University of Florida/Egm6341/s11.TEAM1.WILKS/Mtg3

Page 3-1

NOTE: - page numbering 3-1 defined as meeting number 3, page 1

- T = torque Fig.p.1-1

- HW*

Eq.(3)P.2-1 : "Ordinary" Differential Equation (ODE)

order = highest order of derivative

Nonlinearity = What is linearity? ; use intuition for now, formal definition soon.

System has 3 unknowns:

${\displaystyle y^{1}(t)\ }$

${\displaystyle u^{1}(s,t)\ }$

${\displaystyle u^{2}(s,t)\ }$

${\displaystyle \equiv \ }$ Partial Differential Equations (PDE)

3 equations are coupled ${\displaystyle \Rightarrow \ \ }$ Numerical Methods

Simplify for analytical solution

2nd Order ${\displaystyle \rightarrow \ \ }$ 2nd Order

nonlinear ${\displaystyle \rightarrow \ \ }$ linear

unknown varying coefficient ${\displaystyle \rightarrow \ \ }$ known varying coefficient

Page 3-2

Note: Math structure of coefficient ${\displaystyle c_{i}(Y',t)\ }$ for ${\displaystyle i=0,1,...3\ }$ is known, but not their values until ${\displaystyle u^{1}\ }$ and ${\displaystyle u^{2}\ }$ are known (solved for)

General structure of Linear 2nd order ODEs with varying coefficients (L2_ODE_VC)

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

where ${\displaystyle y''={\frac {d^{2}y}{dx^{2}}}\ }$

${\displaystyle x=\ }$ independant variable

${\displaystyle y(x)=\ }$ dependant variable (unknown function to solve for)

Many applications in engineering are a result of solving PDEs by separation of variables. Some examples include, but are not limited to: Heat, Solids, Fluids, Acoustics and electrmagnetics.

Examples of these types equations are:

the Helmholz equation: ${\displaystyle \Delta \ X+k^{2}X=0\ }$

and the Laplace Equation: ${\displaystyle \Delta \ X=0\ }$

Ref F09 Mtg.28, Ref F09 Mtg.29 , Ref F09 Mtg.30

Page 3-3

In 3_D, ${\displaystyle x=(x_{1},x_{2},x_{3})\ }$

${\displaystyle \displaystyle {\begin{aligned}X(x)=X_{1}(x_{1})X_{2}(x_{2})X_{3}(x_{3})\end{aligned}}}$	(1)

Where the lowercase ${\displaystyle x\ }$ in the first term ${\displaystyle X(x)\ }$ is defined as ${\displaystyle x=(x_{1},x_{2},x_{3})\ }$

and ${\displaystyle X_{1}(x_{1})X_{2}(x_{2})X_{3}(x_{3})\ }$ is the separation of variables

${\displaystyle \displaystyle {\begin{aligned}X(\xi \ )=X_{1}(\xi \ _{1})X_{2}(\xi \ _{2})X_{3}(\xi \ _{3})\end{aligned}}}$	(2)

Where ${\displaystyle \xi \ \ }$ in the first term ${\displaystyle X(\xi \ )\ }$ is defined as ${\displaystyle \xi \ =(\xi \ _{1},\xi \ _{2},\xi \ _{3})\ }$

and ${\displaystyle X_{1}(\xi \ _{1})X_{2}(\xi \ _{2})X_{3}(\xi \ _{3})\ }$ is the separation of variables

Separated equations for ${\displaystyle i=1,2,3\ }$

${\displaystyle \displaystyle {\begin{aligned}{\frac {1}{g_{i}(\xi \ _{i})}}{\frac {d}{d\xi \ _{i}}}\left[g_{i}(\xi \ _{i}){\frac {dX_{i}(\xi \ _{i})}{d\xi \ _{i}}}\right]+f_{i}(\xi \ _{i})X_{i}(\xi \ _{i})=0\end{aligned}}}$	(3)

Page 3-4

Simplify:

${\displaystyle \xi \ _{i}\rightarrow \ x\ }$

${\displaystyle X_{i}(\xi \ _{i})\rightarrow \ y(x)\ }$

${\displaystyle g_{i}(\xi \ _{i})\rightarrow \ g(x)\ }$

${\displaystyle f_{i}(\xi \ _{i})\rightarrow \ a_{0}(x)\ }$

Eq.(3)p.3-3:

${\displaystyle \displaystyle {\begin{aligned}y''+{\frac {g'(x)}{g(x)}}y'+a_{0}(x)y=0\end{aligned}}}$	(1)

Where ${\displaystyle {\frac {g'(x)}{g(x)}}=a_{1}(x)\ }$

Particular case of Eq.(1)p.3-2

Linearity: Let ${\displaystyle F(.)\ }$ be an operator.

${\displaystyle u\ }$ and ${\displaystyle v\ }$ are 2 possible arguments (could be functions) of ${\displaystyle F(.)\ }$

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

Where ${\displaystyle \alpha \ \ }$ and ${\displaystyle \beta \ \ }$ are any arbitrary number.

References