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Instructions for using the Differential Equations Applet
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Go to the applet
Contents
- Enter the right hand side of the equation into the area after
eqn #k: dy/dx=
The number k is used to identify the equation number in error
messages, so that possible problems can be traced to a particular equation.
In the default mode, the applet has only one equation. Equation #0, if present,
contains initialization code. This code is evaluated only once after its
change. It is typically used to define constants and functions.
- Any expression in variables x and y is allowed, using
operators +,-,* and /, and mathematical functions from the following list:
sin, cos, tan, arctan, arcsin, exp, ln, sqrt, abs, min, max (these last
two are binary). Mathematical constants "E", "PI" and "Infinity"
are also available. Examples of valid expressions:
- sin(x)/y
- y^2*sin(x)
- min(x,y)
- abs(x^2-y)
- sin(PI*y)
- exp(y)
- E^y
The applet supports user-defined functions, conditionals and loops described
in this section. - The equation is entered into
the Applet by hitting Return or by pressing the "Submit All" button. Immediately
after submission, the slope field and solutions are going to be plotted.
The following functions are available for use in expressions:
- sin(x)
- Returns sine of x.
- cos(x)
- Returns cosine of x.
- tan(x)
- Returns tangent of x.
- artcan(x)
- Returns inverse tangent of x.
- arcsin(x)
- Returns inverse sine of x.
- abs(x)
- Returns absolute value of x, i.e. x for x>=0 and -x for x< 0.
- exp(x)
- Returns the exponential of x.
- ln(x)
- Returns the natural logarithm of x.
- sqrt(x)
- Returns the square root of x for x>=0. If x< 0
- min(x, y)
- Returns x if x< y and y otherwise.
- max(x, y)
- Returns y if x< y and x otherwise.
- step(x)
- Returns 0 for x < 0 and 1 for x >= 0.
- sgn(x)
- Returns -1 for x < 0, 0 if x=0 and 1 for x > 0.
- iffun(x, y, z)
- Returns y for x is not equal to 0.0, z otherwise. Typically x is written
as a logical expression (see operator reference).
Note: This is a true function, i.e. it evaluates all its arguments
(x, y, and z) before returning its value.
- E
- The base of natural logarithm.
- PI
- The circumference of a circle of diameter 1.
- Infinity
- Positive infinity.
In addition to the ordinary arithmetical operators +, -, *, /, the following
operators are available for use in expressions:
- x^y
- Has a value of x raised to power y.
- x < y
- Has a value of 1.0 if x is less than y and 0.0 otherwise.
- x <= y
- Has a value of 1.0 if x is less or equal to y and 0.0 otherwise.
- x > y
- Has a value of 1.0 if x is greater than y and 0.0 otherwise.
- x >= y
- Has a value of 1.0 if x is greater or equal to y and 0.0 otherwise.
- x == y
- Has a value of 1.0 if x is equal to y and 0.0 otherwise.
- x != y
- Has a value of 1.0 if x is not equal to y and 0.0 otherwise.
- x && y
- Has a value of 1.0 if x and y are different from 0.0. Otherwise it
has a value of 0.0.
- x || y
- Has a value of 1.0 if either x or y is different from 0.0. Otherwise
it has a value of 0.0.
- x; y
- Has a value of y. The expression x is evaluated for its side effects.
Typically x is an assignment, as in
k=7; k^3-k^2-1
Note: In logical expressions the value of 0.0 is equivalent to false
and any value not equal to 0.0 is considered true. Logical operators
return 1.0 when the logical expression is true. This resembles the convention
used by languages like C and Java.
Click your mouse in the canvas area to select an initial condition. A solution
passing through the indicated initial condition will be plotted. You can
also select the initial condition precisely by typing in values of x and
y. This method is described in this section.
Note: If there are more variables than 2, only two variables will
be set based on the position of the mouse. The remaining variables will be
set to 0.
If the number of variables is greater than 2 (including the independent variable)
then you may select the variables that will be used as the abscissa and ordinate
of the plot. This is done by pressing two choice buttons placed along the
left and bottom edge of the plot and selecting one of the list of all variables.
The applet tabulates numerical solutions of the differential equations.
You can view tabulated values for each initial conditions in a window by
clicking on the button labelled "Show table". A window entitled "Solution
table" will pop up with all numerically generated solutions, in a form resembling
tables in standard textbooks on numerical methods. One can use Cut-And-Paste
to copy these tables into a text editor or word processor. The applet generates
a solution table every time it recalculates the solution. Old solution tables
can be erased by pressing the "Clear" button of the table window.
These are entered into areas labeled as "Min. x", "Max. x",
"Min. y" and "Max. y" etc. They are not used in calculations,
until they are entered by either pressing the Return key, or by pressing the
"Submit All" button.
The sizes of the grid in the x- and y- and other directions
are entered in areas marked as "Num. of segs" . They become effective upon
either hitting the Return key (one item at a time) or by pressing the "Submit
All" button, which updates all input related to entering the formula for dy/dx,
the ranges and the number of grid intervals.
