UsEGM6321.F10.TEAM1.WILKS/Mtg3

EGM6321 - Principles of Engineering Analysis 1, Fall 2009

Mtg 3: Thur, 27 Aug 09

Contents

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

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

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

Where:

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

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

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

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

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

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

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

Page 3-2

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

Boundary Value Problem (BVP)

Prescribe:

 {\displaystyle \displaystyle {\begin{aligned}y(a)=\alpha \ \end{aligned}}} (1)

 {\displaystyle \displaystyle {\begin{aligned}y(b)=\beta \ \end{aligned}}} (1)

where ${\displaystyle \alpha \ \ }$ and ${\displaystyle \beta \ \ }$ are known values

Initial Value Problem (IVP)

Prescribe:

 {\displaystyle \displaystyle {\begin{aligned}y(a)=\alpha \ \end{aligned}}} (2)

 {\displaystyle \displaystyle {\begin{aligned}y'(a)=\beta \ \end{aligned}}} (2)

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

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

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

 Linearity of {\displaystyle \displaystyle {\begin{aligned}L(.)\end{aligned}}} (3)


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

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

${\displaystyle L(\alpha \ u+\beta \ v)=\alpha \ L(u)+\beta \ L(v)\ }$

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


Example: Matrix Algebra

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

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

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

Page 3-4

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

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

${\displaystyle (\alpha \ u+\beta \ v)'=\alpha \ u'+\beta \ v'\ }$

linearity allows the use of superposition

${\displaystyle y=y_{H}+y_{P}\ }$

${\displaystyle L(y)=L(y_{H})+L(y_{P})\ }$, where the subscripts H and P stand for homogeneous and particular in respective order.