This project focuses on the Initial Value Problem (IVP) for ordinary differential equations with the application of multipoint recursion schemes. The effectiveness and convergence of these schemes are explored and subsequently applied to examine the properties of a compound pendulum, specifically the dependence of the oscillation period on energy. The chapter then focuses on Newton’s laws of motion, laying the foundation for understanding the motion equation. The project uses a simple pendulum to illustrate the concept, looking at how changes in amplitude affect the period of harmonic oscillations. Numerical methods, such as recursive methods based on local extrapolation, are then employed to derive formulas. The project concludes by discussing the integration of Runge–Kutta methods and implicit schemes to solve the equations. This project ultimately questions the viability of the pendulum as a standard unit of time, adding value to ongoing discussions in physics and mathematics education.

This project explores the phenomenon of coupled harmonic oscillators, which have a broad relevance in fields such as mechanics, atomic and quantum physics, optics, electronics, and biology. The paper expands on a model of a harmonic oscillator, using numerical methods for solving differential equations to analyse systems of coupled oscillators. The study focuses on two illustrative cases: a one-dimensional system of two point masses suspended from springs, and a system of two simple pendulums moving in the same plane. The derived mathematical equations of motion provide a comprehensive framework for understanding the behaviour of such systems. Through computational experimentation, the project aims to elucidate the oscillatory behaviour of these systems depending on different parameters and their coupling, particularly focusing on energy transfer between the oscillators. The Runge–Kutta algorithm is employed for solving the initial value problem (IVP) for the ordinary differential equations (ODE) governing these systems. The project underscores the versatility of the harmonic oscillator model by showing that different physical systems can be described by the same mathematical model.

This project involves the application of molecular dynamics (MD) to a simple two-dimensional planetary system consisting of two planets and a fixed star. The primary focus is to construct a MD code using Newton’s law of universal gravitation as the interaction law and the Verlet algorithm for solving the initial value problem. The project examines the gravitational interaction described by Newton’s laws, focusing on the law of universal gravitation and its application to the planetary system. It further explores the principle of equivalence, the concept of conservative force, and the effective potential energy of the system. The discussion also covers the reduction of a single planet motion to one dimension, which offers insights into the trajectory of the planetary system. Finally, the project outlines the numerical approach using the Verlet algorithm for simulating the motion of the planets. The comprehensive understanding of the gravitational interactions and the computational techniques provide a solid foundation for the study of complex dynamical systems.

This chapter discusses the application of the variational principle and finite element (FE) methods to electrostatic systems, using a cylindrical capacitor as a representative example. The variational principle, which identifies the solution that minimises a system’s total energy, provides a foundation for numerical techniques such as the FE methods. These methods offer flexibility in the selection of points in the independent variable space, which can be adapted according to the expected function behaviour. The system of focus is a capacitor comprising two coaxial metallic cylinders, chosen for its high symmetry and real-world relevance in areas like telecommunication cables. The chapter presents the Poisson’s equation, demonstrates the variational principle, and introduces the FE method. A numerical example of a cylindrical capacitor is also provided, illustrating how to convert the problem into one of functional minimisation using Poisson’s equation and then solve it using the FE method. The Gauss–Seidel iterative minimisation procedure is used in the solution process. The discussion provides a foundation for extending these methods to 2D and 3D systems in subsequent chapters.

This chapter exploresthe Fermi–Pasta–Ulam–Tsingou (FPUT) problem in the context of a one-dimensional chain of interacting point masses. We start by modelling the system as a chain of masses interacting through a force dependent on their relative displacements. Next, we simplify this system to harmonic oscillators under a linear force dependency, further developing it to describe a wave-like behaviour. The chapter discusses dispersion relations and the impact of boundary conditions leading to discretisation of allowed wave modes. A non-linear, second-order interaction is then included, complicating the system’s dynamics and necessitating the use of numerical methods for its solution. We then track the system’s evolution in a multidimensional phase space, leading to observations of seemingly chaotic motion with emerging periodicity. Energy conservation and its flow through the system are crucial aspects of the analysis. A detailed numerical procedure is provided, involving solution of initial value problems, mode projections, and energy computations to explore the complex behaviour inherent in the FPUT problem.

This project uses the procedure of root-finding to resolve the eigenvalue problem of a rectangular quantum well. The procedure is applied to determine the first two to three energy levels of a simplified model of a hydrogen atom, represented by the rectangular quantum well. The project also explores the eigenvalue problem within various physics fields. While the project involves simple mathematical operations, it is rooted in complex physical concepts like quantum mechanics, often unfamiliar to first-year students. The fundamentals of quantum mechanics are introduced, providing enough understanding for successful project execution. The project initially focuses on the central object, the quantum state, and its probabilistic nature in quantum mechanics. The Schrödinger equation, an eigenvalue problem, is used to find state functions. This project explores eigenenergies and eigenfunctions within a rectangular finite quantum well, treating the well as a simplistic 1D model of the hydrogen atom.

