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 chapter focuses on the project of finding the potential for a given distribution of charges in a two-dimensional system, which does not possess any symmetrical properties, an extension of the cylindrical potential problem discussed in the previous chapter. Using a method of minimising a functional, specifically the Gauss–Seidel method of iterative minimisation, the Poisson’s equation is adjusted to a 2D case, neglecting one partial derivative in Cartesian coordinates. We subsequently derive a discretised form of the functional, leading to a multi-variable function, following which the problem can be solved using the Gauss–Seidel iterative method. The numerical method discussed here is the finite elements method (FEM), with an emphasis on the need for a specific sequence for updating values to optimise computation efficiency. The discussion sheds light on the importance of the uniqueness of solutions in electrostatic systems, thereby exploring a fundamental question in electrostatics. The concluding part of the chapter provides an outline of a numerical algorithm for the problem, suggesting potential modifications and points for further exploration.