5. Derive the improved Euler method with step size k for the following ODE d dty=y, y(0) = 1. Compare the numeric solution yn with the exact solution y(t) = et at t = nk. For fixed t≤ 1, show that if nk = t ≤ 1, n = £, we have y(t) - Yn → 0 if k→ 0. This implies that the numeric solution converges to the exact solution. You do not need to prove the convergence rate 0(²).
5. Derive the improved Euler method with step size k for the following ODE d dty=y, y(0) = 1. Compare the numeric solution yn with the exact solution y(t) = et at t = nk. For fixed t≤ 1, show that if nk = t ≤ 1, n = £, we have y(t) - Yn → 0 if k→ 0. This implies that the numeric solution converges to the exact solution. You do not need to prove the convergence rate 0(²).
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.5: Iterative Methods For Solving Linear Systems
Problem 20EQ
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Transcribed Image Text:5. Derive the improved Euler method with step size k for the following ODE
d
=
dtyy, y(0) = 1.
Compare the numeric solution yn with the exact solution y(t) = et at t = nk. For fixed
t≤ 1, show that if nk = t ≤ 1, n =
we have
k'
y(t) - Yn0
if k→ 0. This implies that the numeric solution converges to the exact solution. You do not
need to prove the convergence rate O(k²).
Expert Solution
Step 1im
In this solution we will discuss the improved Euler method with step size for the following ODE:
with .
The given numeric solution is with .
Where is fixed and .
Given that .
We know that iteration of an ODE using improved Euler method is calculated as:
, where is step size.
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