computational modeling for engineering mecn 6040 professor: dr. omar e. meza castillo...
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COMPUTATIONAL MODELING FOR ENGINEERINGMECN 6040
Professor: Dr. Omar E. Meza [email protected]
http://facultad.bayamon.inter.edu/omezaDepartment of Mechanical Engineering
INTRODUCTION TO THE THEORY
OF PDEs
LEARNING OBJECTIVES
1. Be able to distinguish between the 3 classes of 2nd order, linear PDE's. Know the physical problems each class represents and the physical/mathematical characteristics of each.
2. Be able to describe the differences between finite-difference and finite-element methods for solving PDEs.
3. Be able to solve Elliptical (Laplace/Poisson) PDEs using finite differences.
4. Be able to solve Parabolic (Heat/Diffusion) PDEs using finite differences.
DEFINITIONS AND TERMINOLOGY
DIFFERENTIAL EQUATIONAn equation containing the derivative of one or more dependent variables, with respect to one or more independent variables is said to be a differential equation (DE).
DEFINITIONS AND TERMINOLOGY
h
xfhxf
dx
dyh
)()(lim
0
DEFINITION OF A DERIVATIVEIf y=f(x), the derivative of y or f(x) with respect to x is defined as
The derivative is also denoted by y’, dy/dx or f’(x)
THE EXPONENTIAL FUNCTION
dependent variable: y independent variable: x
xexfy 2)(
yedx
xde
dx
ed
dx
dy xxx
22)2()( 22
2
DEFINITIONS AND TERMINOLOGY
Differential Equations are CLASSIFIED by type, order and linearity.
TYPEThere are two main types of differential equation: “ordinary” and “partial”.
DEFINITIONS AND TERMINOLOGY
Ordinary differential equation (ODE) Differential equations that involve only ONE independent variable are called ordinary differential equations.Examples: ,
only ordinary (or total) derivatives
xeydx
dy5 06
2
2
ydx
dy
dx
ydyx
dt
dy
dt
dx 2
DEFINITIONS AND TERMINOLOGY
Partial differential equation (PDE) Differential equations that involve two or more independent variables are called partial differential equations.Examples:
only partial derivatives
t
u
t
u
x
u
22
2
2
2
x
v
y
u
DEFINITIONS AND TERMINOLOGY
ORDERThe order of a differential equation is the order of the highest derivative found in the DE.
second order first order
xeydx
dy
dx
yd
45
3
2
2
DEFINITIONS AND TERMINOLOGY
xeyxxy 2'
3'' xy
first order
second order
DEFINITIONS AND TERMINOLOGY
DEFINITIONS AND TERMINOLOGY
LINEAR OR NONLINEARAn n-th order differential equation is said to be linear if the function
is linear in the variables)1(' ,..., nyyy
0),......,,( )(' nyyyxf
DEFINITIONS AND TERMINOLOGY
there are no multiplications among dependent variables and their derivatives. All coefficients are functions of independent variables.
)()()(...)()( 011
1
1 xgyxadx
dyxa
dx
ydxa
dx
ydxa
n
n
nn
n
n
or
linear first-order ordinary differential equation
linear second-order ordinary differential equation
linear third-order ordinary differential equation
0)(4 xydx
dyx
02 ''' yyy
04)( xdydxxy
xeydx
dyx
dx
yd 53
3
3
PDE'S DESCRIBE THE BEHAVIOR OF MANY
ENGINEERING PHENOMENA:
▪ Wave propagation
▪ Fluid flow (air or liquid)▪ Air around wings, helicopter blade, atmosphere
▪ Water in pipes or porous media
▪ Material transport and diffusion in air or water
▪ Weather: large system of coupled PDE's for momentum, pressure, moisture, heat, …
▪ Vibration
▪ Mechanics of solids: ▪ stress-strain in material, machine part, structure
▪ Heat flow and distribution
▪ Electric fields and potentials
▪ Diffusion of chemicals in air or water
▪ Electromagnetism and quantum mechanics
CLASIFIQUE LAS SIGUIENTES ECUACIONES:
Solución (a)
032
2
2
2
2
2
2
2
2
2
y
u
x
u
y
u
x
u
y
u
x
u(c) (b) (a)
parabólicaACB
C,BAy
u
x
u
:04
;0,030;3
2
2
2
elípticaACB
CBAy
u
x
u
ahiperbólicACB
CBAy
u
x
u
:04
;1,0,1;0
:04
;1,0,1
2
2
2
2
2
2
2
2
2
2
0;
Solución (b)
Solución (c)
