Lecture Notes in Differential Equations - Bruce E. Shapiro

360 LESSON 33. NUMERICAL METHODS
Matlab have an extensive number of functions built in for this purpose, so
in general, you will never actually need to implement the methods we will
discuss in the next several sections if you confine yourself to those environments.
Sometimes, however, the built-in routines don’t provide enough
generality and you will have to go in and modify them. In this case it helps
to understand the basics of the numerical solution of differential equations.
By a numerical solution of the initial value problem
we mean a sequence of values
a corresponding mesh or grid M by
and a grid spacing as
y ′ = f(t, y), y(t 0 ) = y 0 (33.4)
y 0 , y 1 , y 2 , ..., y n−1 , y n ; (33.5)
M = {t 0 < t 1 < t 2 < · · · < t n−1 < t n }; (33.6)
h j = t j+1 − t j (33.7)
Then the numerical solution or numerical approximation to the solution is
the sequence of points
(t 0 , y 0 ), (t 1 , y 1 ), . . . , (t n−1 , y n−1 ), (t n , y n ) (33.8)
In this solution the point (t j , y j ) represents the numerical approximation
to the solution point y(t j ). We can imagine plotting the points (33.8) and
then “connecting the dots” to represent an approximate image of the graph
of y(t), t 0 ≤ t ≤ t n . We will use the convenient notation
y n ≈ y(t n ) (33.9)
which is read as “y n is the numerical approximation to y(t) at t = t n .”
Euler’s Method or the Forward Euler’s Method is constructed as
illustrated in figure 33.1. At grid point t n , y(t) ≈ y n , and the slope of the
solution is given by exactly y ′ = f(t n , y(t n )). If we approximate the slope
by the straight line segment between the numerical solution at t n and the
numerical solution at t n+1 then
y ′ n(t n ) ≈ y n+1 − y n
t n+1 − t n
= y n+1 − y n
h n
(33.10)