This chapter covers the problem of a single quantum well filled with electrons, specifically, the changes that occur when electrons are introduced into an empty quantum well. Utilising the ’jellium’ model, the chapter commences by identifying the energy levels of an empty quantum well composed of infinite dipole planes filled with positively charged jellium. The subsequent introduction of electrons leads to a significant change in the well’s shape due to interactions within a non-uniform electron gas. However, the complexity of this problem is simplified by considering only Hartree interactions, allowing for self-consistent calculations. The density of the negative charge introduced is determined, then added to the uniform density of the jellium, resulting in a new potential energy shape of the well. The iterative process of determining new energy levels, populating them with electrons, and re-evaluating continues until a chosen property converges. The chapter concludes by demonstrating how to model the potential of an empty well and how this potential changes in a self-consistent procedure when the well is filled with electrons.

This project explores the boundary value problem (BVP) for ordinary differential equations concerning the gravitational field inside a star. The study equates the problem to the electric field inside an atom and reduces the partial differential equation of Poisson’s type to a second order ordinary differential equation, utilising high symmetry. The uniqueness of the solution is ensured by applying two conditions at two ends of the independent variable range. The Numerov’s (Cowell’s) algorithm is employed to solve the equation accurately. However, it is identified that numerical solutions can be very sensitive to the value chosen for the second point, necessitating a recursive scheme. The project also introduces the application of Gauss law, Poisson’s equation, and the Numerov–Cowells algorithm in determining the gravitational potential inside a star given a model radial mass density distribution. The study concludes by discussing the possibility of treating the recursive formula as a tridiagonal system of linear equations and solving it with Gaussian elimination with backward substitution algorithm.

This chapter addresses the eigenvalue problem (EVP) with a focus on its application to describing standing waves or stationary quantum systems. A numerical method known as the shooting method is introduced to solve the EVP. Using a cylindrical waveguide (e.g., optical fibre) as a system model, the normal modes of a scalar wave are explored. The wave equation is examined, with considerations made for axial symmetry, boundary conditions, and the influence of refraction coefficients. A significant part of the study is devoted to finding the normal modes and associated wave numbers in an optical fibre. The latter part of the chapter presents the shooting method, a recursive technique to ascertain the eigenvalue in numerical calculations. The applicability of this method is further examined in the context of a quantum well. This chapter offers a thorough exploration of the EVP, highlighting its relevance to real-world research and introducing a robust numerical method for its resolution.

This work introduces and explores the thermal insulation properties of a house wall using the partial differential equation method of finite difference (FD). By applying the steady-state diffusion equation, we delve into how the temperature across the wall depends on the thermal conductivity distribution of insulating material. Our study assumes a quasi-1D case where heat diffusion occurs through the wall. We stipulate that the wall interfaces with heat reservoirs on both sides, thereby stabilising the temperature, and that there are no heat sources within the wall itself. We then employ the FD method to transform the boundary value problem for the differential equation into a system of linear equations. An efficient Gaussian elimination with back substitution algorithm is applied to solve this system. This technique simplifies the problem, requiring only two sweeps of arithmetic operations of the order ’N’ to find the solution. The FD method’s limitation – requiring the domain’s shape and the grid to fit the chosen coordinate system – is acknowledged, hinting towards the next chapter’s discussion on finite elements (FE) methods.

This project looks into the time evolution of a wave function within a two-dimensional quantum well. We start by solving the time-dependent Schrödinger equation for stationary states in a quantum well. Next, we express any wave function as a linear combination of stationary states, allowing us to understand their time evolution. Two methods are presented: one relies on decomposing the wave function into a basis of stationary states and the other on discretisation of the time-dependent Schrödinger equation, incorporating three-point formulas for derivatives. These approaches necessitate confronting intricate boundary conditions and require maintaining energy conservation for numerical accuracy. We further demonstrate the methods using a wave packet, revealing fundamental phenomena in quantum physics. Our results demonstrate the utility of these methods in understanding quantum systems, despite the challenges in determining stationary states for a given potential. This study enhances our comprehension of the dynamics of quantum states in constrained systems, essential for fields like quantum computing and nanotechnology.

This chapter focuses on the numerical simulation of light diffraction by single or multiple slits, which serves to illustrate key principles of wave physics and interference. Students will become acquainted with numerical differentiation and quadrature procedures, particularly in relation to grid parameter convergence. The physics background emphasises wave physics elements, such as the superposition principle and phase difference, as well as their practical applications in real systems. Concepts such as optical paths and coherence are addressed. To understand diffraction phenomena, the Huygens principle is introduced, leading to the diffraction integral formulation for infinite slits. The chapter then explores numerical methods based on local approximations of functions, such as the two-, three-, and five-point schemes for derivatives. This study culminates in the presentation of quadrature schemes, the application of power series expansions for numerical differentiation, and the Simpson algorithm for accurate numerical integration.

Unlike in Chapter 5, this project aims at finding a real mass density distribution of a hydrogen star of given mass. For that purpose an equilibrium condition for the gravitational and pressure-induced forces acting on a mass element is utilised. Using the integral form of Gauss’s law and the equation of state, we establish an integro-differential equation describing the mass density distribution. To numerically solve the integro-differential equation, we adapt the Adams–Bashforth method and implement a linear extrapolation based on known data points. This approach involves modelling the star as a gas under pressure using an exponential form for the equation of state, which helps in avoiding gravitational collapse. The equation of state is derived based on density functional theory data. We also discuss the constraints of this model and the significance of the parameters within it. The chapter concludes by suggesting potential numerical experiments to examine the influence of these parameters and their physical interpretation. This analysis aims to provide a more comprehensive understanding of stellar structure and the behaviour of mass density distribution within stars.