The methods presented in this book primarily focus on differential
equations, both ordinary (ODE) and partial (PDE), while also covering
basic mathematical operations, derivatives, and quadrature, as intro
duced in the second project (Diffraction on a Slit). The book addresses
typical problems such as initial value problems (IVP), boundary value
problems (BVP), and eigenvalue problems (EVP). Most of the methods
discussed are based on variable discretisation and recursive algorithms
for ODEs. For PDEs, the methods of finite difference (FD) and finite
elements (FE) are explained, along with selected matrix methods for
solving systems of equations (FD) and chosen iterative optimisation
methods (FE). In the case of PDEs, system symmetry is utilised to
reduce dimensionality from 3D to 1D, simplifying implementation and
allowing projects to be completed within reasonable time constraints.
Author(s): Paweł Scharoch
Publisher: Cambridge university press
Year: 2024
Language: English
Pages: 130
Pawet Scharoch
M a c i e j P. P o l a k
Rad o staw
Szymon
Katarzyna
H o t o d n i k -
M a t e c k a
@1 CAMBRIDGE
CAMBRIDGE
Contents
Preface
How to Use the Book
First Steps
Basic Mathematical Operations
0.1 Finding Roots of a 1D Function
0.2 Finding Minimum of a 1D Function
0.3 Exercises
Rectangular Finite Quantum
Well - Stationary Schrodinger Equation in 1D
1.1 Physics Background: Chosen Ideas of Quantum Mechanics
1.2 Problem: Eigenenergies and Eigenfunctions in Rectangular Finite Quantum Well
1.3 Numerical Methods: Finding Roots of Characteristic Functions
1.4 Exercises
Diffraction of Light on a Slit
2.1 Physics Background: Elements of Wave Physics
2.2 Problem: Diffraction of a Wave on a Slit
2.3 Numerical Methods: Schemes Based on Local Approximations of a Function
2.4 Exercises
Pendulum as a Standard
of the Unit of Time
3.1 Physics Background: Newton’s Laws of Motion, Equation of Motion
3.2 Problem: Simple Pendulum as a Standard of the Unit of Time
3.3 Numerical Methods: Recursive Methods Based on Local Extrapolation of One-Step Integral Integrand
3.4 Exercises
Planetary System
4.1 Physics Background: Law of Universal
Gravitation
4.2 Problem: Motion of Planets in the Field of a
Fixed Star
4.3 Reduction of a Single Planet Motion in a Central Field to 1D
4.4 Numerical Method: Verlet Algorithm
4.5 Exercises
Gravitation inside a Star
5.1 Physics Background: Gauss’s Law, Poisson’s Equation
5.2 Problem: Gravitational Field Due to a
Continuous Mass Density Distribution
5.3 Numerical Method: Numerov-Cowells
Algorithm
5.4 Exercises
Normal Modes in a Cylindrical Waveguide
6.1 Physics Background: Wave Equation, Standing Waves
6.2 Problem: Normal Modes in an Optical Fibre
6.3 Numerical Method: Shooting Method
6.4 Exercises
Thermal Insulation Properties of a Wall
7.1 Physics Background: Steady-State Diffusion
7.2 Problem: Steady-State Diffusion of Heat through the Wall
7.3 Numerical Method: Finite Difference Method
7.4 Exercises
Cylindrical Capacitor
8.1 Physics Background: Variational Principle for Electrostatic Systems
8.2 Problem: Cylindrical Capacitor
8.3 Numerical Method: Finite Elements (FE) Method
8.4 Exercises
Advanced Projects
Coupled Harmonic Oscillators
9.1 Problem: Equations of Motion of Coupled Oscillators
9.2 Exercises
The Fermi-Pasta-Ulam-Tsingou Problem
10.1 Problem: Dynamics of a One-Dimensional Chain of Interacting Point Masses
10.2 Exercises
Hydrogen Star
11.1 Problem: Mass Density Distribution in a Cold Hydrogen Star
11.2 Numerical Method
11.3 Exercises
Rectangular Quantum Well Filled with
Electrons - The Idea of Self-Consistent Calculations
12.1 Problem: Quantum Well Filled with Electrons and Charge Neutralising Jellium
12.2 Exercises
Time-Dependent Schrodinger Equation
13.1 Problem: Time Evolution of a Wave Function
in 2D Quantum Well
13.2 Exercises
Poisson’s Equation in 2D
14.1 Problem: Variational Computational
Approach to a 2D Electrostatic System
14.2 Numerical Method: Finite Elements (FE)
Method
14.3 Exercises
Appendix A
Supplementary Materials
A.1 Euler Representation of a Complex Number
A.2 Local Representation of a Function as a Power Series
A.2.3 Lagrange Polynomials
A.3 Wilberforce’s Pendulum
A.4 Dispersion Relation for FPU Problem
A.5 Equivalence of Variational Principle and Poisson’s Equation in Electrostatics
A.6 First and Second Uniqueness Theory
A.6.1 First Uniqueness Theory
A.6.2 Second Uniqueness Theory
A.7 Discretisation of 2D Laplace’s Functional
A.8 Density of Star
A.9 Energy and Pressure of a Cubic Hydrogen
Atom Lattice as a Function of Unit Cell Volume
Further Reading
Index
