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

✍ Scribed by Harendra Singh (editor), Jagdev Singh (editor), Sunil Dutt Purohit (editor), Devendra Kumar (editor)

• Publisher: CRC Press
• Year: 2021
• Tongue: English
• Leaves: 337
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

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