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IntroductionIn this lecture we shall continue to use MATLAB programming to solve some applied

dynamics problems. A summary of some fundamental MATLAB features follow next.

An introduction to Matlab may be found at

https://sites.google.com/a/nd.edu/jay-brockman/multimedia-2#TOC-MATLAB-Basics-

Tutorials

Take a look at these flash videos in your own time.

MATLAB ProgrammingThis section describes some of the basic features of MATLAB programming. For a more

detailed study of MATLAB, the reader is referred to the MATLAB help files (invoked by

pressing the F1 key or typing help at the MATLAB command) or any introductory book on

MATLAB programming for engineers and scientists.

MATLAB is a high-level language for technical computing. It integrates computation,

visualization, and programming in an environment where problems and solutions are

expressed in mathematical notation. Typical application areas include:

Mathematical and computational algorithm development Numerical modelling and simulation Data analysis and visualisation Scientific and engineering graphics Application development, including graphical user interface building Data acquisition and control in real time

MATLAB stands for Matrix Laboratory; hence, the basic storage unit for any MATLAB

variable is the matrix also known as an array.

Command LineMATLAB (like the old DOS or UNIX terminal) is a command-line system. When you open

MATLAB for the first time, you see the Command Window with the command line prompt

>> and flashing cursor position. Individual MATLAB commands may be typed at the

command line prompt. Or multiple commands may be written if they are separated by

commas or semi-colons.

WorkspaceThe Workspace window should be visible in the MATLAB window (if not, go to

Desktop/workspace in the command list at the top).

The workspace is the repository for any MATLAB constants or named variables that the user

creates during the course of a MATLAB session. For example, to create a variable, letscall

itx, and give it a value of 5 type:

x=5

https://sites.google.com/a/nd.edu/jay-brockman/multimedia-2#TOC-MATLAB-Basics-Tutorialshttps://sites.google.com/a/nd.edu/jay-brockman/multimedia-2#TOC-MATLAB-Basics-Tutorialshttps://sites.google.com/a/nd.edu/jay-brockman/multimedia-2#TOC-MATLAB-Basics-Tutorialshttps://sites.google.com/a/nd.edu/jay-brockman/multimedia-2#TOC-MATLAB-Basics-Tutorialshttps://sites.google.com/a/nd.edu/jay-brockman/multimedia-2#TOC-MATLAB-Basics-Tutorials

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at the command line. MATLAB echoes the details of the variable just created. Take a look in

the workspace window and you will see a new variable with the name x and a value of 5.

Because we are working in the Matrix Laboratory, x is a 1x1 dimensions array.

To stop the automatic echo, use a semi-colon. Type

y=9;

and see what happens.

To create larger arrays or matrices we use square brackets, commas (or spaces will do) and

semi-colons. The square brackets indicate that the values are to be stored in an array.

Commas or spaces are used to distinguish individual entries along the rows. Semi-colons are

used to distinguish a new row. Typing

A = [1 2 ; 3 4];

at the command line will create a 2x2 array in the workspace.

We can transpose a matrix (flip it about its diagonal) by using the apostrophe. Create a new

array B which is the transpose of A by typing.

B=A'

The transpose function is useful for turning a row array into a column array and vice versa.

Try

C = [1 2 3 4 5 6]

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D = [1 2 3 4 5 6]

E = [1;2;3]

F = [1;2;3]

Note that we can use the = command to relate variable; hence, D=C and F=E would haveworked too.

Note that the command = is used to assign a value to a parameter. It does not mean equal

to more later.

To find out what variables are in the workspace at any time type the command who at the

command prompt. The workspace can be saved and retrieved for later use. The command

clear is useful for removing all variables from the workspace.

MATLAB Path

When you open MATLAB you see a window called current directory. This currentdirectory has a location on the hard drive as C:\MATLAB7\work (depending on the version

of MATLAB installed). The current directory is always \work by default. You can

change the working directory at any other location. MATLAB files in the current directory

are always visible to MATLAB. If MATLAB files are stored outside of the current directory

then they must be in file locations that are designated on the MATLAB path. Otherwise,

MATLAB will not be able to run these files. To see the path type path at the command

prompt or go to file\set path on the command list at the top. By taking the latter approach,

one can add or remove file locations to the MATLAB path.

