4.2.5 Imposition of the Boundary Conditions

The set of boundary conditions at the boundary points x=0 and x=L, presented in equation (4.2), can be imposed on the assembled set of finite element equations (4.24), as depicted in (4.4). The Dirichlet boundary condition (φ(x=0)=0) implies that c₁=0, while the Neumann boundary condition requires that

\[\begin{align} \left(a\frac{du^{2}_\text{2}}{dx}\right)\Bigg\vert _{x=x_\text{3}}=\beta_\text{l} \quad \text{at} \quad x_\text{3}=L. \end{align}\] (4.25)

The application of the boundary conditions of the problem on the assembled set of equations reduces equation (4.24) to

\[\begin{equation} \begin{bmatrix} K^{1}_\text{22}+K^{2}_\text{11} & K^{2}_\text{12} \\[2pt] K^{2}_\text{21} & K^{2}_\text{22} \\ \end{bmatrix} \begin{Bmatrix} c_\text{2} \\[2pt] c_\text{3} \\ \end{Bmatrix} = \begin{Bmatrix} f^{1}_\text{2}+f^{2}_\text{1} \\[2pt] f^{2}_\text{2} \\ \end{Bmatrix} +\begin{Bmatrix} 0 \\[2pt] \beta_\text{l} \\ \end{Bmatrix}. \end{equation}\] (4.26)

In this way, the assembled matrix contains two equations in two unknowns c₂ and c₃, which are the nodal values of the solution.