1. Use Euler’s method to approximate the solutions for each of the following initial-valueproblems.
, with h = 0.5
, with h = 0.5
(c) dy dt = −y + ty1/2, 2 ≤ t ≤ 3, y(2) = 2, with h = 0.25
, with h = 0.25
2. The actual solutions to the initial-value problems in problem 1 are given here. Computethe actual error
3. Given the initial-value problem
with exact solution .
(a) Use Euler’s method with h = 0.05 to approximate the solution, and compare it with the actual values of y(t).
(b) Use the answers generated in part (a) and linear interpolation to approximate the following values of y(t), and compare them to the actual values.
(I) y(1.052) (II) y(1.555) (III) y(1.978)
4. Let h > 0 and let xj = x0 + jh (j = 1,2,...,n) be given nodes. Consider the initial value problem , with
for all x ∈ [x0,xn] and for all y.
i. Using error analysis of the Euler’s method, show that there exists an h > 0 such that
for some ξ ∈ (xn−1,xn), where en = y(xn)−yn with yn obtained using Euler method. ii. Applying the conclusion of (i) above recursively, prove that
|en| ≤ |e0| + nh2Y, where, Y . (1)
5.
The solution of the initial value problem
is y(x) = sinx. For λ = −20, find the approximate value of y(3) using the Euler’s method with h = 0.5. Compute the error bound given in (1), and Show that the actual absolute error exceeds the computed error bound given in (1). Explain why it does not contradict the validity of (1).