Command History

The other window usually visible on the MATLAB window is called Command History. Thiswindow stores the list of previous MATLAB commands from the current session and

previous sessions. The up and down arrow keys ( and ) may be used at the command

prompt to toggle through previous commands.

MATLAB FilesFiles that contain code in the MATLAB language are called M-files. You create M-files

using a text editor, then use them as you would any other MATLAB function or command.

There are two kinds of M-files:

1. Scripts, which do not accept input arguments or return output arguments. Theyoperate on data in the workspace.

2. Functions, which can accept input arguments and return output arguments. Internalvariables are local to the function.

If you're a new MATLAB program, just create the M-files that you want to try out in the

current directory. As you develop more of your own M-files, you will want to organize them

into other directories and personal toolboxes that you can add to your MATLAB path. If you

duplicate function or script file names, MATLAB executes the one that occurs first in the

search path.

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MATLAB script filesScripts files (or batch files) are lists of commands that are operated in a sequence. The

sequence or flow is controlled by the code, but the default sequence is to complete each

command one at a time... more on this topic later.

Last week we wrote a few of our own script files to solve for axle loads. An important pointabout file name, they should be a single string of characters (no spaces allowed) followed by

the extension .m. To run the script file you can use the MATLAB editor/debug commands

or type the file name without the .m extension at the MATLAB command line prompt.

You may call a script file command from within another script file.

MATLAB Function filesFunctions are M-files that can accept input arguments and return output arguments. The

names of the M-file and of the function should be the same. Functions operate on variables

within their own workspace, separate from the workspace you access at the MATLAB

command prompt. A good example is provided by the command rank. The M-file rank.m isavailable in the directory toolbox/matlab/matfun

You can see a file with the command type followed by the filename.

In our case insert typerank

Here is the file as written in MATLAB.

function r = rank(A,tol)

% RANK Matrix rank.

% RANK(A) provides an estimate of the number of linearly

% independent rows or columns of a matrix A.

% RANK(A,tol) is the number of singular values of A

% that are larger than tol.

% RANK(A) uses the default tol = max(size(A)) * norm(A) * eps.

s = svd(A);

if nargin==1

tol = max(size(A)') * max(s) * eps;

end

r = sum(s > tol);

The first line of a function M-file starts with the keyword function. It gives the function name

and order of arguments. In this case, there are up to two input arguments and one output

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argument. The next several lines, up to the first blank or executable line, are comment lines

that provide the help text. These lines are printed when you type help rank

The rest of the file is the executable MATLAB code defining the function. The variable s

introduced in the body of the function, as well as the variables on the first line, r, A and tol,

are all local to the function; they are separate from any variables in the MATLAB workspace.

This example illustrates one aspect of MATLAB functions that is not ordinarily found in

other programming languages -- a variable number of arguments. The rank function can be

used in several different ways. rank(A)

r = rank(A)

r = rank(A,1.e-6)

Many M-files work this way. If no output argument is supplied, the result is stored in the

default variable ans. If the second input argument is not supplied, the function computes a

default value.

Flow ControlUsing programming statements we can control the flow or sequence of commands in

MATLAB.

The most widely used conditional flow statement is the ifstructure.

The first of these to look at is the if-endstructure

It can be written in code seen in figure 1 below. The figure shows how the commands are

types into the program, and the flowchart symbolically shows the flow. Or the sequence, in

which the commands are executed. As the program executes, it reached the if statement.

If the conditional expression is true (1), the program continues to execute the commands that

follow the if statement all the way down to the end statement.

If the conditional expression is false (0), the program skips the commands that follow the if

statement all the way down to the end statement, and continues with the commands that

follow the end.

So, if we had the following problem of calculating a workers pay. Lets say that a worker is

paid according to his hourly wage up to 40 hours, and 50% more for over time above the 40

hours. Write a program in a script file that calculates the pay to a worker. The program has to

ask the user to enter the number of hours and the hourly wage. The program is then to display

the pay.