The right mouse button, on machines with 2 and 3 mouse buttons, or pressing
the mouse button while the META key is depressed on machines with a single
mouse button, will initiate a selection of a rectangle in the canvas. By dragging
the mouse, the rectangular region is selected. When the mouse button is released,
the range of the two coordinates is reduced to the selected rectangle. The
zoom operation can be undone by pressing middle mouse button on system with
three mouse buttons (or where the "wheel" can be used as middle mouse button)
or by pressing the ALT key simultaneously with the mouse button.
- Slopes
- If checked, slopes are displayed.
- Solutions
- If checked, solutions with selected initial conditions are displayed.
- Init. Conditions
- If checked, every initial condition is marked and labeled.
- Euler/ModifiedEuler/Midpoint/Runge-Kutta/RK4/Runge-Kutta-Fehlberg/RKF
- Selects the algorithm used in calculating solutions.
- Step
- Displays and sets the integration step. Integration step is set automatically
in the following circumstances:
- the number of intervals (fields following "Num. of intervals")
is set;
- the "Submit All" button is pressed.
The automatic setting is calculated by dividing the range of the independent
variable by the number of intervals. The automatic setting of the step can
be overridden, though, by filling in the value of the step and hitting the
Return key.
The Applet maintains a list of initial conditions whose solutions will
be plotted. You can:
- Add an initial condition by setting the fields following the label
"Add init. cond.:". There is a field corresponding to each variable.
- Remove all initial conditions and start anew by pressing the "Clear
All" button.
- Initial conditions can be added by clicking them with the mouse.
All solutions displayed by the applet can be tabulated by pressing the
"Show table" button in the initial condition entry group of buttons. Tables
are displayed in a separate window. The table can be edited or copied into
another application (e.g. text editor). This is achieved by first selecting
the data to be copied (typically with a mouse) and then with the usual Control-C
(Cut) and Control-V (Paste) key combination, or other system-specific Cut/Paste
keys. The table data can also be printed directly from the applet by selecting
"Pring" from the "File" menu. Further necessary information on printing from
the applet is contained in this section.
The best general advice to avoid problems with printing (due to
various bugs in major browsers and Java implementations) is to
accept the certificate. The dialog to accept the certificate
should pop up automatically when the page containing the applet is
loaded for the first time. Another method primarily geared towards
older Netscape browsers, is given here.
It is possible to
print without accepting the certificate, but some browsers will hang
during the process.
The page containing the Applet can be printed with all major browsers.
However, there are some browser peculiarities. The problem is especially aggravating
with Microsoft Internet Explorer, because the applet window is reset to its
original state (i.e, your data is erased ):- ) before printing. Thus, in order to print,
manipulating the security settings of the browser is required, or downloading
a certificate file, which certifies the applet's authenticity. Unfortunately,
the number of various browser version combinations and strategies is too
numerous to discuss in detail. The included HTML files should take care of
making the choice for various Netscape and Internet Explorer versions. The
good news is that for versions 6.0+ of the two major browsers the only required
action to print is accepting the certificate. Other strategies are explained
below.
If you simply use the browsers "Print" button to print the version of the
Applet embedded into the Web page, Internet Explorer
will initialize the Applet before printing, and thus any changes, initial
conditions, etc. will be lost. Therefore you must use the built-in
"Print" button of the Applet, or the "Print" from the "File" menu of the Applets's window if running
the Applet in a separate frame. You must also accept
my security certificate when the Applet loads first by pressing OK when this dialog pops up.
Editing HTML
Another approach to printing is by editing the Applet's page with a text
editor to enter the Applet parameters directly.
Just edit the Applet's Web page with your favorite
HTML editor or a text editor (my preferred way).
Often you
will find it convenient to detach the applet frame by pressing the
"Frame" button. The Applet will run in a separate frame which will
behave nearly like an application that you installed on your
computer. Most importantly, you will be able to resize the window,
even to cover the entire screen, and to use additional applet features
accessible through the menu system.
In order to save the applet data (the formulas you typed in, the initial conditions, algorithms settings etc.)
follow this procedure:
- Detach the applet from the browser window if necessary, by pressing the "Frame" button.
- From the "File" menu, select "Save".
- When a file selection dialog pops up, select the name of the file to save the data to.
- Confirm the selection by pressing "OK" in the file dialog pop-up window, or in a similar, system-specific manner.
The data will be written to the file upon completion of the last step. The old content of the file, if any, will be lost.
In order to load the applet from the file you just created, follow these steps:
- Start the applet in the usual manner, in a framed mode.
For instance, you can select this link.
It does not matter what the content of the applet window is at this point.
- From the "File" menu, select "Load".
- From the resulting file dialog, select the name of the file containing previously saved data.
- Confirm the selection. During this step you will overwrite the current applet data with
the data stored in the file, so make sure you want to do this.
The data will be loaded from the previously saved file upon completion of the last
step. This will be quite obvious to you under most circumstances, as
the applet will significantly change visually, in a manner
corresponding to the saved state.
Errors
These are typically generated by typing in an expression (see section on
entering differential equations) which does not conform to
the syntax rules of the Applet. The most recent error message appears
in the box labeled "Last error". It will hopefully identify the
problem by pointing to the column in which the first error occurs. If
not, by pressing the button labelled "Show All Errors" all error
messages can be examined and used to correct the problem.
