Consider the differential equation
y = f (t, y), a ≤ t ≤ b, y(a) = α.
a. Show that
for some ξ , where ti < ξi < ti+2.
b. Part (a) suggests the difference method
wi+2 = 4wi+1 − 3wi − 2hf (ti ,wi), for i = 0, 1, . . . , N − 2.
Use this method to solve
y = 1 − y, 0≤ t ≤ 1, y(0) = 0,
With h = 0.1. Use the starting values w0 = 0 and w1 = y(t1) = 1 − e−0.1.
c. Repeat part (b) with h = 0.01 and w1 = 1 − e−0.01.
d. Analyze this method for consistency, stability, and convergence.
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