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If

Statement

Commands

True

False

Flowchart

End

..

.. MATLAB program

.

if conditional expression

..

..

..

end

..

.. MATLAB program

.

Figure 1: the structure of the if-else conditional statement

%Script file: pay.m%%Purpose

% This program calculates the pay of a worker for a given number of% hours. If the number of hours is over 40 hours, then a weighting of% 50% extra is given on base salary for the extra hours%%Record of revisions:% Date Programmer Description of Change% ==== ========== =====================% 25/12/20212 Gerard Nagle Original code%%Define Variables:% t -Number of hours Worked% h -Hourly wage% Pay -Calculated Pay

t=input('Please enter the number of hours worked ');h=input('please enter the hourly wage in ');Pay=t*h;ift>40

Pay=Pay+(t-40)*0.5*h;endfprintf('The workers pay is $%5.2f', Pay)

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Note onfprintf

The fprintf can be used to display output (text and data) on to the MATLAB command

window or save it to a file. This command allows the output to be formatted. Also text and

data can be intermixed and displayed on the same line.

fprintf(text as string %-5.2f additional text, variable name)

The % sign marks the

spot where the number

is inserted within the text

Formatting elements.

(These define the format

of the number)

The name of the

variable whose value is

displayed

And the formatting elements are;

-5.2f

Flag

(optional)

Field width and precision

(optional)

Conversion character

(required!)

Where the flag is;

- (minus sign) Left justify within the field+ (plus sign) Prints a character (+ or -) in front of the number)

0 (zero) adds zeros if the number is shorter than the field.

Conversion character

E or e is exponential using upper or lower case

f is fixed point notation

i is integer

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if-else-endThe ifstatement evaluates a logical expression and executes a group of statements when the

expression is true. The first line if an if statement with a conditional expression. If the

conditional expression is true (1), the program

If

Statement

Commands

Group 1

True

False

Flowchart

End

..

.. MATLAB program

.

if conditional expression

..

..

..

else

..

..

..

end

..

..MATLAB program

Commands

Group 2Group 2 of

MATLAB commands

Group 1 of

MATLAB commands

A water tank has the geometer as shown in the figure below. The lower part is a cylinder, and

the upper part is an inverted frustum cone. There is a float in the tank that indicates the level

of the water. We have to write a function that determines the volume of the water in the tank

from the height of the water in the tank.

Diameter 25 m

19 m

14m

Diameter 46 m

rh

h

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%Function file: watervol.m%%Purpose% This programme calculates the volume of water in a water tower.% The dimensions of the tower are as in that attached diagram.%%%Record of revisions:% Date Programmer Description of Change% ==== ========== =====================% 25/12/2012 Gerard Nagle Original code%%Define Variables:% h -Height of Water in tower% rh -radius of upper inverted frustum cone

functionv = watervol(h)ifh

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If

Statement

Commands

Group 1

True

False

Flowchart

End

Commands

Group 2

Commands

Group 3

elseif

Statement

True

False

..

.. MATLAB program

.

if conditional expression

..

..

..

elseif

..

..

..

else

..

..

..

end

..

..MATLAB program

Group 2 of

MATLAB commands

Group 1 of

MATLAB commands

Group 3 of

MATLAB commands

%Script file: calc_roots.m

%Script file: calc_roots.m%%Purpose% This program solves for the roots of a quadratic equation of the form% a*x**2 + b*x + c = 0. It calculates the answers regardless if the type% of roots that the equation possesses% %%%Record of revisions:% Date Programmer Description of Change% ==== ========== =====================% 25/12/20212 Gerard Nagle Original code%%Define Variables:% a -Coefficient of the x^2 term% b -Coefficient of the x term% c -Constant term of the equation% discriminant -Discriminant of the equation% imag_part -Imag part of the equation (for complex roots)% real_part -Real part of the equation (for complex roots)% x1 -First Solution of equation (for real roots)

