## Linear Algebra and Its Applications, Exercise 1.7.4

Exercise 1.7.4. For the differential equation with $f(x) = 4\pi^2 \sin 2\pi x$ the corresponding difference equation is $\begin{bmatrix} 2&-1&0 \\ -1&2&-1 \\ 0&-1&2 \end{bmatrix} \begin{bmatrix} u_1 \\ u_2 \\ u_3 \end{bmatrix} = \frac{\pi^2}{4} \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}$

for $h = \frac{1}{4}$. Solve the above equation for $u_1, u_2, u_3$ and compare their values to the true solution $u = \sin 4\pi x$ at $x = \frac{1}{4}$ $x = \frac{1}{2}$ and $x = \frac{3}{4}$.

Answer: We do Gaussian elimination on the difference matrix and the right-hand side: $\begin{bmatrix} 2&-1&0 \\ -1&2&-1 \\ 0&-1&2 \end{bmatrix} \quad \frac{\pi^2}{4} \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix} \rightarrow \begin{bmatrix} 2&-1&0 \\ 0&\frac{3}{2} &-1 \\ 0&-1&2 \end{bmatrix} \quad \frac{\pi^2}{4} \begin{bmatrix} 1 \\ \frac{1}{2} \\ -1 \end{bmatrix}$ $\rightarrow \begin{bmatrix} 2&-1&0 \\ 0&\frac{3}{2} &-1 \\ 0&0&\frac{4}{3} \end{bmatrix} \quad \frac{\pi^2}{4} \begin{bmatrix} 1 \\ \frac{1}{2} \\ -\frac{2}{3} \end{bmatrix}$

Solving for $u_3$ we have $\frac{4}{3}u_3 = \frac{\pi^2}{4}(-\frac{2}{3}) \rightarrow u_3 = \frac{3}{4} \frac{\pi^2}{4} (-\frac{2}{3}) = -\frac{\pi^2}{8} \approx -1.23$

Solving for $u_2$ we have $\frac{3}{2}u_2 - u_3 = \frac{\pi^2}{4} \frac{1}{2} \rightarrow \frac{3}{2}u_2 = \frac{\pi^2}{8} - \frac{\pi^2}{8} = 0 \rightarrow u_2 = 0$

Finally we solve for $u_1$: $2u_1 - u_2 = \frac{\pi^2}{4} \rightarrow 2u_1 = \frac{\pi^2}{4} + 0 \rightarrow u_1 = \frac{\pi^2}{8} \approx 1.23$

For the exact solution $u(x) = \sin 2\pi x$ we have $u(\frac{1}{4}) = \sin 2 \pi \frac{1}{4} = \sin \frac{\pi}{2} = 1$ $u(\frac{1}{2}) = \sin 2 \pi \frac{1}{2} = \sin \pi = 0$ $u(\frac{3}{4}) = \sin 2 \pi \frac{3}{4} = \sin \frac{3\pi}{2} = 1$

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition , Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books .

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