Table of Contents

Preface

Introduction

1. Finite Difference Method for the Solution of Partial Differential Equations (PDE)
1.1 FDM for Elliptic Equations
1.2 FDM for Parabolic Equations
1.2.1 Forward-Difference Method
1.2.2 Backward-Difference Method
1.2.3 Crank-Nicolson Method
1.2.4 Stability Analysis of Finite Difference Method
1.2.5 Exercises
1.3 FDM for Hyperbolic Partial Differential Equations
1.3.1 Stability Analysis

2. Finite Element Method for the Solution of Partial Differential Equations (PDE)
2.1 The Development of the Finite Element Method
2.2 Application of Finite Element Method to Partial Differential Equations
2.2.1 Discretization
2.2.2 Elements
2.2.3 2-D Elements
2.2.4 Shape Functions
2.2.5 Triangular Elements Shape Functions
2.2.6 Finite Element Method Formulation for Laplace Equation
2.2.7 Weak Form for Laplace Equation
2.2.8 Variational Form for Laplace Equation

3. The Numerical Integration Methods
3.1 Newton-Type Integration Methods
3.1.1 Trapezium Rule
3.1.2 Simpson Method
3.2 Gauss-Type Integration Methods
3.2.1 Gauss-Legendre Method
3.3 Two-Dimensional Numerical Integration

4. Finite Element Method Solution of Laplace Equation

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5. One-Dimensional Wave Equation
5.1 Finite Element Solution of One-Dimensional Wave Equation
5.2 Central Difference Approximation

6. Finite Element Simulation of the One- and Two-Dimensional Wave Equation in Einstein and de Sitter Space-Time
6.1 Application of the Numerical Method for One-Dimensional Problem
6.2 Application of the Numerical Method for Two-Dimensional Problem

7. Root Finding Approximations
7.1 Newton’s Method
7.1.1 Application of Newton Method
7.2 Secant Method

8. Newton’s Method for Nonlinear System of Equations in n􀀀Dimension
8.1 Newton Method
8.2 Quasi-Newton Method
8.3 Broyden Method

9. Finite Difference=Galerkin Finite Element Simulation of the Semi-Linear Wave Equation with Scale-Invariant Damping, Mass and Power Nonlinearity
9.1 Application of the GFEM
9.2 Application of the FDM
9.3 Application of Newton Method

10. Higgs Boson in de Sitter Space-Time
10.1 Mathematical Model of the Higgs Boson in de Sitter Space-Time
10.2 Finite Element Method for Higgs Boson Equation
10.2.1 Newton Method
10.3 Numerical Results

References

Index

Author’s ORCID iD

Harun Selvitopi – 0000-0001-5958-7625