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Consider $\frac{d^{2} u}{d x^{2}}=f(x)$ with u(0) = 0 and u(L) = 0. (a) Using the centered difference approximation for the second derivative and dividing the length L into three equal mesh lengths, derive a system of linear equations for an approximation to u(x). Use the notation $x_{i}=i \Delta x, f_{i}=f\left(x_{i}\right),$ and $u_{i}=u\left(x_{i}\right).$ (Note: $x_{0}=0, x_{1}=\frac{1}{3} L, x_{2}=\frac{2}{3} L, \left.x_{3}=L .\right)$ (b) Write the system as a matrix system Au = f. What is A? (c) Solve for $u_{1}$ and $u_{2}.$ (d) Show that a “Green’s function” matrix G can be defined: $u_{i}=\sum_{j} G_{i j} f_{j} \quad(\boldsymbol{u}=\boldsymbol{G} \boldsymbol{f}).$ What is G? Show that it is symmetric, $G_{i j}=G_{j i}.$

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Complete.

$18 \ \mathrm{mi}=$ $\underline{\qquad}\mathrm{ft}$

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$18$ mi $=18\cdot 5280$ ft $=95040$ ft

1 mi = 5280 ft

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Complete. 18 mi=18 \ \mathrm{mi}=18 mi= ‾ft\underline{\qquad}\mathrm{ft}​ft

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Complete.

$18 \ \mathrm{mi}=$ $\underline{\qquad}\mathrm{ft}$