A simple example of publishing reports

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The Euler approximation of the differential equation on [0,2]:

Define the grid size, interval's partition, and the initial value

h=0.1;                       % grid size
x=0:h:2;                     % specify the partition of the interval [0,2]
clear y;                     % clear possibly existing variable y
y(1)=1;                      % initial condition: x1=0, thus y(1) corresponds to y(x1)=1

Define the function

f=inline('3*exp(-4*x)-2*y');

Compute the Euler approximation

size(x);                     % size of x to be used in determining the size of vector i
for i=1:20 y(i+1)=y(i)+h*f(x(i),y(i));end

The exact solution

Y_exact=2.5*exp(-2*x)-1.5*exp(-4*x);

The differences between exact and approximate solutions

error = abs(Y_exact-y)'

error =

         0
    0.0587
    0.0793
    0.0794
    0.0696
    0.0560
    0.0418
    0.0289
    0.0180
    0.0092
    0.0025
    0.0024
    0.0058
    0.0080
    0.0093
    0.0099
    0.0100
    0.0097
    0.0092
    0.0086
    0.0079

Plot of the exact and approximate (dashed curve) solutions

plot(x,y,'--',x,Y_exact)

Published with MATLAB® 7.8