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

Description

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. Cover Half Title Series Page Title Page Copyright Page Contents Preface About the Author Chapter 1: Stability and Convergence Analysis of Numerical Scheme for the Generalized Fractional Diffusion-Reaction Equation 1.1. Introduction 1.2. Fractional Derivatives Review 1.3. Existence and Uniqueness Via Banach Fixed Theorem 1.4. Numerical Scheme of the Fractional Diffusion Reaction Equation 1.5. Stability Analysis of the Numerical Approximation 1.6. Convergence Analysis of the Numerical Approximation 1.7. The Graphics with the Numerical Scheme 1.8. Conclusion References Chapter 2: Studying on the Complex and Mixed Dark-Bright Travelling Wave Solutions of the Generalized KP-BBM Equation 2.1. Introduction 2.2. The SGEM 2.3. Applications of SGEM and Mathematical Analysis 2.3.1. Investigation of Generalized KP-BBM Equation 2.4. Conclusions References Chapter 3: Abundant Computational and Numerical Solutions of the Fractional Quantum Version of the Relativistic Energy–Momentum Relation 3.1. Introduction 3.2. Analytical Explicit Wave Solutions 3.2.1. Extended exp ( –f(X)) Expansion Method 3.2.2. Extended Fan Expansion Method 3.2.3. Extended (G0 G ) Expansion Method 3.2.4. Improved F-expansion Method 3.2.5. Modified Khater Method 3.3. Stability 3.4. Numerical Solutions 3.4.1. Semi-Analytical Solutions 3.4.2. Numerical Solutions 3.4.2.1. Cubic B-Spline 3.4.2.2. Quantic–B–spline 3.4.2.3. Septic B-Spline 3.5. Figures Representation 3.6. Conclusion References Chapter 4: Applications of Conserved Schemes for Solving Ultra-Relativistic Euler Equations 4.1. Introduction 4.2. The URE Equations 4.2.1. The (p, u) Subsystem 4.3. The Numerical Schemes 4.3.1. Cone Grid Scheme 4.3.2. The Structure of Numerical Solutions 4.4. Numerical Results 4.5. Conclusions References Chapter 5: Notorious Boundary Value Problems: Singularly Perturbed Differential Equations and Their Numerical Treatment 5.1. Introduction 5.2. Layer Adapted Meshes 5.2.1. A Priori Refined Meshes 5.2.1.1. Bakhvalov-Type Meshes 5.2.1.2. Shishkin-Type Meshes 5.2.1.3. Comparison Between Bakhvalov Mesh and Shishkin Mesh 5.2.2. A Posteriori Refined Meshes 5.2.3. Error Estimates and the Construction of A Monitor Function 5.2.3.1. Constructing a Monitor Function from a Priori Error Estimates 5.2.3.2. Constructing a Monitor Function from a Posteriori Error Estimates 5.2.4. Numerical Experiments for Mesh Adaptation on a Test Problem 5.3. Concluding Remarks 5.3.1. Future Directions References Chapter 6: Review on Non-Standard Finite Difference (NSFD) Schemes for Solving Linear and Non-linear Differential Equations 6.1. Introduction 6.2. Non-standard Finite Difference (NSFD) Schemes 6.2.1. Comparison between Standard and Non-Standard Finite Difference Methods 6.2.2. Applications of NSFD scheme 6.2.2.1. Applications to Modelled ODEs 6.2.2.2. Applications to Modelled PDEs 6.2.2.3. Applications to Modelled Fractional Differential Equations 6.3. Conclusions and Scope References Chapter 7: Solutions for Nonlinear Fractional Diffusion Equations with Reaction Terms 7.1. Introduction 7.2. Reaction Diffusion Problem 7.2.1. Linear Case 7.2.2. Nonlinear Case 7.3. Numerical Method 7.3.1. Linear Case 7.3.2. Nonlinear Case 7.4. Final Remarks Acknowledgments References Chapter 8: Convergence of Some High-Order Iterative Methods with Applications to Differential Equations 8.1. Introduction 8.2. Local Convergence Analysis 8.3. Application 8.4. Numerical Example 8.5. Conclusion References Chapter 9: Fractional Derivative Operator on Quarantine and Isolation Principle for COVID-19 9.1. Introduction 9.2. Mathematical Analysis of the Dynamics 9.2.1. Uniqueness and Continuous Dependence of the Solution 9.2.2. Equilibrium for the Dynamics 9.3. Derivation of the Numerical Method 9.4. Numerical Results and Discussions 9.5. Conclusion Acknowledgement References Chapter 10: Superabundant Explicit Wave and Numerical Solutions of the Fractional Isotropic Extension Model of the KdV Model 10.1. Introduction 10.2. Analytical Explicit Wave Solutions 10.2.1. Exp(􀀀f(X))-Expansion Method 10.2.2. Extended Fan-Expansion Method 10.2.3. Extended (G' / G)-Expansion Method 10.2.4. Extended Simplest Equation Method 10.2.5. Extended Tanh(X)-Expansion Method 10.2.6. Modified Khater Method 10.3. Stability 10.4. Numerical Solutions 10.4.1. Semi-Analytical Solutions 10.4.2. Numerical Solutions 10.4.2.1. Cubic BSpline 10.4.2.2. Quantic BSpline 10.4.2.3. Septic BSpline 10.5. Figures and Tables Representation 10.6. CONCLUSION References Chapter 11: A Modified Computational Scheme and Convergence Analysis for Fractional Order Hepatitis E Virus Model 11.1. Introduction 11.2. Elemental Definitions and Formulae 11.3. Mathematical Description Of Hev Model 11.4. q-HASTM: Basic Methodology 11.5. Uniqueness and Convergence Analysis for q-HASTM 11.6. q-HASTM Solution for the Fractional Hev Model 11.7. Numerical Simulations 11.8. Concluding Remarks and Observations References Index