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Numerical Methods for Solving Differential Equations

Heun's Method

Mathematica Implementation

(continued fromÌýlast page...)

We will test ourÌýheunÌýprogram by having it create numerical solutions to the same differential equations that we solved using our Euler program in the last lab. Recall that the command-syntax used with your Euler program for this task was:

euler[f,{x,x0,xn},{y,y0},steps]

so for your new program, (since we made no changes to the input format when we adapted it to the Heun method, other than its name,) you would use a command of the form:

heun[f,{x,x0,xn},{y,y0},steps]

Of course, for such a command to work you must have evaluated yourÌýheunÌýprogram during your current session with theÌýMathematicaÌýkernel! Use it to solve each of the initial value problems given in the table below. For each exercise, do the following:

1. Find the numerical solution, in the form of a list, on theÌýx-interval spanning from the initial condition point up to the specifiedÌýx-value.

2. Read this solution list into the solution name specified for the exercise.

3. Use the number of steps specified.

4. Redisplay your solution in tidy columns with a command of the formÌýMatrixForm[Solution Name], inserting the appropriate name for the exercise inside the brackets.

5. Plot your solution using a command of the formÌýListPlot[Solution Name], inserting the appropriate name for the exercise inside the brackets.

6. Check your solution and its graph by clicking on theÌýMathematicaÌýAnswer button provided.Ìý(Only do this once you've tried the problem yourself.)

Exercises

Solution Name	Differential Equation	Initial Condition	Solve up to	Number of Steps	Check Solution
heunex1	²â′Ìý=Ìýy	y(0)Ìý=Ìý1	xÌý=Ìý1	10
heunex2	²â′Ìý=Ìý³æÌý-Ìý²â	y(0)Ìý=Ìý0.5	xÌý=Ìý1	20
heunex3	²â′Ìý=Ìýx2+1/y2	y(5)Ìý=Ìý1	xÌý=Ìý6	30
heunex4	²â′Ìý=Ìý(³æÌý-Ìý²â)2	y(0)Ìý=Ìý0.5	xÌý=Ìý0.5	20
heunex5	²â′Ìý=Ìý³æÌý²âÌý+Ìýy1/2	y(0)Ìý=Ìý1	xÌý=Ìý1	20

If you've finished the above exercises, then you may eitherÌýreturn to the beginningÌýof this laboratory, jump to theÌýTable of ContentsÌýof all of the differential equations laboratories, or simply quit.

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If you're lost, impatient, want an overview of this laboratory assignment, or maybe even all three, you can click on the compass button on the left to go to the table of contents for this laboratory assignment.