University of Florida/Egm6321/F10.TEAM1.WILKS/Mtg15

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Ex.1 P.14-4  :



${\displaystyle N_{J}(x)=\ }$ finite element basis function associated with node J = "hat" function.



${\displaystyle N_{J}\in C^{0}\ }$ , but ${\displaystyle N_{J}\notin C^{1}\ }$

Ex.1 P.14-4 : Cubic Spline (Bexler, cubic Hermitian)

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Cubic ${\displaystyle \Rightarrow \ \ }$ 4 Coefficients ${\displaystyle \Rightarrow \ \ }$ 4 degrees of freedom per element ${\displaystyle \Rightarrow \ \ }$ 2 degrees of freedom per node (each element has 2 nodes)



HW: ${\displaystyle N_{J}^{\alpha \ }(x)\in C^{1}\ }$, but ${\displaystyle N_{J}^{\alpha \ }\notin C^{2}\ }$, for ${\displaystyle \alpha \ =1,2\ }$

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${\displaystyle L_{2}(y_{H}^{\alpha \ })=0\ }$ , ${\displaystyle \alpha \ =1,2\ }$

${\displaystyle L_{2}(y_{P})=f(x)\ }$

${\displaystyle y=Ay_{H}^{1}+By_{H}^{2}+y_{P}\ }$ , where A and B are constants

Where this can be rewritten for x as: ${\displaystyle x=Ax_{H}^{1}+Bx_{H}^{2}+x_{P}\ }$ , where A and B are constants

${\displaystyle L_{2}(y)=f\Leftrightarrow y=L_{2}(f)\ }$

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Euler Equations: Special Homogeneous Ln_ODE_VC

${\displaystyle a_{n}x^{n}y^{(n)})+a_{n-1}x^{n-1}y^{n-1}+...+a_{1}xy'+a_{0}y=0\Leftrightarrow \sum _{i=0}^{n}a_{i}x^{i}y^{(i)}=0\ }$

Where ${\displaystyle y'=y^{(1)}\ }$ and ${\displaystyle y=y^{(0)}\ }$

Two methods of solution:

Method 1: Transfer of variables ${\displaystyle x=e^{t}\ }$

Method 2: Method of undetermined coefficients ${\displaystyle y=x^{r}\ }$

(Or Trial Solution) K.etal(2003)

References