This question is inspired by Gilbert Strang's Course on Computational Science and Engineering, MIT 18.085.(adsbygoogle = window.adsbygoogle || []).push({});

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?

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# I Difference Equation Boundary Conditions0.

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