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 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.

Chapter 4 starts out with a physics motivation, as well as a mathematical statement of the problems that will be tackled in later sections. An extensive section discusses the conditioning of linear-algebra problems: borrowing ideas and examples from matrix perturbation theory, this teaches students what to look out for. Roughly half of the remaining chapter is dedicated to the solution of linear systems of equations, employing methods of varying sophistication: Gaussian elimination, LU decomposiion, pivoting, and the Jacobi iterative method. The second half addresses the eigenvalue problem, again with a variety of methods, including the power method, eigenvalue shifting, and the QR method. Crucially, this includes explicit mathematical derivations of these approaches. A brief introduction to the singular value decomposition is also given, including both an existence proof and a programming implementation. The chapter is rounded out by an extensive physics project, which studies the eigenvalue problem of interacting spins, and a problem set. The physics project patiently builds up matrix quantum mechanics, allowing students to tackle problems of increasing difficulty.

We study the solution of square systems of linear equations with a nonsingular matrices, we provide sufficient conditions for the classical Gaussian elimination with pivoting to proceed without failure. We introduce the LU factorization of a matrix, and several of its variants, like Cholesky factorization.

Vectors and matrices provide a mathematical framework for formulating and solving linear systems of algebraic equations, which have applications in all areas of engineering and the sciences.Solution methods include Gaussian elimination and the matrix inverse.Vectors play a key role in representing various quantities in mechanics as well as providing the bases for vector spaces.Linear transformations allow one to alter such vector spaces to ones that are more convenient for the task at hand.