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% x2 -Second Solution of equation (for real roots)

%prompt the user for the coefficients of the equationdisp ('This program solves for the roots of a quadratic ');disp ('equation of the form A*X^2 + B*X + C = 0.');a=input('Enter the coefficient A: ');b=input('Enter the coefficient B: ');c=input('Enter the coefficient C: ');

% Calculate the discriminantdiscriminant = b^2 -4 * a * c;

%solve for the roots, depending on the values of the discriminantifdiscriminant > 0 % there are two real roots, so we have

x1 = ( -b + sqrt(discriminant))/(2 *a);x2 = ( -b - sqrt(discriminant))/(2 *a);disp ('This equation has two real roots:')fprintf ('x1 = %f\n', x1);fprintf ('x2 = %f\n', x2);

elseifdiscriminant == 0 % there is a repeated root, so we havex1 = ( -b )/(2 *a);disp ('This equation has two identical roots:')fprintf ('x1 = x2 %f\n', x1);

else % there is are complex roots, so we havereal_part = ( -b )/(2 *a);imag_part = sqrt ( abs ( discriminant ) ) /( 2 * a );disp ('This equation has two complex roots:')fprintf ('x1 = %f +i %f\n', real_part, imag_part );fprintf ('x2 = %f -i %f\n', real_part, imag_part );

end

Loop ControlsSometimes we want MATLAB to commands or complete sections of code. This can be

achieved by using the while and for statements.

for-endstatementThe for loop repeats a group of statements a fixed, predetermined number of times. A

matching end delineates the statements.

for n = 3:32

compute these commands

end

The variable n starts at a value of 3 and increases automatically for each iteration by 1 until it

reaches 33 whereupon it exits the loop without executing the commands within the loop.

The variable n could be made to increment by a different amount.

For n = 3:2:32

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Would make n increase by 2 after each iteration.

It is a good idea to indent the loops for readability, especially when they are nested.

for i = 1:m

for j = 1:n

H(i,j) = 1/(i+j);

end

end

Example

lets use afor-end loop in a script file to calculate the sum of the first n terms of the series

1

1

2

kn

kk

K

. Check the script file for 4n and 20n

for 4n , the sum is -0.125 and for 20n the sum is -0.222216

%Script file: sum_n_terms.m%%Purpose% This program calculates the sum of the n terms of the equation given in% the text%%%%Record of revisions:% Date Programmer Description of Change% ==== ========== =====================% 25/12/20212 Gerard Nagle Original code%%Define Variables:% n -Number of terms% S -The sum output

n = input('Enter the number of terms ');S=0;ticfork=1:n

S=S+(-1)^k*k/(2^k);endtocfprintf('The sum of the series is: %f',S)

%Function file: Tsin.m

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..

.. MATLAB program

.

for conditional expression

..

..

..

end

Group of

MATLAB commands

for

conditional statement

START

MATLAB program

Group of

MATLAB

commands

END

The function sin x can be written as a Taylor series by the function

2 1

0

1sin

2 1 !

k k

k

xx

k

Write a user defined function file that calculates sin x by using the Taylors series. For the

function name and argument, use Tsin x,ny . The input arguments are the angle x in

degrees, and n the number of terms in the series.

%Function file: Tsin.m%%Purpose% This program calculates the sin(x) with input arguments of x in degrees% and n, the number of terms in the Tatylor series%%%%Record of revisions:% Date Programmer Description of Change% ==== ========== =====================% 25/12/2012 Gerard Nagle Original code%%Define Variables:% x -Angle in degrees% n -Number of terms% y -The output

functiony=Tsin(x,n)xr = x*pi/180;y = 0;fork = 0:n-1

y = y + (-1)^k*xr^(2*k+1)/factorial(2*k+1);end

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While StatementThe while loop repeats a group of statements an indefinite number of times under the control

of a logical expression. The logical expression is tested and if it is true then the loop restarts

repeats the statements.

while expression

Repeat this code each time the expression is true

end

..

.. MATLAB program

.

while conditional expression

..

..

