# One-Group Model

In this chapter, various methods have been discussed to investigate the one-group model of neutron diffusion problems. These problems may be solved by analytical methods for simple cases only. When analytical methods may not be used to solve, then we may utilize numerical methods such as the finite difference method (FDM) and the finite element method (FEM).

## Analytical Method

In physical problems, we may have some specified conditions known as the initial orboundary conditions, which are to be considered with the governing differential equation.The solution of the differential equation satisfies these initial boundary conditions. Hence,the differential equation, together with these initial or boundary conditions, forms an initial value or boundary value problem. The problems which are governed by ordinary differential equations are solved in twosteps. First, the general solution is to be found and then we determine the arbitrary constants from the initial values. But the same process is not applicable to problems involvingpartial differential equations. In partial differential equations, the general solution contains arbitrary functions, which are difficult to be adjusted to satisfy the given boundaryconditions. In the following sections, a few methods are discussed for the boundary valueproblems (linear partial differential equations).