Let us consider standard coupled differential equations in their compact forms:
where Cj, i,j = 1, 2, ..., n and ф, , i = 1,2 , ... , n are the coefficients and flux of the system.
To find the approximate flux (ф), Galarkin's weighted residue method has been used here. In this method, we need the residue equation, which can be obtained from Equation 7.7. Here, the residue equation is
Using linear interpolation (linear element discretization), the weight functions or the shape functions are obtained. In vector form, these functions will be
where l is the length of each element of the domain. Then, the shape functions are multiplied with the residue equations and integrated over the domain.
If we consider only two equations having two dependent variables, then the residue will be
Now, Equation 7.9 is multiplied with the shape functions and it gives
Integrating Equation 7.10 over the domain Q, we get
Now, adjusting the boundary condition Equation 7.11 can be solved (which is a set of algebraic equations) and unknowns are obtained. This concept is used to investigate the multigroup neutron diffusion equation, which is discussed in the following section.