Higgs Boson: A Mathematical Survey with Finite Element Method


Harun Selvitopi, PhD – Associate Professor, Mathematics, Erzurum Technical University, Erzurum, Turkey

Series: Computational Mathematics and Analysis
BISAC: COM072000; MAT007020; SCI040000
DOI: https://doi.org/10.52305/TFFL2482

In this book, the finite difference, finite element and the root finding approximations i.e. Newton, Quasi Newton and Broyden methods has been presented. We also consider the finite difference/finite element hybrid method to solve the wave equation in Einstein and de Sitter space-time. The mathematical model of the Higgs Boson equation in de Sitter space-time has been presented as a chapter. Finally, the finite difference/finite element solution using Newton linearization method has been presented.

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Table of Contents



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

viii Contents

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



Author’s ORCID iD

Harun Selvitopi – 0000-0001-5958-7625

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