You can download
and any of the applet HTML documents:
in a directory of your choice on your local machine. This Jar file contains
all the necessary Java classes to run the applet. The entire package can be
also downloaded as a zip archive. After downloading,
make a new directory (UNIX users) or folder (Windows, Mac users) where you
want the applet files to reside. Move the downloaded file into that directory
or folder and unzip it there. In order to run the applet, you must add the
Jar file to your Java CLASSPATH. Windows 95/98 users would probably do it
in their AUTOEXEC.BAT file. UNIX users and Windows NT users would set the
environment variable. You can also run the Java applet as an application,
if you have a stand-alone Java interpreter. The syntax for running the applet
is:
java -classpath directory_prefix/(or\)JOdeApplet.jar com.rychlik.jode.JOde "dimension=2" "autonomous=true" ...
With the newer versions of Java you will be able to start the applet
by executing the jar:
java -jar JOdeApplet.jar "dimension=2" "autonomous=true" ...
You can configure the applet by using several "parameter=value" options.
All optiona are described in the parameters section.
For instance "dimension=2" and "autonomous=2" set the number of differential
equations to 2, and the equations are autonomous, i.e. time independent,
and thus there is no need to display the independent variable axis.
All settings of the Applet can be set in the containing HTML document by
passing parameters to the Applet. Currently, the applet recognizes these parameters, corresponding closely to the quantities
which can be set from the graphical user interface (GUI).
The precedence of operators closely resembles that of C or Java. The following
table lists operators according to precedence, highest precedence first:
- *, /
- +, -
- <, >, <=, >=, ==, !=
- &&, ||
- =
- ;
User-defined functions
The applet supports user defined functions, conditionals and flow-control.
They may be defined in equation #0 or in any other equation. For instance,
C=7; function f(x, y) = x + y + C
defines a function which, when called with two arguments, returns their sum
plus 7. It is worth noting that every statement of the JOde language is an
expression. Function definition expression above returns a value of 0 when
evaluated. The function is defined as a side effect.
Semi-colon
Semi-colon (;) is used as a separator in expressions. The resulting sequence
of expressions is evaluated in order and the value of the last espression
is returned. For example
x=1; y=2; x+y
is an expression whose value is 3.
Conditionals
JOde language supports the "if" statement. For instance, the following expression
x=2; if(x<3) 7 else 1
returns 7.
Loops
JOde language supports the "while" statement. If the body of the while loop
is not evaluated, the value returned is 0. Otherwise, the value is that of
the last evaluation of the body. For instance,
x=1; while(x<10) x=x+2
evaluates to 11. JOde supports " continue" and "break" statements in a way
similar to C and Java. However, "continue" and "break" with no argument causes
the value of the entire loop to be 0. If given an argument to " break", the
entire loop evaluates to that argument. If "continue" is called with an argument,
and the next test evaluates to false (i.e. 0.0) then the value of the entire
loop will be the value given as an argument to "continue". Thus
x=1; while(x<10) x=x+2; if(x==5) break x
returns 5 and
x=1; while(x<10) x=x+2; if(x==5) break
return 0. In the following example:
x=1; while(x<2) if(x==1) x=3; continue 7; 5
the value of the expression is 7. The semi-colon has very low precedence,
and thus in the following example
x=1; while(x<2) if(x==1) x=3; continue (7; 5)
the return value is 5 because of the explicit parenthesis.
This software is written in Java, of course. The documentation of a Java program
is typically generated as HTML with the help of Javadoc. For those interested
in the architecture of our program, here it is.
The files saved by JOde are in the generic format produces by object
serialization (Serialization API). It is customary to use the ".ser"
extension for these files, and we follow this convention. The saved files
will be compatible with future minor revisions of the applet. A major
revision may make the saved applets obsolete within a few years. Therefore,
if a substantial amount of work was put into a given ODE example, the
compatible version of the software (i.e. the file jode.zip) should be kept
around.
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A major addition has been the ability to save the applet to a file and load
the saved files.
-
Switching between framed and unframed versions of the applet can be
done by pressing the "Frame" button. Therefore, the links to
the framed versions were dropped. The old "*Framed*.html" files
still exist in the distribution for backwards compatibility of
the Web site.
-
The classes in the JAR file now belong to packages. Thus, you need
to update your old HTML files which use the applet by replacing
"JOdeApplet.class" with
"com/rychlik/jode/JOdeApplet.class". Also, the command to
run the applet from command line changed (see this section for details).
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Jason Miller's tutorial on using
the Applet is part of the distribution now. He also contributed the
style sheet used for all pages included with this package.
-
The performance improved significantly, using the new JDK 1.4.1
from Sun. The reson for the
improvement is not any drastic changes in the code but the fact that
the "-O" option actually works now. The byte code produced should be
compatible with Java 1.1 specification, which means that the
applet should work with older browsers. However, I suspect that the
mileage may vary in this respect.
Go to the applet
Enjoy,
Marek Rychlik (rychlik@u.arizona.edu)
