Difference Equation Boundary Conditions0.

[MisterX]

This question is inspired by Gilbert Strang's Course on Computational Science and Engineering, MIT 18.085.

Consider the three matrices
Fixed-Fixed
$$K=\begin{bmatrix}
2 &-1 & 0 &0 \\
-1&2 & -1 &0 \\
0 & -1 &2 & -1 \\
0 & 0 & -1 & 2 \\
\end{bmatrix} $$
Free-Fixed
$$T=\begin{bmatrix}
1 &-1 & 0 &0 \\

-1&2 & -1 &0 \\

0 & -1 &2 & -1 \\

0 & 0 & -1 & 2 \\
\end{bmatrix} $$
Free-Free
$$B=\begin{bmatrix}

1 &-1 & 0 &0 \\

-1&2 & -1 &0 \\

0 & -1 &2 & -1 \\

0 & 0 & -1 & 1 \\

\end{bmatrix} $$
$$ \mathbf{u} =\begin{bmatrix}
u_1 \\ u_2 \\ u_3 \\ u_4
\end{bmatrix} $$
The problem is to solve ##(\text{scale})A\mathbf{u} = \mathbf{b} ## where##A=K,T, \text{ or } B ## and ##\mathbf{b} ## is (I presume) an arbitrary source vector. This should be the finite difference solution corresponding to
$$\frac{\partial^2 }{\partial x^2} u(x) = b(x)$$
subject to some boundary conditions.
I what way do these matrices correspond the boundary conditions described when trying to solve the equation? I think continuous constraints may take the form of specific values or slopes of ##u## at the boundaries. How does that get translated into a finite difference? Also, could anyone explain his comment here?

 

[jvicens]

I would first like to clarify that the matrices shown here are known as stiffness matrices in the field of computational science and engineering. These matrices represent the stiffness of a system and are used in finite difference methods to solve differential equations.

In this case, the differential equation being solved is the second-order partial differential equation ##\frac{\partial^2}{\partial x^2} u(x) = b(x)##, where ##u(x)## represents the unknown function and ##b(x)## represents the source term. The boundary conditions for this equation can be imposed in different ways, which are reflected in the different stiffness matrices shown.

The Fixed-Fixed matrix, ##K##, corresponds to the boundary conditions where the function ##u(x)## is fixed at both ends of the domain. This means that the values of ##u## at the first and last grid points are known and cannot change. This is similar to fixing the endpoints of a string or beam in a physical system.

The Free-Fixed matrix, ##T##, corresponds to the boundary conditions where the function ##u(x)## is free at one end and fixed at the other end of the domain. This means that the value of ##u## at the first grid point is known and cannot change, but the value at the last grid point is free to vary. This is similar to fixing one endpoint of a string or beam while allowing the other endpoint to move.

The Free-Free matrix, ##B##, corresponds to the boundary conditions where the function ##u(x)## is free at both ends of the domain. This means that the values of ##u## at the first and last grid points are free to vary. This is similar to having a string or beam that is not fixed at either end and can move freely.

In terms of finite difference methods, these boundary conditions are enforced by using different types of grid points at the boundaries. For example, for the Fixed-Fixed matrix, the first and last grid points are known and are not included in the system of equations. For the Free-Fixed matrix, the first grid point is known and the last grid point is included in the system of equations. And for the Free-Free matrix, both the first and last grid points are included in the system of equations.

As for Gilbert Strang's comment, he is referring to the fact that the stiffness matrices for different boundary conditions will have different properties, such as symmetry or positive definit