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ECIV 301
Programming & Graphics
Numerical Methods for Engineers
Lecture 31
Ordinary Differential Equations
Fig 23.1FORWARD FINITE DIFFERENCE
Fig 23.2BACKWARD FINITE DIFFERENCE
Fig 23.3CENTERED FINITE DIFFERENCE
Data with Errors
Pendulum
W=mg
02
2
l
sinmg
dt
dm
02
2
l
sing
dt
d
OrdinaryDifferentialEquation
ODEs
02
2
l
sing
dt
dNon Linear
Linearization
Assume is small
sin 02
2
l
g
dt
d
ODEs
02
2
l
g
dt
dSecond Order
ydt
d
Systems of ODEs
0
l
g
dt
dy
Application of ODEs in Engineering Problem SOlving
ODE
15810450 234 x.xxx.y
5820122 23 .xxxdx
dy
ODE - OBJECTIVES
Cx.xxx.y 5810450 234
5820122 23 .xxxdx
dy
dx.xxxy 5820122 23
15810450 234 x.xxx.y
Undetermined
ODE- Objectives
15810450 234 x.xxx.y
Initial Conditions
10 y
ODE-Objectives
y,xfdx
dy
Given
.C.Iknowny,f 0
Calculate
xy
Runge-Kutta MethodsNew Value = Old Value + Slope X Step Size
hyy ii 1
Runge Kutta Methods
hyy ii 1
Definition of yields different Runge-Kutta Methods