Homotopy-Based Methods in Water Engineering

Description

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 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