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Lecture 23: Sets of Simultaneous First Order Ordinary Differential Equations, Slides of Numerical Methods in Engineering

A part of the ce 341/441 lecture notes from fall 2004. It covers sets of simultaneous first order ordinary differential equations, their numerical solutions using vector notation, and the importance of efficient methods for large systems. The document also discusses boundary value problems and their solutions using matrix methods and shooting methods.

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2011/2012

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CE 341/441 - Lecture 23 - Fall 2004
p. 23.1
LECTURE 23
SETS OF SIMULTANEOUS FIRST ORDER O.D.E.’S
Using vector notation:
All procedures discussed for single o.d.e.’s apply
However for both R.K. type methods and multi-step methods we must complete the
computation for the entire vector before moving on to the next step of the procedure due
to the coupling inherent to the system
d
dt
-----
y1
y2
˙
˙
˙
yn
f1y1y2ynt,,,,()
f2y1y2ynt,,,,()
˙
˙
˙
fny1y2ynt,,,,()
=
dy
dt
------fyt,()=
pf3
pf4
pf5

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Download Lecture 23: Sets of Simultaneous First Order Ordinary Differential Equations and more Slides Numerical Methods in Engineering in PDF only on Docsity!

p. 23.

LECTURE 23SETS OF SIMULTANEOUS FIRST ORDER O.D.E.’S • Using vector notation:• All procedures discussed for single o.d.e.’s apply• However for both R.K. type methods and multi-step methods we must complete the

computation for the entire vector before moving on to the next step of the procedure dueto the coupling inherent to the system

d ----- dt

y^1 y^2 ˙ ˙ ˙ y^ n

f^^1

y 1

y^2

y

n^

t

,^

,^

,^

(^

f^^2

y 1

y^2

y

n^

t

,^

,^

,^

(^

f^ n

y 1

y^2

y

n^

t

,^

,^

,^

(^

d y ----- dt

-^

f^

y t ,(

p. 23.

  • Notes:
    • It is especially important to use efficient methods when dealing with the large

systems of o.d.e.’s associated with the differentially time dependent equationswhich result from application of F.D./F.E. methods to discretize spatial variation.

  • If the o.d.e.’s are linear (both for single and multiple equations), no iterative proce-

dure is required for closed methods!

Boundary Value Problems • Boundary value problems must be 2nd order o.d.e.’s or higher• Example:

with b.c.’s

and

  • Consider 2 types of methods
    • Matrix formulation methods• Shooting methods

d

2 y d x

2

Ay

B

y^

)^

y L (

)^

p. 23.

INSERT FIGURE NO. 107

y

x=L

x

dy dx

x = 0

=U

y

L^

(U)

x

x=O

p. 23.

  • Therefore we must solve the problem

This is now a root solving problem

use the secant method (discrete derivative form of the Newton-Raphson algorithm to

find the solution)

  • Provide 2 estimates for the root of equation• Find solutions for

and

  • Now a new estimate for U can be obtained• Continue this procedure until convergence
    • Note that shooting methods are only practical in one dimensional applications and are

not practical in 2 and 3 dimensional b.v.p.’s.

y^ L

U (

)^

y^ L

U (

)^

U

o U

oo   

y^ L

U

o (^

)^

y^ L

U

oo (^

U

1

U

o

y^ L

U

o (^

y^ L

U

o (^

)^

y^ L

U

oo (^

[^

]/

U

o^

U

oo

(^