The Computable Differential Equation Lecture ... - Bruce E. Shapiro

Chapter 3
Approximate Solutions
3.1 The Forward Euler Method
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 (3.1)
y 0 , y 1 , y 2 , ..., y n−1 , y n ; (3.2)
M = {t 0 < t 1 < t 2 < · · · < t n−1 < t n }; (3.3)
h j = t j+1 − t j (3.4)
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 ) (3.5)
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 (3.5) 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 ) (3.6)
which is read as “y n is the numerical approximation to y(t) at t = t n .”
Euler’s Method is constructed as follows. 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
(3.7)
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