Study guide: Solving nonlinear ODE and PDE problems

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Details: without Dirichlet condition equation

$ N_x=2 $ and Dirichlet condition not as a separate equation: $$ \left(\begin{array}{cc} A_{0,0}& A_{0,1}\\ A_{1,0} & A_{1,1} \end{array}\right) \left(\begin{array}{c} u_0\\ u_1 \end{array}\right) = \left(\begin{array}{c} b_0\\ b_1 \end{array}\right) $$ $$ \begin{align*} A_{0,0} &= \frac{1}{2\Delta x^2}(-\dfc(u_{1}^{-}) + 2\dfc(u_{0}^{-}) -\dfc(u_{1}^{-})) + a\\ A_{0,1} &= -\frac{1}{2\Delta x^2}(\dfc(u_{0}^{-}) + \dfc(u_{1}^{-}))\\ A_{1,0} &= -\frac{1}{2\Delta x^2}(\dfc(u_{0}^{-}) + \dfc(u_{1}^{-}))\\ A_{1,1} &= \frac{1}{2\Delta x^2}(-\dfc(u_{0}^{-}) + 2\dfc(u_{1}^{-}) -\dfc(u_{2})) + a\\ b_0 &= f(u_0^{-})\\ b_1 &= f(u_1^{-}) \end{align*} $$

Note: subst. $ u_{-1} $ by Neumann condition formula, subst. $ u_2 $ by $ D $