Project

Project 1

This assignment follows the standard form for a project submission. You need to include an

introduction, primary discussion, and summary. Include graphs, tables, and images, as necessary,

to improve the clarity of your discussion. Your project needs to be both correct and well written.

Communication remains a critical component of our modern, technological society.

A few notes about format: you MUST use MS Word for your project and use Equation Editor for all

mathematical symbols, e.g. 𝑧(𝑡) = sin(𝑡) + 1

ln(𝑡) .

If you have any questions about the requirements for this project, ask before you submit.

This project introduces approximations to Ordinary Differential Equations using numerical

methods. You will program three numerical solvers: Euler’s method, Improved Euler’s method, and

4th order Runge-Kutta (RK4). You are required to write your own numerical methods in either

MATLAB or MS Excel. You are not allowed to use numerical solvers written by anyone else.

Problem 1: Consider the following Initial Value Problem (IVP) where 𝑦 is the dependent variable

and 𝑡 is the independent variable:

𝑦 ′ = sin(𝑡) ∗ (1 − 𝑦) with 𝑦(0) = 𝑦0 and 𝑡 ≥ 0

Note: the analytic solution for this IVP is:

𝑦(𝑡) = 1 + (𝑦0 − 1)𝑒 cos(𝑡)−1

Part 1A: Approximate the solution to the IVP using Euler’s method with the following conditions:

Initial condition 𝑦0 = − 1

2 ; time step ℎ =

1

16 ; and time interval 𝑡 ∈ [0,20]

+ Derive the recursive formula for Euler’s method applied to this IVP

+ Plot the Euler’s method approximation

+ Plot the absolute error between the approximation and the exact solution using a semilog plot

Part 1B: Approximate the solution to the IVP using the Improved Euler’s method with the following

conditions: Initial condition 𝑦0 = − 1

2 ; time step ℎ =

1

16 ; and time interval 𝑡 ∈ [0,20]

+ Derive the recursive formula for the Improved Euler’s method applied to this IVP

+ Plot the Improved Euler’s method approximation

+ Plot the absolute error between the approximation and the exact solution using a semilog plot

Part 1C: Approximate the solution to the IVP using the RK4 method with the following conditions:

Initial condition 𝑦0 = − 1

2 ; time step ℎ =

1

16 ; and time interval 𝑡 ∈ [0,20]

+ Plot the RK4 method approximation

+ Plot the absolute error between the approximation and the exact solution using a semilog plot

Problem 2: Consider the following Initial Value Problem (IVP) where 𝑦(𝑡) is the dependent

function:

𝑦 ′ = 𝑦 − 𝑦 2 + 1.14 cos(𝑒𝑡/2) with 𝑦(0) = 𝑦0 and 𝑡 ≥ 0

Part 2A: Approximate the solution to the IVP using the Improved Euler’s method with the following

conditions: Initial condition 𝑦0 = 1; time steps ℎ = 1

8 ,

1

16 ,

1

32 ,

1

64 ; and time interval 𝑡 ∈ [0,20]

Project 1

+ Plot the Improved Euler’s method approximation for all 4 time steps

+ Discuss the results of these approximations

Part 2B: Approximate the solution to the IVP using the RK4 method with the following conditions:

Initial condition 𝑦0 = 1; time steps ℎ = 1

8 ,

1

16 ,

1

32 ,

1

64 ; and time interval 𝑡 ∈ [0,20]

+ Plot the RK4 approximation for all 4 time steps

+ Discuss the results of these approximations

Grading Rubric

Projects provide you with an opportunity to improve your Mathematical skills as well as your

communication. For this project you will need to correctly solve the problems and effectively

communicate your ideas and solutions. This assignment will be evaluated across the areas of

Validity, Readability, and Fluency.

Validity – Validity corresponds to the validity of your arguments. It addresses the extent to which

your method is appropriate, your calculations are correct, and your analysis is accurate.

Readability – If your written work is not readable it cannot be assessed. Since the ability to

communicate Mathematics is a focal point for this class, special attention will be paid to the

readability of your work.

Fluency – Mathematics is a concise and precise language, and we wish to enhance your

fluency. Therefore, part of every assessment will focus on your ability to incorporate correct,

established notation and terminology into your written work

Evaluation

criteria

Descriptive adjectives Scoring

Validity quality methods, correct solutions, proper

conclusions, complete reasoning 40%

Readability organization, presentation, format, clarity,

effectiveness 35%

Fluency proper notation, proper terminology,

appropriate definitions, conciseness 25%

Course and Learning Objectives

This Project supports the following Course Objectives:

CO-2: Solve a variety of linear and nonlinear Ordinary Differential Equations (ODEs).

CO-5: Analyze the qualitative behavior of single variable, differential time models and their

functional solutions.

CO-6: Synthesize several solution techniques to determine their appropriate application.