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Equation (10) written for \( i=1,\ldots,Nx-1= 1,2 \) becomes $$ \begin{align} \frac{u^{n}_1-u^{n-1}_1}{\Delta t} &= \dfc\frac{u^{n}_{2} - 2u^n_1 + u^n_{0}}{\Delta x^2}\\ \frac{u^{n}_2-u^{n-1}_2}{\Delta t} &= \dfc\frac{u^{n}_{3} - 2u^n_2 + u^n_{1}}{\Delta x^2} \end{align} $$
(The boundary values \( u^n_0 \) and \( u^n_3 \) are known as zero.)
Collecting the unknown new values on the left-hand side and writing as \( 2\times 2 \) matrix system: $$ \left(\begin{array}{cc} 1+ 2F & - F\\ - F & 1+ 2F \end{array}\right) \left(\begin{array}{c} u^{n}_1\\ u^{n}_{2}\\ \end{array}\right) = \left(\begin{array}{c} u^{n-1}_1\\ u^{n-1}_2 \end{array}\right) $$
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