07 - Inhomogeneous Boundary Conditions for FAT

Dealing with Inhomogeneous Boundary Conditions for FAT

Consider homogeneous and inhomogeneous problems

$$\begin{array}{rlrl}Ly& =0& & (H)\\ Ly& =f\ne 0& & (N)\end{array}$$

for $0<x<a$

Suppose the linear homogenous boundary conditions $(NBC)$ are of form

$$\begin{array}{c}\begin{array}{c}{B}_{1}y={\alpha }_{1}y(a)+{\alpha }_2{y}^{\prime }(a)+{\beta }_{1}y(b)+{\beta }_2{y}^{\prime }(b)={\gamma }_1\\ B_{2}y={\alpha }_{3}y(a)+{\alpha }_4{y}^{\prime }(a)+{\beta }_{3}y(b)+{\beta }_4{y}^{\prime }(b)={\gamma }_2\end{array}\}\text{(NBC)}\end{array}$$

where $({\alpha }_1,{\alpha }_2,{\beta }_1,{\beta }_2)$ and $({\alpha }_3,{\alpha }_4,{\beta }_3,{\beta }_4)$ are linearly independent and constants ${\gamma }_1$ and ${\gamma }_2$

Let $v(x)$ be a twice differentiable function that satisfies conditions $(NBC)$
Let

$$\tilde{y}(x)=y(x)-v(x)$$

Hence $\tilde{y}$ satisfies

$$L\tilde{y}=f-Lv=\tilde{f}$$

with homogeneous boundary conditions

$$B_1\tilde{y}=0=B_2\tilde{y}$$

Hence can apply Fredholm Alternative (Linear ODE Version)