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

50 CHAPTER 3. APPROXIMATE SOLUTIONS
e.g.,
y 2 = y 1 + (t 2 − t 1 )f(t 1 , y 1 ) (3.52)
y 2 = y 1 + (t 2 − t 1 )f(t 1 , y 1 ) (3.53)
. (3.54)
y k+1 = y k + (t k+1 − t k )f(t k , y k ) (3.55)
. (3.56)
From figure (3.2) we observe that tan α = M; hence, since f is bounded,
|f(t 1 , y 1 )| ≤ M = tan α (3.57)
and therefore
|y 1 − y 0 | ≤ |t 1 − t 0 ||f(t 0 , y 0 )| ≤ |t 1 − t 0 | tan α (3.58)
Thus the point (t 1 , y 1 ) is inside of the triangle OPQ. By a similar argument, each
of (t 2 , y 2 ), (t 3 , y 3 ), ... are also in the triangle OPQ.
The curve we have just drawn is admissible, continuous, and piecewise differentiable,
with derivative given by
y ′ (t) = f(t k−1 , y(t k−1 )), t k−1 < x < t k , k = 1, 2, . . . (3.59)
Subtracting f(t, y(t) from both sides of the equation and taking absolute values,
|y ′ (t) − f(t, y(t))| = |f(t k−1 , y(t k−1 )) − f(t, y(t))| (3.60)
By equations (3.46), (3.55) and the boundedness of f,
|y k+1 − y k | = |(t k+1 − t k )f(t k , y k )| (3.61)
≤ min(δ, δ/M)M (3.62)
≤ δ (3.63)
Hence by uniform continuity (equation (3.44)), because |y k+1 − y k | < δ,
|f(t k−1 , y(t k−1 )) − f(t, y(t))| ≤ ɛ (3.64)
From equation (3.60)
|y ′ (t) − f(t, y(t))| ≤ ɛ (3.65)
which holds true for all t ≠ t k , k = 1, 2, ..., n − 1.
Therefore the function we have constructed satisfies all of the conditions of definition
3.1, hence it is an ɛ-approximate solution. Since we have constructed an
ɛ-approximate solution, we conclude that an ɛ-approximate solution exists.
Math 582B, Spring 2007
California State University Northridge
c○2007, B.E.Shapiro
Last revised: May 23, 2007