Creating and Incorporating Dynamic Applets for Differential Equations the Easy Way

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1 Creating and Incorporating Dynamic Applets for Differential Equations the Easy Way Robert Decker University of Hartford June Decker Three Rivers Community College uhaweb.hartford.edu/rdecker

2 A set of java classes (programs) that allows you to create java applets (mathlets) without knowing any java. You create a one page set of inputs that create your applet. Compile the applet (Eclipse recommended) Create.htm file and post to website. The Mathlet Toolkit: Creating Dynamic Applets for Differential Equations and Dynamical Systems in IJTME

3 Get students to make connections, discover new properties, see solutions from different points of view (must be dynamic/interactive). Can be used as tool for doing homework. Demonstrate during lecture. Provide method to deliver projects and other interactive documents (text and applets together).

4 Plot more than one coordinate system at a time (make connections, multiple representations) Plot functions, parametric functions, first order differential equations, differential equations systems, data points, iterated maps and systems of maps, iterated function systems, bifurcation diagrams and lyapunov exponent graphs for iterated maps. Combine different function types in one applet Control graphs interactively Include text/math with applets to make interactive document

5 Interactive initial conditions for differential equations. Click in a graph window to set an initial condition; double click to keep it (or click Keep IC); click and drag to change interactively. All initial conditions can be cleared with the Clear IC button. If the Click ICs checkbox is not checked, you cannot change initial conditions as described above. This can be useful if you want more control over the precision of the initial conditions Sliders control parameters interactively; changes are reflected immediately. The run button on the slider creates an animation. The range of parameter values can be changed using <<>> or >><<

6 Any value in a text box can be changed/edited. Hit return/enter after making the change to see it reflected in the graph(s) At the bottom of each coordinate system are small square buttons that indicate which function(s) are plotted, and which variables on which axes. For example 2xy means function 2 is plotted with x on the horizontal axis and y on the vertical. Click on one of these buttons to change these items, or to change whether lines or points are plotted (or both). In each function box there is a Display values button. This will generate a new window with a scrollable list of function values (the values that are being plotted)

7 The range of values for plotted functions is set in the function definition part of the applet. The coordinate system ranges for the horizontal and vertical axes are set in the coordinate system parts. Thus the points plotted are not tied to the ranges of the coordinate systems as the are for a graphing calculator. You can zoom on a graph using the zoom buttons or by right clicking and dragging a rectangle. A parameter can be used in a text box, such as the max value of an independent variable (shows an animated graph)

8 Go to my website uhaweb.hartford.edu/rdecker and click on The Mathlet Toolkit for detailed instructions I recommend Eclipse for editing and compiling the applets (website instructions are tied to Eclipse). You make changes in one file only, the driver file. Specify functions (differential equations, etc.), coordinate systems, parameters, and which functions are plotted on which coordinate systems (and how). Create.htm file. Enhance.htm file with text, graphics, etc. Post to website or Blackboard (some versions are easier than others).

9 Layout +Functions (DE s, points, etc) + Coordinate Systems + Parameters =Applet

10 public static final int NUMBER_OF_FUNCTIONS_HORIZONTALLY = 1; public static final int NUMBER_OF_COORDINATE_SYSTEMS_HORIZ ONTALLY = 2; public static final int NUMBER_OF_PARAMETERS_HORIZONTALLY = 3;

11 new DifferentialEquationGUI( "a*y*(1-y/b)-d", /*DE*/ "t", /*indep var (variable 1 for plotting)*/ "y", /*dep var (variable 2 for plotting)*/ -10, /*indep var min*/ 10, /*indep var max*/ 0.1, /*indep var step size*/ 0, /*initial indep var*/ 0.5, /*initial dep var*/ RK2, /*method*/ Color.BLUE, /*color*/ FIELD_ON, /*slope field*/ 15 /*field resolution*/ ),

12 new CoordinateSystem( -1, /*horizontal axis min*/ 10, /*horizontal axis max*/ -2, /*vertical axis min*/ 8, /*vertical axis max*/ Array2ArrayList2D(new int[][]{ {1,INDEP_VAR,DEP_VAR1,LINES_ON,POINTS_OFF} /*{function,horizontal var,vertical var, lines, points}*/ ), })

13 new Parameter( "a", /*parameter name*/ 1, /*initial parameter value*/ -2, /*parameter range min*/ 4, /*parameter range max*/ 2 /*slider precision; -1=tens,0=ones,1=tenths,2=hundreths etc*/ ), new Parameter( "b", /*parameter name*/ 4, /*initial parameter value*/ -10, /*parameter range min*/ 10, /*parameter range max*/ 2 /*slider precision; 0=ones 1=tenths 2=hundreths etc*/ ),

14 The differential equation y' = a(6 y) can be used as a model for a warming or cooling object. The solution by separation of variables is ax y = 6 + ce or using the initial condition y ( 0) = y0 it becomes y = + ( y 6) e 6 0 Experiment with the parameters to see how the approximate solution differs from the exact solution for different parameter values ax

15 Investigate y = ay(1 y / b) d which can be used to model the number of fish in a lake with initial population growth rate a, carrying capacity b, and fishing rate d. Explain what is happening for d=0 and various a and b using many well-chosen solution curves. Fix a and b and increase the parameter d continuously and observe/explain what is happening Find the d value beyond which no fish can survive in the long term (called the bifurcation value). Explain how you came up with your answer.

16 An equation for a damped pendulum is x + cx + bsin( x) = 0 It can be rewritten in system form as x = y, y' = cy bsin( x) Investigate this equation for various values of the damping constant c. You may set the constant b=1. Explain your phase portraits in terms of the pendulum. What do you see when c gets large? Drag your initial condition around to see what is happening.

17 dy Investigate = ay( 1 y) using Euler, Rk2, and dx Rk4 Fix the step size at 1 (too large practically speaking) Increase the parameter a continuously and observe/explain what is happening Results obtained by one of my students (using computer algebra after making conjectures based on the graphs)

18 Robert Decker University of Hartford June Decker Three Rivers Community College uhaweb.hartford.edu/rdecker

Direction Fields; Euler s Method

Direction Fields; Euler s Method It frequently happens that we cannot solve first order systems dy (, ) dx = f xy or corresponding initial value problems in terms of formulas. Remarkably, however, this