QuestionQuestion

Numerical Solutions to Ordinary Differential Equations
Problems:
1. Write a user defined function euler(f, a, b, N, α) that outputs a vector w = (wi) = (w0, w1, . . . , wN ) of Euler approximations at the points tᵢ
2. Consider the IVP below

dx             x
--- = 3t − ---               1 ≤ t ≤ 6
dt             t

x(1) = 2

The solution to this IVP is x(t) = t² + 1/t.

(a) Determine the rate of convergence to its solution (demonstrate numerically) using the classical Runge-Kutta scheme
(b) Approximate the solution using your euler function with stepsize h = .02 and RK4 with stepsizes h = .02 and h = .2. Plot the solution function and these three approximations on one figure. Any observations?
(c) Determine the rate of convergence to its solution (demonstrate numerically)using the classical Runge-Kutta scheme if the initial value is x(1) = 1 instead. The exact solution in this case is x(t) = t²
(d) Is there a difference in the rate of convergence? If so, explain why that might be.

Solution PreviewSolution Preview

This material may consist of step-by-step explanations on how to solve a problem or examples of proper writing, including the use of citations, references, bibliographies, and formatting. This material is made available for the sole purpose of studying and learning - misuse is strictly forbidden.

function [t, w] = euler( f,a,b,N,alpha )
%[ w] = euler( f,a,b,N,alpha )
%INPUTS
% f -- function f(t,x)
% a -- begin point interval
% b -- end points interval
% N -- number of steps
% alpha -- initial condition x(a)
%OUTPUTS
% t -- the vector of times t_i
% w -- the vector of solutions at the times t_i

h=(b-a)/N;
w(1)=alpha;
for i=1:N+1;...
$33.00 for this solution

PayPal, G Pay, ApplePay, Amazon Pay, and all major credit cards accepted.

Find A Tutor

View available Numerical Analysis Tutors

Get College Homework Help.

Are you sure you don't want to upload any files?

Fast tutor response requires as much info as possible.

Decision:
Upload a file
Continue without uploading

SUBMIT YOUR HOMEWORK
We couldn't find that subject.
Please select the best match from the list below.

We'll send you an email right away. If it's not in your inbox, check your spam folder.

  • 1
  • 2
  • 3
Live Chats