..

end

Group of

MATLAB commands

while

conditional statement

START

MATLAB program

Group of

MATLAB

commands

END

The function xf x e can be represented by the Taylor series0 !

nx

n

xe

n

. We need to

write a script file that calculates xe by using the Taylor series. The program calculates xe by

adding terms of the series and stopping when the absolute value of the terms that was added

last is smaller than 0.0001. We are required to use a while-endloop, but limit the number of

passes to 30 (this is an arbitrary number). If in the 30th pass, the value of the term that is

added is not smaller than 0.0001, then the program stops and displays a message that more

than 30 terms are needed.

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%Script file: taylor_series_exp(x).m%%Purpose% This program solves for the exp(x) by using the Taylor series% represenation. Thr program cacluates exp(x) by adding terms of the

% series until the absolute values of the last term added is smaller% than 0.0001%%Record of revisions:% Date Programmer Description of Change% ==== ========== =====================% 25/12/20212 Gerard Nagle Original code%%Define Variables:% x -Value of exp(x) to be calaculated% n -Value of incremenation of Taylor series% an -Value of each term in the Taylor series at each loop% S -The Sum of the Taylor series

%prompt the user for x in the exp(x)('This program solves for exp(x) by the Taylor Series Expansion ');x = input ('Enter x ')

%Declare the initial variablesn = 1;an = 1;S = an;

%Start of the while loop%The conditional statement has two relational and one logical operatorwhile(abs(an) >= 0.0001) && (n = 30disp('Maore than 30 terms are needed')

elsefprintf('exp(%f) = %f',x,S)fprintf('\nThe number of terms used is:%i',n)end

Logical and relational expressionsThe following are the possible relational expressions used for controlling flow

A < B (is A less than B)

A > B (is A greater than B)

A

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A >= B (is A greater than or equal to B)

A = = B (is A equal to B)

A ~= B (is A not equal to B)

The logical operators are AND, OR, and NOT

A & B (is A and B true)

A | B (is A or B true)

~A (NOT A)

The following truth table determines the responses from each logical test (1 is true and 0 is

false)

Flowcharts

Flowcharts give useful graphical representations of algorithms. We can study the logic of an

algorithm if we draw its flowchart.

In the flowchart we use flow lines and shapes to distinguish the control of the algorithm.

Typical symbols used in programming are given below

Start or end of the program,

subroutine, or function

A process such as a computation or

an assignment

A decision process, where a choice is

made

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Alternative to MATLAB

GNU Octave is open-source code that is Matlab portable. It will have a different interface to

Matlab, but should run the code with little or no changes required. For those who cannot getregular access to DIT computers, download GNU active for free.

Exercise #5An electric motor has a power rating of 6 kW and can produce a peak torque of 27.5 Nm.

Write a function that in Matlab that returns the maximum available torque from the motor

(Nm) for any given speed (rad/s).

Use this function in a Matlab script file that plots the motor torque-speed characteristic.

A subroutine or external function that

is documented elsewhere

When it is inconvenient to connect

two points use connector with a

reference number

A counting or iterative loop, such as

a for loop

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Ordinary Differential Equations

In mathematics, an ordinary differential equation is one which involves the derivative of a

function, the function itself, and (sometimes) the independent variable. As an example

consider the equation

( )

Where u is the dependent variable and is a function of time, t. The equation tells us that the

time rate of change of the function (or first derivative with respect to time) is dependent on

the value of u at that time and on the independent time value.

The differential equation is called ordinary because u is a function of one variable only and

the derivatives are complete, or ordinary, differentials. The order of the differential is given

by the order of the highest derivative in the equation; hence, for our example, the equation is

a first-order differential equation.

The solution of a differential equation is a function that satisfies the equation and also initial

conditions: this is known as a starting-value problem. The requirement to know the initial

conditions is determined by the order of the equation a first order equation requires one

initial condition, a second order equation requires two, and so on.

