University of Florida/Egm6321/F10.TEAM1.WILKS/Mtg3

Mtg 3: Thur, 27 Aug 09

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To find class homepage, go to Wikiversity: Main Page (http://en.wikiversity.org/wiki/Wikiversity:Main_Page) and search for--> user:egm6321.f09

My Wiki address is: http://clesm.mae.ufl.edu/wiki.vq/index.php/Main_Page

From Eq.(1)p.2-3: If $P(x)\ne 0\mathrm{\forall }x$, divide throughout by $P(x)$ to get:

$\begin{array}{c}1y^{″}+\frac{Q}{P}{y}^{\prime }+\frac{R}{P}y=\frac{F}{P}\end{array}$

Where:

$\mathrm{\forall }$ is defined as "for all"

$a_2(x)=1$,

$a_1(x)=\frac{Q}{P}$,

$a_0(x)=\frac{R}{P}$, and

$f(x)=\frac{F}{P}$

$\mathrm{\forall }{x}_0$ such that $P(x_0)\ne 0$ then $x_0$ is a regular point

Any $x_0$ such that $P(x_0)=0$ is a regular point

File:EGM6321 F10 TEAM1 WILKS xc mtg3 1.svg

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2nd order--> need 2 conditions to solve for 2 constraints

Boundary Value Problem (BVP)

Prescribe:

$\begin{array}{c}y(a)=\alpha \end{array}$

$\begin{array}{c}y(b)=\beta \end{array}$

where $\alpha$ and $\beta$ are known values

Initial Value Problem (IVP)

Prescribe:

$\begin{array}{c}y(a)=\alpha \end{array}$

$\begin{array}{c}{y}^{\prime }(a)=\beta \end{array}$

where $\alpha$ and $\beta$ are known values

Solve IVP by ODE from p3-1 Eq(1) or initial condition p3-2 Eq(2)

Two points:

1) Existence and uniqueness of solution

File:EGM6321 F10 TEAM1 WILKS Exc mtg3 2.svg

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2) Superposition based on linearity of differential operation L(.)

$\begin{array}{c}{L}_2(.)=\frac{d^2(.)}{d{x}^2}+a_1\frac{d(.)}{dx}+a_0(.)\end{array}$

$\begin{array}{c}{L}_2(y)=y^{″}+a_1{y}^{\prime }+a_{0}y\end{array}$

Where the 2 in $L_2(y)$ is defined as 2nd order

Linearity of $\begin{array}{c}L(.)\end{array}$

$\mathrm{\forall }u,v$ in a function of x 


and $\mathrm{\forall }\alpha \beta$ belonging to $ℝ$ (scalars, real numbers);


$L(\alpha u+\beta v)=\alpha L(u)+\beta L(v)$


Where $ℝ$ is defined as a set of real numbers

Example: Matrix Algebra

$𝐀ϵℝ{}^{nxm}$ matrix with n rows and m columns of real numbers

$\mathrm{\forall }𝐮,𝐯ϵℝ{}^{mx1}$ is a column matrix

$\mathrm{\forall }\alpha \beta ϵℝ$

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Clearly: $𝐀(\alpha 𝐮+\beta 𝐯)=\alpha 𝐀𝐮+\beta 𝐀𝐯$

Example:
$\frac{d}{dx}(.)$ is a linear operation

$(\alpha u+\beta v{)}^{\prime }=\alpha u^{\prime }+\beta v^{\prime }$

linearity allows the use of superposition

$y=y_H+y_P$

$L(y)=L(y_H)+L(y_P)$, where the subscripts H and P stand for homogeneous and particular in respective order.

References

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