MA 625 Assignment 1

MA 625: Assignment 1

This assignment is due Tuesday, February 2 in class. Please e-mail your Matlab script(s) to

douglas@ccs.uky.edu

before class. Please use the Subject part of your e-mail message(s) to identify what you are sending. Please bring to class the pencil and paper part and the code part.

Part I

Consider the following simple initial value problem (IVP):

y_x = y, 0<x<1,

y(0) = 1,

The solution is y(x)=e^x.

Write Matlab programs to solve IVP numerically using Euler's Method and one of the modified Euler's or Heun's Methods at n equally spaced mesh points x_i, where x₀=0 and x_n=1. For each numerical method answer parts (a) and (b) separately.
(a) How many mesh points do you need to bound the error at any point x_i so that it is less than or equal to 8%, 4%, and 2%?
(b) Based on (b), how many mesh points do you need to bound the error by 1%? Prove your answer.

Part II

Prove the Existence and Uniqueness Theorem on page 3 of the class lecture transparencies.

Cheers,
Craig C. Douglas