Math1360: Computer Lab I
Introduction
Today we will learn how to use Matlab to solve some of the problems we have covered so far.
As the first step you have to download to your account several subroutines I have listed on my
website. Use a web browser, and go to the link
http://www.math.pitt.edu/~swigon/math1360.html
click on the files for the Lab and download them to your home directory.
You will need the following files:
flow.m – subroutine for plotting the 1-D flow diagram
euler.m – subroutine for numerical solution of ODE by Euler method
impreuler.m - subroutine for numerical solution of ODE by improved Euler method
rungekutta45.m – subroutine for numerical solution of ODE by Runge-Kutta method
Fixed points
Fixed points of the first-order differential equation dx/dt = f(x) are the solutions of the equation
f(x) = 0. Matlab contains several subroutines that can help us locate fixed points by
1) plotting the function f(x)
ezplot('x^2')
ezplot('-x^3+x^2+4*x-4',[-10 10])
2) finding the solutions of f(x) = 0
fsolve(inline('x^2-1'),2)
3) plotting the flow of dx/dt = f(x)
flow(inline('x^2-1'),[-2 2])
– plots the function x2
– plots the function –x3 + x2 + 4x – 4 over
the interval –10 < x < 10
– finds a solution of the equation x2 – 1 = 0
that is closest to the initial guess x = 2
– plots the flow diagram for dx/dt = x2 – 1
in the interval –2 < x < 2
Problems
For the following ODEs determine all fixed points (if possible) and their stability.
a) dx/dt = 3*sin(x) – x
b) dx/dt = x2 + e-x
use exp(-x) for e-x
x
c) dx/dt = e – cos(x)
d) dx/dt = x sin(1/x)
Note
Plotting two functions on the same graph:
ezplot('x^2',[-2 2])
hold on
ezplot('cos(x)',[-2 2])
Numerical solution of ODEs
Methods for numerical solution of dx/dt = f(x) with initial condition x(t0) = x0 provide estimates
of the values of x = x(t) at discrete points t1, t2, t3, etc. The simplest estimate is provided by the
first term in Taylor expansion (linearization) about x(t):
x(t + t) ≈ x(t) + f(x(t)) t
This defines the Euler's algorithm:
xn+1 = xn + f(xn) (tn+1 – tn)
The points tn are usually assumed equidistant with tn+1 – tn = t. Solution found using Euler
method can be visualized by plotting xn versus tn.
Example:
Find numerically approximate solutions of the initial value problem dx/dt = x, x(0) = 1, for
various values of t and compare them with the exact solution.
1) Use the euler subroutine to compute approximate solution with x(0) = 1 and t = 0.1
on the interval [0, 5]. The values of tn are stored in the matrix T and xn in the matrix X.
[T,X] = euler(inline('x'),[0 5],1,0.01);
2) Plot the graph of xn versus tn in red
plot(T,X,'r')
3) On the same figure plot in blue the graph of the exact solution x(t) = et
hold on
ezplot('exp(t)',[0 5])
4) Observe how the error is changing with increasing t by plotting on the same graph
numerical solutions for t = 0.05, 0.1, 0.5.
[T,X] = euler(inline('x'),[0 5],1,0.05);
plot(T,X,'r')
[T,X] = euler(inline('x'),[0 5],1,0.1);
plot(T,X,'r')
[T,X] = euler(inline('x'),[0 5],1,0.5);
Problems
Follow the same procedure for the following initial value problems:
a) dx/dt = x (3 – x), x(0) = 0.1, – 1, 5
b) dx/dt = x2, x(0) = – 2, 2
c) dx/dt = –9x, x(0) = 5.
(What is the smallest value of t at which numerical instability occurs?)
More accurate algorithm is the improved Euler method in which first a trial step k is taken and
then a slope f(k) is found at that point. The real step is taken with a slope equal to the average of
f(xn) and f(k):
k = xn + f(xn) t
xn+1 = xn + t [f(xn) + f(k)]/2
Improved Euler method is included in the file impreuler.m
Example
Use the improved Euler method to calculate an approximate solution of the initial value problem
dx/dt = x, x(0) = 1, for various values of t and compare them with solutions obtained using the
Euler method.
1) Use the impreuler subroutine to compute approximate solution with x(0) = 1 and t =
0.1 on the interval [0, 5].
[T,X] = impreuler (inline('x'),[0 5],1,0.1);
2) Plot the graph of xn versus tn in red
plot(T,X,'r')
3) On the same figure plot in green the solution obtained by using Euler's method
hold on
[T,X] = euler(inline('x'),[0 5],1,0.1);
plot(T,X,'g')
5) On the same figure plot in blue the graph of the exact solution x(t) = et
hold on
ezplot('exp(t)',[0 5])
6) Which method is more accurate? Repeat for different values of t.
The improved Euler method is of order t2 while the original Euler method is of order t. This
means that improved Euler method tends to make smaller error for a given stepsize.
Problem
The Euler method showed oscillatory numerical instability in solving the equation dx/dt = –9x,
x(0) = 5 for certain values of t. Does the improved Euler method show a similar instability ? Or
a different type of instability ? (Hint: obtain numerical solution for various values of t and
observe the results.)
Fourth-order Runge-Kutta method is frequently used as the method of choice in first attempts for
a numerical solution of a first-order differential equation. As the name suggest the method is of
order t4 and hence even more accurate than improved Euler method. (For description see the
book.) It is included in the file rungekutta45.m.
Problem
Examine whether the numerical instabilities of Euler and improved Euler methods have been
resolved in the design of the Runge-Kutta method. (Hint: obtain numerical solution of the
problem dx/dt = –9x, x(0) = 5 for various values of t and observe the results.)