MAT 514 Project 2

Project Description

The goal of this project is to compare the performance of three numerical methods of solving differential equations: Euler method, Modified Euler, and Runge-Kutta (RK4).

Consider the initial value problem

y'=m*cos(m*x)*y, with y(0)=k

Here m is 10+(1st digit of your SUID), k=2+(2nd digit of your SUID).

Your goal

is to find the exact solution of this IVP and, most importantly, compare it to the numerical approximations. The comparison should be made on the interval 0≤x≤5.

For each of three methods find a critical step size h such that the numerical solution is an acceptable approximation when step size is h, but is not acceptable if step size is 10*h. Use your judgment to decide what is an acceptable approximation.

The results of this investigation will consist of 6 screenshots: Euler with critical step size, Euler with 10*(critical step size), and so on. These should be combined into one file using software such as Microsoft Word (or its free alternative, OpenOffice Writer), and accompanied by a paragraph with your comparison of three methods. The file should be submitted via Blackboard by 10PM on Monday, October 3.

Technical Details

For this project I recommend using the applet Solution Verifier although you are free to use your favorite math software package instead.

How to use the applet

"eqn #1 dy/dx=" is m*cos(m*x)*y with m as above

"Function #1: y=" is the solution you found yourself. This will be plotted in green color. The numerical approximation will appear in black.

Choose the window Min x =0, Max x =5, Min y =0. For Max y enter the number 3*k where k is as above

Ignore "Num of segs": this line refers to the slope field which we don't need here.

In the line "Show", uncheck "Slopes" to get rid of the slope field.

"Add init. cond.:" enter x=0, y=k.

The applet has two Submit buttons: one for equation and one for initial value.

Use Clear All to remove plots that are not needed.

Avoid moving the mouse pointer over the graphing window, because this changes the initical conditions.