User:Eml5526.s11.team01.roark/Mtg03

[[media: Fe1.s11.mtg3.djvu| Mtg 3]]: Mon, 10 Jan 11 [[media: Fe1.s11.mtg3.djvu| Page 3-1]]


 * Students who just joined
 * Example of computation for insight: Vu-Quoc et al. on granulated flows (1999, PRS and 2000, CMAME)

Elastic Bar (continued)

$$\Omega =\left] 0,L \right[=\left( 0,L \right)$$

$$\partial \Omega =\left\{ x=0, \right.\left. x=L \right\}$$


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$${{\Gamma }_{u}}={{\Gamma }_{g}}=\left\{ x= \right.\left. 0 \right\}$$


 * $$\displaystyle (Ess. B.C.) $$




 * }


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$${{\Gamma }_{h}}=\left\{ x= \right.\left. L \right\}$$


 * $$\displaystyle (Nat. B.C.) $$




 * }

$$\partial \Gamma ={{\Gamma }_{g}}\bigcup {{\Gamma }_{h}}$$

Connected to abstract notation 60-minute program Part 1, Part 2

Linear 2nd order ODE w/avarying coeff (L2-ODE-VC)

x is the independent variable, y(x)=unknown

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$$\underbrace{{{a}_{2}}\left( x \right)}_{coeff}\underbrace_{{{y}^{(2)}}:=\frac{{{\partial }^{2}}y}{\partial {{x}^{2}}}}+{{a}_{1}}(x){y}'+{{a}_{0}}(x)y=g(x)$$


 *  (1)




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where g(x) is the forcing function. “Self adjoint” L2-ODE-VC


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$$\underbrace_{\frac{d}{dx}\left[ {{a}_{2}}(x)\frac{dy}{dx} \right]={{a}_{2}}{y}''+{{a}_{2}}^{\prime }{y}'}+{{a}_{1}}(x){y}'+{{a}_{0}}(x)y=g(x)$$


 *  (2)


 * }

Compare (2) to (1) [[media: Fe1.s11.mtg2.djvu| p. 2-2]]:

$${{a}_{2}}(x)=E(x)A(x)$$

$${{a}_{1}}(x)={{\left[ E(x)A(x) \right]}^{\prime }}$$

$${{a}_{0}}(x)=0$$

$$g(x)=-f(x)$$

Note:

Self adjoint different operator <-> sym. square matrices (“stiffness” matrices).

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All engineering problems here are self adjoint except advanced differential equations.

End Note