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[[media: Fe1.s11.mtg11.djvu| Mtg 11]]: (EDGE Lect 8) [[media: Fe1.s11.mtg11.djvu| Page 11-1]]
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Discrete WRF

HW 2.8:

Show [[media: Fe1.s11.mtg10.djvu| (1) p. 10-4]] <=>[[media: Fe1.s11.mtg10.djvu| (2) p. 10-4]]

End HW 2.8

Issues: $$\displaystyle {{u}^{h}}$$ in [[media: Fe1.s11.mtg10.djvu| (2) p. 10-3]] must satisfy boundary conditions (essential, [[media: Fe1.s11.mtg4.djvu| (3) p. 4-4]] and natural, [[media: Fe1.s11.mtg4.djvu| (4) p. 4-4]])

Ex: G1DM1.0/D1

Essential boundary conditions: [[media: Fe1.s11.mtg9.djvu|(5) p. 9-1]] => $$\displaystyle {{u}^{h}}(1)=4,$$ [[media: Fe1.s11.mtg10.djvu|(2) p. 10-3]] => $$\displaystyle \underset{j=1}{\overset{n}{\mathop \sum }}\,{{d}_{j}}{{b}_{j}}\left( 1 \right)=4$$

Natural boundary conditions: [[media: Fe1.s11.mtg9.djvu|(1)&(2) p. 9-2]] =>$$\displaystyle -2\frac{d{{u}^{h}}(x=0)}{dx}=12,$$ negative because $$\displaystyle n\left( 0 \right)=-1$$. This and [[media: Fe1.s11.mtg10.djvu|(2) p. 10-3]] yields$$\displaystyle ~\underset{j=1}{\overset{n}{\mathop \sum }}\,{{d}_{j}}\underbrace{{{{{b}'}}_{j}}(0)}_{\frac{\partial {{b}_{j}}(0)}{\partial x}}=-6$$

[[media: Fe1.s11.mtg11.djvu| Page 11-2]] More Generally:

Essential boundary condition:

Natural boundary condition: $$\displaystyle {{\left. \left( n(x)\cdot {{a}_{2}}(x)\frac{\partial {{u}^{h}}(x)}{\partial x} \right) \right|}_}=h$$ (1)&(2) are 2 additional equations, which together with [[media: Fe1.s11.mtg10.djvu|(3) p. 10-5]] or [[media: Fe1.s11.mtg10.djvu|(4) p. 10-5]] can be used to find $$\displaystyle {{d}_{nx1}}={{\{{{d}_{j}}\}}_{nx1}}$$

Select n linearly independent equations starting with (1)&(2), plus (n-2) equations from [[media: Fe1.s11.mtg10.djvu|(3) $$\displaystyle \equiv $$ (4)p. 10-5]]