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

EGM6321 - Principles of Engineering Analysis 1, Fall 2009

Mtg 3: Thur, 27 Aug 09

Page 3-1

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 $P(x)\neq \ 0\ \forall x$, divide throughout by $P(x)\$ to get:

$\displaystyle {\begin{aligned}1y''+{\frac {Q}{P}}y'+{\frac {R}{P}}y={\frac {F}{P}}\end{aligned}}$

(1)

Where:

$\forall \$ is defined as "for all"

$a_{2}(x)=1\$,

$a_{1}(x)={\frac {Q}{P}}$,

$a_{0}(x)={\frac {R}{P}}$, and

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

$\forall x_{0}\$ such that $P(x_{0})\neq \ 0\$ then $x_{0}\$ is a regular point

Any $x_{0}\$ such that $P(x_{0})=0\$ is a regular point

Page 3-2

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

Boundary Value Problem (BVP)

Prescribe:

$\displaystyle {\begin{aligned}y(a)=\alpha \ \end{aligned}}$

(1)

$\displaystyle {\begin{aligned}y(b)=\beta \ \end{aligned}}$

(1)

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

Initial Value Problem (IVP)

Prescribe:

$\displaystyle {\begin{aligned}y(a)=\alpha \ \end{aligned}}$

(2)

$\displaystyle {\begin{aligned}y'(a)=\beta \ \end{aligned}}$

(2)

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

Page 3-3

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

$\displaystyle {\begin{aligned}L_{2}(.)={\frac {d^{2}(.)}{dx^{2}}}+a_{1}{\frac {d(.)}{dx}}+a_{0}(.)\end{aligned}}$

(1)

$\displaystyle {\begin{aligned}L_{2}(y)=y''+a_{1}y'+a_{0}y\end{aligned}}$

(2)

Where the 2 in $L_{2}(y)\$ is defined as 2nd order

Linearity of $\displaystyle {\begin{aligned}L(.)\end{aligned}}$

(3)

$\forall u,v\$ in a function of x 


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


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


Where $\mathbb {R} \$ is defined as a set of real numbers

Example: Matrix Algebra

$\mathbf {A} \epsilon \ \mathbb {R} \ ^{nxm}$ matrix with n rows and m columns of real numbers

$\forall \mathbf {u} ,\mathbf {v} \epsilon \ \mathbb {R} \ ^{mx1}\$ is a column matrix

$\forall \alpha \,\beta \,\epsilon \ \mathbb {R} \ \$

Page 3-4

Clearly: $\mathbf {A} (\alpha \ \mathbf {u} +\beta \ \mathbf {v} )=\alpha \ \mathbf {A} \mathbf {u} +\beta \ \mathbf {A} \mathbf {v}$

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

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

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.