Advanced Numerical Methods for Differential Equations: Applications in Science and Engineering (Mathematics and its Applications)

"Mathematical models are used to convert real-life problems using mathematical concepts and language. These models are governed by differential equations whose solutions make it easy to understand real-life problems and can be applied to engineering and science disciplines. This book presents numerical methods for solving various mathematical models. This book offers real-life applications, includes research problems on numerical treatment, and shows how to develop the numerical methods for solving problems. The book also covers theory and applications in engineering and science. Engineers, mathematicians, scientists, and researchers working on real-life mathematical problems will find this book useful"-- Provided by publisher Mathematical models are used to convert real-life problems using mathematical concepts and language. These models are governed by differential equations whose solutions make it easy to understand the real-life problems. The applications can be applied to engineering and science disciplines. Cover 1 Half Title 2 Series Page 3 Title Page 4 Copyright Page 5 Contents 6 Preface 12 About the Author 16 Chapter 1: Stability and Convergence Analysis of Numerical Scheme for the Generalized Fractional Diffusion-Reaction Equation 18 1.1. Introduction 19 1.2. Fractional Derivatives Review 20 1.3. Existence and Uniqueness Via Banach Fixed Theorem 22 1.4. Numerical Scheme of the Fractional Diffusion Reaction Equation 24 1.5. Stability Analysis of the Numerical Approximation 25 1.6. Convergence Analysis of the Numerical Approximation 27 1.7. The Graphics with the Numerical Scheme 29 1.8. Conclusion 31 References 31 Chapter 2: Studying on the Complex and Mixed Dark-Bright Travelling Wave Solutions of the Generalized KP-BBM Equation 34 2.1. Introduction 34 2.2. The SGEM 36 2.3. Applications of SGEM and Mathematical Analysis 37 2.3.1. Investigation of Generalized KP-BBM Equation 37 2.4. Conclusions 39 References 52 Chapter 3: Abundant Computational and Numerical Solutions of the Fractional Quantum Version of the Relativistic Energy–Momentum Relation 56 3.1. Introduction 57 3.2. Analytical Explicit Wave Solutions 60 3.2.1. Extended exp ( –f(X)) Expansion Method 60 3.2.2. Extended Fan Expansion Method 66 3.2.3. Extended (G0 G ) Expansion Method 79 3.2.4. Improved F-expansion Method 81 3.2.5. Modified Khater Method 84 3.3. Stability 88 3.4. Numerical Solutions 89 3.4.1. Semi-Analytical Solutions 89 3.4.2. Numerical Solutions 90 3.4.2.1. Cubic B-Spline 91 3.4.2.2. Quantic–B–spline 93 3.4.2.3. Septic B-Spline 94 3.5. Figures Representation 95 3.6. Conclusion 98 References 98 Chapter 4: Applications of Conserved Schemes for Solving Ultra-Relativistic Euler Equations 104 4.1. Introduction 104 4.2. The URE Equations 106 4.2.1. The (p, u) Subsystem 107 4.3. The Numerical Schemes 109 4.3.1. Cone Grid Scheme 109 4.3.2. The Structure of Numerical Solutions 110 4.4. Numerical Results 112 4.5. Conclusions 121 References 122 Chapter 5: Notorious Boundary Value Problems: Singularly Perturbed Differential Equations and Their Numerical Treatment 126 5.1. Introduction 127 5.2. Layer Adapted Meshes 129 5.2.1. A Priori Refined Meshes 130 5.2.1.1. Bakhvalov-Type Meshes 130 5.2.1.2. Shishkin-Type Meshes 132 5.2.1.3. Comparison Between Bakhvalov Mesh and Shishkin Mesh 133 5.2.2. A Posteriori Refined Meshes 136 5.2.3. Error Estimates and the Construction of A Monitor Function 139 5.2.3.1. Constructing a Monitor Function from a Priori Error Estimates 139 5.2.3.2. Constructing a Monitor Function from a Posteriori Error Estimates 140 5.2.4. Numerical Experiments for Mesh Adaptation on a Test Problem 142 5.3. Concluding Remarks 145 5.3.1. Future Directions 149 References 149 Chapter 6: Review on Non-Standard Finite Difference (NSFD) Schemes for Solving Linear and Non-linear Differential Equations 152 6.1. Introduction 152 6.2. Non-standard Finite Difference (NSFD) Schemes 159 6.2.1. Comparison between Standard and Non-Standard Finite Difference Methods 160 6.2.2. Applications of NSFD scheme 160 6.2.2.1. Applications to Modelled ODEs 161 6.2.2.2. Applications to Modelled PDEs 164 6.2.2.3. Applications to Modelled Fractional Differential Equations 165 6.3. Conclusions and Scope 166 References 168 Chapter 7: Solutions for Nonlinear Fractional Diffusion Equations with Reaction Terms 172 7.1. Introduction 172 7.2. Reaction Diffusion Problem 174 7.2.1. Linear Case 174 7.2.2. Nonlinear Case 179 7.3. Numerical Method 182 7.3.1. Linear Case 183 7.3.2. Nonlinear Case 190 7.4. Final Remarks 198 Acknowledgments 200 References 200 Chapter 8: Convergence of Some High-Order Iterative Methods with Applications to Differential Equations 204 8.1. Introduction 204 8.2. Local Convergence Analysis 206 8.3. Application 215 8.4. Numerical Example 217 8.5. Conclusion 219 References 219 Chapter 9: Fractional Derivative Operator on Quarantine and Isolation Principle for COVID-19 222 9.1. Introduction 222 9.2. Mathematical Analysis of the Dynamics 227 9.2.1. Uniqueness and Continuous Dependence of the Solution 230 9.2.2. Equilibrium for the Dynamics 231 9.3. Derivation of the Numerical Method 233 9.4. Numerical Results and Discussions 236 9.5. Conclusion 239 Acknowledgement 240 References 240 Chapter 10: Superabundant Explicit Wave and Numerical Solutions of the Fractional Isotropic Extension Model of the KdV Model 244 10.1. Introduction 245 10.2. Analytical Explicit Wave Solutions 246 10.2.1. Exp(􀀀f(X))-Expansion Method 246 10.2.2. Extended Fan-Expansion Method 250 10.2.3. Extended (G' / G)-Expansion Method 257 10.2.4. Extended Simplest Equation Method 260 10.2.5. Extended Tanh(X)-Expansion Method 262 10.2.6. Modified Khater Method 264 10.3. Stability 269 10.4. Numerical Solutions 270 10.4.1. Semi-Analytical Solutions 270 10.4.2. Numerical Solutions 272 10.4.2.1. Cubic BSpline 272 10.4.2.2. Quantic BSpline 273 10.4.2.3. Septic BSpline 275 10.5. Figures and Tables Representation 275 10.6. CONCLUSION 292 References 292 Chapter 11: A Modified Computational Scheme and Convergence Analysis for Fractional Order Hepatitis E Virus Model 296 11.1. Introduction 296 11.2. Elemental Definitions and Formulae 298 11.3. Mathematical Description Of Hev Model 299 11.4. q-HASTM: Basic Methodology 302 11.5. Uniqueness and Convergence Analysis for q-HASTM 306 11.6. q-HASTM Solution for the Fractional Hev Model 308 11.7. Numerical Simulations 319 11.8. Concluding Remarks and Observations 326 References 327 Index 330 Computational,Methods;,Non-Linear;,Numerical,Analysis;,Integral,Equations;,Mathematical,Modelling;,Fractional,Differential,Equations Computational Methods,Non-Linear,Numerical Analysis,Integral Equations,Mathematical Modelling,Fractional Differential Equations