.7.2. Convergence Theorem of OHAM-Based Solution
7.8. Concluding Remarks
References
Further Reading
Chapter 8: Spatially Varied Flow Equations
8.1. Introduction
8.2. Governing Equation
8.3. HAM-Based Solution
8.3.1. General Methodology
8.3.2. HAM-Based Solution for Subcritical Flow
8.3.2.1. Frictionless Case
8.3.2.2. Frictional Case
8.4. HPM-Based Analytical Solution
8.4.1. Frictionless Case
8.4.2. Frictional Case
8.5. OHAM-Based Analytical Solution
8.5.1. Frictionless Case
8.5.2. Frictional Case
8.6. Convergence Theorems
8.6.1. Convergence Theorem of the HAM-Based Solution
8.6.2. Convergence Theorem of the OHAM-Based Solution
8.7. Results and Discussion
8.7.1. Validation of the HAM-Based Solution
8.7.2. Validation of the HPM-Based Solution
8.7.3. Validation of the OHAM-Based Solution
8.7.4. Comparison with Gill’s (1977) Solution
8.8. Concluding Remarks
Appendix: Gill’s (1977) Perturbation-Based Solution
A: Frictionless Case
B: Frictional Case
References
Further Reading
Chapter 9: Modeling of a Nonlinear Reservoir
9.1. Introduction
9.2. Governing Equation and Analytical Solution
9.3. HAM-Based Solution
9.4. Homotopy Perturbation Method (HPM)-Based Solution
9.5. Optimal Homotopy Asymptotic Method (OHAM)-Based Solution
9.6. Convergence Theorems
9.6.1. Convergence Theorem of the HAM-Based Solution in Eq. (9.27)
9.6.2. Convergence Theorem of the OHAM-Based Solution in Eq. (9.50)
9.7. Results and Discussion
9.7.1. Validation of the HAM-Based Solution
9.7.2. Validation of the HPM-Based Solution
9.7.3. Validation of the OHAM-Based Solution
9.8. Concluding Remarks
References
Further Reading
Chapter 10: Nonlinear Muskingum Method for Flood Routing
10.1. Introduction
10.2. Governing Equation
10.3. Analytical Solutions
10.3.1. Analytical Solution of Eq. (10.5)
10.3.2. HAM-Based Analytical Solution for Eq. (10.6)
10.3.3. HPM-Based Analytical Solution for Eq. (10.6)
10.3.4. OHAM-Based Analytical Solution for Eq. (10.6)
10.4. Convergence Theorems
10.4.1. Convergence Theorem of the HAM-Based Solution for Eq. (10.38)
10.4.2. Convergence Theorem of the OHAM-Based Solution for Eq. (10.65)
10.5. Results and Discussion
10.5.1. Validation of the HAM-Based Solution
10.5.2. Validation of the HPM-Based Solution
10.5.3. Validation of the OHAM-Based Solution
10.6. Concluding Remarks
References
Further Reading
Chapter 11: Velocity and Sediment Concentration Distribution in Open-Channel Flow
11.1. Introduction
11.2. Governing Equation and Analytical Solutions
11.2.1. HAM-Based Solution
11.2.2. HPM-Based Solution
11.2.3. OHAM-Based Solution
11.3. Convergence Theorems
11.3.1. Convergence Theorem of the HAM-Based Solution
11.3.2. Convergence Theorem of the OHAM-Based Solution
11.4. Results and Discussion
11.4.1. Selection of Parameters
11.4.2. Numerical Convergence and Validation of the HAM-Based Solution
11.4.3. Validation of the HPM-Based Solution
11.4.4. Validation of the OHAM-Based Solution
11.5. Concluding Remarks
References
Further Reading
PART IV: Partial Differential Equations (Single and System)
Chapter 12: Unsteady Confined Radial Ground-Water Flow Equation
12.1. Introduction
12.2. Governing Equation
12.3. Theis Solution
12.4. HAM Solution
12.5. HPM-Based Solution
12.6. OHAM-Based Solution
12.7. Convergence Theorems
12.7.1. Convergence Theorem of the HAM-Based Solution
12.7.2. Convergence Theorem of the OHAM-Based Solution
12.8. Results and Discussion