Numerical Integration Methods

For many straight-forward ODEs we can find an analytical solution to the problem, that is,

we can find a closed-form exact solution that satisfies the ODE and by adjusting some

constants we can satisfy the initial conditions. For example, the following ODE and IC

()

has the following analytical solution

However, in many cases, as the ODE becomes more complicated, the analytical solution is

unknown or too difficult to solve. In this instance a numerical solution may be your only

choice of solution.

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Our procedure for solving an ODE numerically applies to first-order ODEs. If our underlying

equation is a higher-order ODE then we substitute for derivatives with parameters until we

get a set of first order ODEsmore on this topic later.

Euler Integration

Euler integration is the simplest form of integration method. It is called a single step or

explicit method of integration, where we estimate the value of our function at the next time

using the data at the current time step. If our first order ODE is given as

( ) ()

Or ,y F x y

Let us consider the problem of approximating a continuous function y f x on 0x

which satisfies the differential equation

,y F x y

On 0x , with the initial condition

0y

In which is a given constant. Taking Eulers method, we go back to the definition of a

derivative

0limh

f x h f xy x

h

For small 0h , then we can say that a reasonable quotient approximation for y x is

f x h f x

y xh

Which can be restated as

,f x h f x

F x yh

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It can be rewritten as

,f x h f x hF x y

Or equivalently as

,y x h y x hF x y x

Which enables us to approximate y x h in terms of y x and ,F x y x . The equation

above is a corner stone of Eulers method.

Since a computer cannot calculate indefinably, we let 0x be replaced by 0 x L , in

which L is a positive constant. The value of L is determined by the problem under

consideration. Regardless, L is a fixed, positive constant. The interval 0 x L is divided

into n equal parts, each a length of h , by the points , 0,1,2,...,ix ih i n . The valueL

hn

is called the grid size. The points ix are called grid points. Let , 0,1,2,...,i iy y x i n so

that the initial condition (IC) implies 0y . Next, at each of the grid points

0 1 2 1, , ,..., ,nx x x x approximate the difference equations by the notation

1 , 0,1,2, ..., 1,i i i iy y

F x y i nh

Or, in explicit recursive form

1 , 0,1,2,..., 1,i i i iy y hF x y i n

Then, beginning with

0y

Set 0i and determine1

y .

Knowing1

y , set 1i and determine 2y

Knowing 2y , set 2i and determine 3y

And so on

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The resulting discrete function

0 1 2, , ,..., ny y y y

Is called the numerical solution

So, lets look at a problem.

We have the initial value problem,

y y x 0 1y

This is a linear problem that can be solved exactly to have the solution 1 2 xY x x e

As there is a solution to the problem, there is no need to actually solve the equation, but we

will do so to show the technique.

So, for Eulers method, 1L and 0.2h . Then 0 0.0,x 1 0.2,x 2 0.4,x 3 0.6,x

4 0.8,x 5 1.0x and the difference equation is approximated by

1 , 0,1,2,3,4,

0.2

i ii i

y yy x i

Or, if we rearrange, we get

1 0.8 0.2 , 0,1,2,3,4,i i iy y x i

Since 0 1y , we can calculate the equation above

1 0 0

2 1 1

3 2 2

4 3 3

5 4 4

0.8 0.2 0.8 1.000 0.2 0.0 0.800

0.8 0.2 0.8 0.800 0.2 0.2 0.680

0.8 0.2 0.8 0.680 0.2 0.4 0.624

0.8 0.2 0.8 0.624 0.2 0.6 0.619

0.8 0.2 0.8 0.619 0.2 0.8 0.655

y y x

y y x

y y x

y y x

y y x

Thus the numerical approximation with 0.2h is

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0.0 1.000

0.2 0.800

0.4 0.680

0.6 0.6240.8 0.619

1.0 0.655

y

y

y

yy

y

The exact solution is, rounded to three decimal places

0.0 1.000

0.2 0.837

0.4 0.7410.6 0.698

0.8 0.699

1.0 0.736

y

y

yy

y

y

This can be seen in the plot below.

y

x

nx nx h

ny

1ny

True y value from

analytical solution

y value from

Eulers Method

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With 0.1h , find the numerical solution on 0 1x by Eulers Method for

2 42 ,y y x x 0 0y

And compare your results to the exact solution 2y x

With 0.1h , find the numerical solution on 0 2x by Eulers Method for

3 38 2,y y x 0 0y

And compare your results to the exact solution 2y x

Modified Euler

We can improve upon the Euler integration by getting the arithmetic average of the slope at

the beginning and the end of the interval.

