Professor Kate Poirier | OL67 | Fall 2020

Project #2: Numerical Methods

Due Sunday, October 11

Numerical methods provide us a way to approximate solutions of initial value problems without actually having to solve them. In Chapter 3, we see three of these methods for initial value problems of the form y'=f(x,y), y(x_0)=y_0:

  • Euler’s method
  • Improved Euler’s method
  • Runge-Kutta method

For Part 1 of Project 2, you will create your own calculator to approximate the solution of one initial value problem using all four of these methods, each with different step sizes. You will not perform your calculations by hand; instead you should use spreadsheet software like Excel or Google Sheets. You may write your own computer program instead of using a spreadsheet if you prefer.

Here is an example of the type of spreadsheet you will create (though this spreadsheet does not conform to all the instructions below).

Instructions

This is not a group project. You may help each other, but each student must submit their own work.

  1. Choose one of the problems below and submit a comment on this post with the number corresponding to your choice (no more than three people may choose the same problem).
  2. Solve the problem using each of the three methods. Use a spreadsheet or write your own computer program to perform these calculations. If using a spreadsheet, include calculations for the different methods on different sheets. Make sure to organize your work and indicate clearly the calculations for different step sizes. Your solution must also display any intermediate values found while carrying out each method:
    • Euler’s Method: display the slope f(x,y) as well as x and y values at each stage,
    • Improved Euler’s: also display the values of k_1 and k_2 at each stage
    • Runge-Kutta: also display the values of k_1, k_2, k_3 and k_4 at each stage
    • You can display other values as well, if you wish.
  3. Email your finished product (Excel spreadsheet, link to Google sheet, link to computer program, etc) by Sunday, October 11, 11:59pm to your professor.

Problem choices

  1. Use step sizes h = 0.1, h=0.05, h=0.025 to find approximate values of the solution of the initial value problem at x=0,0.1, 0.2, 0.3, \dots, 1.0.

        \[(3y^2+4y)y' + 2x + \cos(x) = 0, y(0)=1\]

  2. Use step sizes h = 0.1, h=0.05, h=0.025 to find approximate values of the solution of the initial value problem at x=1.0,1.1, 1.2, 1.3, \dots, 2.0.

        \[ y'+ \frac{(y+1)(y-1)(y-2)}{x+1} = 0, y(1)=0\]

  3. Use step sizes h = 0.1, h=0.05, h=0.025 to find approximate values of the solution of the initial value problem at x=0, 0.1, 0.2, 0.3, \dots, 1.0.

        \[y'+3y=7e^{-3x}, y(0)=6 \]

  4. Use step sizes h = 0.1, h=0.05, h=0.025 to find approximate values of the solution of the initial value problem at equally spaced points in the interval [2,3]

        \[y'-2y = \frac{1}{1+x^2}, y(2)=2\]

  5. Use step sizes h = 0.2, h=0.1, h=0.05 to find approximate values of the solution of the initial value problem at equally spaced points in the interval [0, 2]

        \[y'-2xy = x^2, y(0)=3\]

  6. Use step sizes h = 0.2, h=0.1, h=0.05 to find approximate values of the solution of the initial value problem at equally spaced points in the interval [1, 3]

        \[y'+ \frac{1}{x}y = \frac{\sin(x)}{x^2}, y(1)=2\]

  7. Use step sizes h = 0.05, h=0.025, h=0.0125 to find approximate values of the solution of the initial value problem at equally spaced points in the interval [0, 1.5]

        \[y'+y=\frac{e^{-x}\tan(x)}{x}, y(1)=0\]

  8. Use step sizes h = 0.2, h=0.1, h=0.05 to find approximate values of the solution of the initial value problem at equally spaced points in the interval [0, 2]

        \[y'+ \frac{2x}{1+x^2}y = \frac{e^x}{(1+x^2)^2}, y(0)=1\]

  9. Use step sizes h = 0.05, h=0.025, h=0.0125 to find approximate values of the solution of the initial value problem at equally spaced points in the interval [1, 1.5]

        \[xy'+(x+1)y=e^{x^2}, y(1)=2\]

  10. Use step sizes h = 0.1, h=0.05, h=0.025 to find approximate values of the solution of the initial value problem at equally spaced points in the interval [0,1]

        \[y'+3y=xy^2(y+1), y(0)=1\]

  11. Use step sizes h = 0.1, h=0.05, h=0.025 to find approximate values of the solution of the initial value problem at equally spaced points in the interval [0,1]

        \[y'-4y = \frac{x}{y^2(y+1)}, y(0)=1\]

  12. Use step sizes h = 0.1, h=0.05, h=0.025 to find approximate values of the solution of the initial value problem at equally spaced points in the interval [2,3]

        \[y'+2y = \frac{x^2}{1+y^2}, y(2)=1\]

Grading

To receive full credit, your submission must include:

  • functioning solutions for all three methods,
  • all appropriate intermediate values at each stage,
  • answers that are correct up to 5 decimal places at each stage,
  • work that is clearly labelled and organized.

Technical help

The Atrium Learning Center offers free self-paced instruction in Microsoft Office (which includes Excel). Information is here:

24 Comments

  1. Richard Li

    Number 5

  2. Raisa Ratri

    Number 10

  3. Sheyla Criollo

    number 5

  4. Jennifer Agudelo

    Number 10

  5. Oscar

    2

  6. Sierra Morales

    Problem 1

  7. Brian Enriquez

    Problem 8

  8. Jian Hui

    Number 5

  9. Ryjll Morris

    Number 5

  10. julio dominguez

    Number 1

  11. Kelon McKoy

    Problem 1

  12. Jason Zhu

    Number 2

  13. Jason Zhu

    Number 2

  14. Wilfrido V.

    Number 3

  15. aron singh

    number 3

  16. Omrit P

    Number 3

  17. Ariel Itshaik

    number 4

  18. Dief

    Problem 2

  19. Dief

    Sorry professor, I did 10 rather than 2

  20. Trevon

    Problem #3

  21. Trevon

    Problem # 4 actually

  22. Tushar Shorma

    number 10

  23. Mushrat

    no: 12

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