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(Solved) How do I write the matlab code for these problems? I am having


How do I write the matlab code for these problems? I am having trouble getting through them.?


BME 113L: INTRODUCTION TO NUMERICAL METHODS IN BIOMEDICAL ENGINEERING

 

LAB REPORT

 

Lab #7 Chapter 13: Eigenvalues

 

Last name, First name:

 

EID:

 

Lab Section: Tuesday, Wednesday, Thursday, Friday

 

Problem 1. From textbook Problem 13.2 (Power Method)

 

Use the power method to manually determine the largest eigenvalue and corresponding eigenvector for

 


 

[

 


 

2??

 

8

 

10

 

8

 

4??

 

5

 

10

 

5

 

7??

 


 


 

Write down your answers on a piece of paper and take a photo of it (or scan it) for submission. Validate your

 

answer with the eig function in Matlab.

 

Things to discuss

 

Assume that square matrix A has eigenvalues -1, -2, -3. What?s your best guess of the eigenvalues of the

 

inverse of matrix A? Why?

 

MATLAB code:

 

Results:

 

Discussion:

 


 

Problem 2. From textbook Problem 13.11

 

A system of two homogeneous linear ordinary differential equations with constant coefficients can be written as

 


 

d y1

 

=?5 y 1 +3 y 2

 

dt

 


 

d y2

 

=100 y 1?301 y 2

 

dt

 


 

y 1 ( 0 )=50, y 2 ( 0 ) =100

 


 

.

 


 

The solutions for such equations have the form

 


 

y i=c e

 

where

 


 

?t

 


 

c

 


 

and

 


 

?

 


 

are constants to be determined. Convert the system into an eigenvalue problem by

 


 

substituting this solution and its derivative into the original equations. The resulting eigenvalues and

 

eigenvectors can then be used to derive the general solution to the differential equations. For example, for the

 

two-equation case, the general solution can be written in terms of vectors as

 


 

{ y }=c 1 { v 1 } e ? t +c 2 {v 2 }e ? t

 

1

 


 

where

 


 

{ vi } =?

 


 

the eigenvector corresponding to the

 


 

2

 


 

i

 


 

th

 


 

eigenvalue (

 


 

?i

 


 

) and the

 


 

c'

 


 

s are unknown

 


 

coefficients that can be determined with the initial conditions.

 

(a) Use MATLAB to solve for the eigenvalues and eigenvectors. Print them in the command window.

 

(b) Employ the results of (a) and the initial conditions to determine the general solution (analytical expression),

 

and develop a MATLAB plot of the solution for

 


 

MATLAB code:

 

Results:

 

NO discussion

 


 

t=0

 


 

to

 


 

1

 


 

.

 


 

Problem 3. From textbook Problem 13.12

 

Water flows between the North American Great Lakes as depicted in Fig. 1. Based on mass balances, the

 

following differential equations can be written for the concentrations in each of the lakes for a pollutant that

 

decays with first-order kinetics:

 


 

Figure 1. The North American Great Lakes. The arrows indicate how water flows between the lakes.

 


 

d c1

 

=?( 0.0056+k ) c 1

 

dt

 


 

d c2

 

=?( 0.01+ k ) c 2

 

dt

 


 

d c4

 

=0.33597 c 3?( 0.376+k ) c 4

 

dt

 

where

 


 

k =?

 


 

d c3

 

=0.01902 c 1+ 0.01387 c2 ?( 0.047+k ) c 3

 

dt

 


 

d c5

 

=0.11364 c 4?(0.133+k )c 5

 

dt

 


 

the first-order decay rate (/yr), which is equal to 0.69315/(half-life). Note that the constants in

 


 

each of the equations account for the flow between the lakes. Due to the testing of nuclear weapons in the

 

atmosphere, the concentrations of strontium-90 (90Sr) in the five lakes in 1963 were approximately

 

T

 


 

{ c }= {17.7 30.5 43.9 136.3 30.1 }

 


 

in unites of Bq/m3. Assuming that no additional

 


 

90

 


 

Sr entered the system

 


 

thereafter, use MATLAB and the approach outlined in Problem 2 to compute and plot the concentrations in

 

each of the lakes from 1963 through 2011. Note that 90Sr has a half-life of 28.8 years.

 

Things to discuss

 

(1) Describe how to use the eigenvectors and eigenvalues to determine the general solution for the

 

concentrations of 90Sr in each of lakes (analytical expression).

 

(2) Use the plot you generate to discuss the changes of concentrations in each of the lakes and the

 

relationships between lakes.

 

MATLAB code:

 

Results:

 

Discussion:

 


 

 


Solution details:

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