12.8.1. Numerical Convergence and Validation of the HAM-Based Solution
12.8.2. Comparison of the HAM Solution with Series Approximations
12.8.3. Validation of the HPM-Based Solution
12.8.4. Validation of the OHAM-Based Solution
12.9. Concluding Remarks
References
Further Reading
Chapter 13: Series Solutions for Burger’s Equation
13.1. Introduction
13.2. Governing Equation
13.3. HAM-Based Solution
13.4. HPM-Based Solution
13.5. OHAM-Based Solution
13.6. Convergence Theorems
13.6.1. Convergence Theorem of the HAM-Based Solution
13.6.2. Convergence Theorem of the OHAM-Based Solution
13.7. Results and Discussion
13.7.1. Numerical Convergence and Validation of the HAM-Based Solution
13.7.2. Validation of the HPM-Based Solution
13.7.3. Validation of the OHAM-Based Solution
13.8. Concluding Remarks
References
Further Reading
Chapter 14: Diffusive Wave Flood Routing Problem with Lateral Inflow
14.1. Introduction
14.2. Governing Equation
14.2.1. Diffusive Wave Equation without Lateral Inflow
14.2.2. Diffusive Wave Equation with Lateral Inflow
14.3. HAM-Based Solution
14.4. HPM-Based Solution
14.5. OHAM-Based Solution
14.6. Convergence Theorems
14.6.1. Convergence Theorem of the HAM-Based Solution
14.6.2. Convergence Theorem of the OHAM-Based Solution
14.7. Results and Discussion
14.7.1. Selection of Expressions and Parameters
14.7.2. Numerical Convergence and Validation of the HAM Solution
14.7.3. Validation of the HPM-Based Solution
14.7.4. Validation of the OHAM-Based Solution
14.8. Concluding Remarks
References
Further Reading
Chapter 15: Kinematic Wave Equation
15.1. Introduction
15.2. Governing Equation
15.3. Solution Methodologies
15.3.1. Numerical Solution
15.3.2. HAM-Based Solution
15.3.3. HPM-Based Solution
15.3.4. OHAM-Based Solution
15.4. Convergence Theorems
15.4.1. Convergence Theorem of the HAM-Based Solution
15.4.2. Convergence Theorem of the OHAM-Based Solution
15.5. Results and Discussion
15.5.1. Numerical Convergence and Validation of the HAM-Based Solution
15.5.2. Validation of the HPM-Based Solution
15.5.3. Validation of the OHAM-Based Solution
15.6. Concluding Remarks
References
Further Readings
Chapter 16: Multispecies Convection-Dispersion Transport Equation with Variable Parameters
16.1. Introduction
16.2. Governing Equation
16.3. HAM-Based Solution
16.4. HPM-Based Solution
16.5. OHAM-Based Solution
16.6. Convergence Theorems
16.6.1. Convergence Theorem of the HAM-Based Solution
16.6.2. Convergence Theorem of the OHAM-Based Solution
16.7. Results and Discussion
16.7.1. Selection of Expressions and Parameters
16.7.2. Numerical Convergence and Validation of the HAM Solution
16.7.3. Validation of the HPM-Based Solution
16.7.4. Validation of the OHAM-Based Solution
16.8. Concluding Remarks
References
Further Reading
PART V: Integro-Differential Equations
Chapter 17: Absorption Equation in Unsaturated Soil
17.1. Introduction
17.2. Governing Equation
17.3. Philip’s Solution
17.4. HAM Solution
17.5. HPM-Based Solution
17.6. OHAM-Based Solution
17.7. Convergence Theorems
17.7.1. Convergence Theorem of the HAM-Based Solution
17.7.2. Convergence Theorem of the OHAM-Based Solution
17.8. Results and Discussion
17.8.1. Numerical Convergence and Validation of the HAM-Based Solution
17.8.2. Validation of the HPM-Based Solution
17.8.3. Validation of the OHAM-Based Solution
17.9. Concluding Remarks
References
Further Reading
Index
