Computational Project 1, Math 3350
Dr. Kevin Long
DUE Monday, 11 Feb
In this project you will compute approximate solutions for several initial value problems using both Eulerâs method
and the improved Eulerâs method. Youâll then compare your approximate solutions to exact solutions, and measure
how the error depends on the step size.
For each of the four problems
1. dy
dx =âαy, y(0) = 1, α =1
4
2. dy
dx =ysin x, y(0) = 2
3. dy
dx =ÎČy2cos x, y (0) = 4
3, ÎČ =1
2
4. dy
dx =âαy + sin x, y(0) = 0, α =1
4
do the following steps.
1. Compute the exact solution yexact, using whatever method is appropriate to that problem.
2. Check your exact solution, verifying that it satisfies both the initial conditions and the differential equation.
3. Write a Matlab function to evaluate the right-hand side (RHS) f(x, y)for the given differential equation.
4. Using your RHS function together with the simple IVP driver and Euler stepper functions provided on the course
web page, compute approximate numerical solutions to the IVP using Eulerâs method using 16, 32, 64, and 128
steps. Solve on the interval xâ[0,2Ï].
(a) Plot on a single figure the exact solution and the four approximate numerical solutions to this problem. Use
different symbols for each. Use the Matlab legend function to show the association between graphical
symbols and number of steps.
(b) Compute the error |yexact âynum|at the end point x= 2Ïfor each approximate solution.
(c) Assuming the error varies as hp, estimate the order of accuracy pfrom your results.
5. Repeat step 4 above using the improved Euler method.
Suppose that design specifications require your simulations to be accurate to 10â5at the final step. Using your results,
estimate the number of Euler steps and improved Euler steps needed to reach that accuracy for this problem. Estimate
the cost difference (measured in number of RHS function evaluations) for using one method over the other. Which
method would you recommend using?
Document this work with a written report. Please refer to the web page for guidelines on writing your report
and an example of a project report.
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