On this page you will find information about the numerical
project. This includes the project problems, example code
and files before the project is due, and solutions after
the due date.
Math 555: Differential Equations I |
Click here for a printable PDF.
Problem I. Consider the initial value problem (IVP),
$$
\begin{cases}
y' = 1 - 20y, \\
y(0) = 0.
\end{cases}
$$
1. Perform Euler's method on the interval $[0,2]$ with a step size of $h = 0.1$. Is this
a good approximation of the solution? What is happening?
2. Does using the Improved Euler's method or Runge-Kutta method with the
same step-size fix this problem? Explain. Include a graph with all three methods plotted
on the same set of axes in your submission.
Problem II.
Consider the IVP,
$$
\begin{cases}
ty' + 2y = t^2e^{t}, \\
y(1) = 0.
\end{cases}
$$
1. Apply the Fundamental Existence and Uniqueness Theorem (FEUT) to verify that a
solution exists.
2. Find the exact solution, $\varphi$, analytically.
3. Use Euler's Method, Improved Euler's Method, and the Runge-Kutta Method to approximate
the solution curve on the interval $[1,3]$ with a step-size of $h = 0.1$. Plot each
approximation together with the exact solution $\varphi$ on the same set of axes. Plot the
exact errors, $\epsilon_k = \varphi(t_k) - y_k$, for each method on the same set of axes.
Make sure the graphs and errors for each method are clearly labeled.
4. Repeat II.3 with a step-size of $h = 0.05$.
Problem III.
Consider the IVP,
$$
\begin{cases}
\dfrac{dy}{dt} = \dfrac{3t^2}{3y^2 - 4}, \\
y(0) = 0.
\end{cases}
$$
1. Apply the FEUT to show that a solution exists.
2. Use the Runge-Kutta method with various step-sizes to estimate the maximum $t$-value,
$t = t^*>0$, for which the solution is defined on the interval $[0,t^*)$. Include a
few representative graphs with your submission, but not the lists of points.
3. Find the exact solution to the IVP and solve for $t^*$ analytically.
How close was your approximation from the previous question?
4. The Runge-Kutta method continues to give data for $t > t^*$. Does this data have
any meaning or significance? Explain.
Problem IV.
Consider the IVP,
$$
\begin{cases}
y' = 1 - t - y \\
y(0) = 0.
\end{cases}
$$
1. Solve for the exact solution $\varphi$ analytically, and compute $\varphi(1)$.
2. Find the first 5 Picard iterations, $\varphi_1, ..., \varphi_5$, starting with initial
guess $\varphi_0(t) = 0$. Plot them together
with the exact solution $\varphi$ on the same set of axes. Compute $\varphi_i(1)$ for each
approximation and compare with the exact value.
3. Use Euler's method, Improved Euler's method, and the Runge-Kutta method with step
sizes of $h = 0.05$ and $h = 0.01$ to approximate $y(1)$. Compare each of these values
with the exact value $\varphi(1)$.
Sample code and notes can be found here. You can change and run the Sage and Octave code from that page, or download an Excel file to modify on your own computer.
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