Instructions for using the Differential Equations Applet (JOde)
Go to the
applet
- 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 triangle.
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 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, 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.
Microsoft Internet Explorer
The default security settings do not allow unsigned applets to print.
In order to print from the applet in Internet Explorer 5.5, one has
the following method:
- Allow the applet Web server (i.e. the Web server from which you
downloaded this applet) to print. This is done as follows:
- Select the "Internet Options/Security" menu.
- One needs to add this Web server to the list of
trusted sites. Click on "Trusted sites" image.
Add this server to the list of trusted servers.
Close the dialog after doing so.
-
Again, select "Trusted sites" icon on the "Internet Options/Security"
menu. Click on the "Custom level" button. The "Security Settings"
dialog will pop up. Following "Virtual Machine/Java settings"
check the radio button labelled "Custom". Subsequently,
press button "Java Custom Settings" at the bottom.
A new dialog "Trusted sites" will pop up. Find "Printing"
and enable it.
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 framed version of the Applet
and select "Print" from the "File" menu of the
Applets's window. You must also accept my security certificate when
the Applet loads first by pressing OK when this
dialog pops up.
Netscape
Netscape up to version 4.79 will correctly print the Applet embedded into the Web page . However, at least some
versions (4.51) of the Communicator will crash if you try to
print from the framed version. Thus,
you should use the embedded version. Unfortunately, the embedded
Applet cannot be resized without editing the containing Web
page. In order to enable printing in the framed version, you
should
- import my X509 certificate,
using Netscape Security menu, into your Signers list.
- authorise printing after selecting
"Print" from the applet menu.
Netscape 6.0 and Mozilla
The applet should print with these browsers after accepting
the certificate. The dialogue to do so will pop up automatically.
Appletviewers
You can also print the Applet using Sun's appletviewer.
Microsoft viewer JView does not allow printing.
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).
Errors
These are typically generated by typing in an expression (see
"Entering Differential Equation" section) 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 JOde "dimension=2" "autonomous=true" ...
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.
Go to
the applet
Enjoy,
Marek Rychlik (rychlik@u.arizona.edu)