' '

, 1 3

, 12

p n

c n n

y yy y h e h

The procedure in this process is two-fold, we use the Euler method to the predict the first

value of yp,n+1and yp,n+1. We then use the above equation to calculate the corrected value of

yc,n+1. This method is called a predictor-corrector method and the order of the local error is

increased to three, which means that reductions in the step will yield more accurate results

that the basic Euler method.

Exercise #6

Take for example, the following ODE and IC

dyx y

dx , 0 1y

Which has the following analytical solution

2 1x

y e x

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We can use the analytical solution (or exact solution) to check the accuracy of our numerical

integration scheme. Note that an analytical solution is not always available and the only

method is to use a numerical method.

Let us use a table to get the answers. We shall use a integration step, h= 0.02.

n xn yn yn hyn Exact y(x)

1 0 1.0000 1.0000 0.0200 1.0000

2 0.02 1.0200 1.0400 0.0208 1.020403

3 0.04 1.041622

4 0.06 1.063673

Complete the table.

Write a piece of Matlab code that completes the integration table above

Exercise #7 Modified Euler

The procedure in this process is two-fold, we use the Euler method to the predict the first

value of yp,n+1and yp,n+1. We then use the above equation to calculate the corrected value of

yc,n+1. This method is called a predictor-corrector method and the order of the local error is

increased to three, which means that reductions in the step will yield more accurate results

that the basic Euler method.

We shall repeat the previous exercise using the modified Euler method.

n xn yn yn hyn yn+1 yn+1

' '

, 1

2

p ny yh

1 0 1.0000 1.0000 0.0200 1.0200 1.0400 0.0204

1.0204** 1.0404 0.0204

2 0.02 1.0204 1.0404 0.0208 1.0412 1.0812 0.0212

1.0416 1.0816 0.0212

3 0.04 1.0416 1.0816 0.0216 1.0632 1.1232 0.0220

1.0636 1.1236 0.0221

4 0.06 1.0637 1.1237 0.02247 1.1462

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* The first row shows the value given by the Euler method (predicted value)

** The second row in shows the value calculated by the Modified Euler (corrected value)

Complete the table and find estimated value for x = 0.08.

Write a piece of Matlab code that completes the integration table above

Runge-Kutta

A further advance on the Euler method is to use the Runge-Kutta method. Here we shall

present the fourth-order Runge-Kutta method: this one of the most widely used integration

methods.

1 1 2 3 41

2 26

n ny y k k k k

1 ,n nk hf x y

2 , 1

1 1,

2 2n n

k hf x h y k

3 , 2

1 1,

2 2n n

k hf x h y k

4 3,n nk hf x h y k

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Exercise #8

Using Matlab solve the equation

dyx ydx , 0 1y

Solve for 10x , this time take 0.1h . Compare with the exact solution.

Hint: Solution (by hand) for 0.1x (step 1)

Calculate1

k

1 ,n nk hf x y

1 0.1 0 1 0.100k

Calculate 2k

2 , 1

1 1,

2 2n n

k hf x h y k

21 1

0.1 0 0.1 1 0.1 0.1 0.05 1.05 0.1102 2

k

Calculate 3k

3 , 2

1 1,

2 2n nk hf x h y k

31 1

0.1 0 0.1 1 0.110 0.1 0.05 1.055 0.11052 2

k

Calculate 4k

4 3,n nk hf x h y k

4 0.1 0 0.1 1 0.1105 0.1 0.1 1.1105 0.12105k

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Calculate1ny

1 1 2 3 41

2 26

n ny y k k k k

11 1

1.0 0.1 2 0.110 2 0.1105 0.12105 1.0 0.66205 1.110346 6

ny