University of California Santa Barbara Math Question Python Code Worksheet Please write down this two question and you will still need use python to show it. All problems requiring coding must
be integrated in a single jupyter notebook file and the rest of the
problems in a single pdf file. (2) You must write your own python code individually. Do not copy
codes. 1. Consider the initial-value problem
su'(t) + 3tu + 22 = 0,
, u(1)=0.5.
Let h=0.1, 0.05 and 0.025, and solve the differential equation respectively to determine u(2)
using
(a) Forward Euler Method.
(b) Backward Euler Method.
() Trapzoidal Method.
(a) Leapfrog Method (use forward Euler Method to compute Ul).
Denote the corresponding solutions as 10.1(2), 40-05 (2) and 20.025 (2) respectively, and compute
the following ratio for the above four methods
20.1(2) 20.05 (2)
2,0.05 (2) 20.025 (2)
2. Consider the following problem
u'(t) = -e “u(t)
2(0)=1
(a) Compute the exact solution for the above initial value problem.
(b) Use your codes of Forward Euler, Backward Euler, Trapzoidal Method and Leapfrog
Method developed for Problem 1 to compute u(1) using h = 0.1, 0.05 and 0.025, and
denote numerical solutions as 401(1), 20.05(1) and 20.025(1).
(c) Compute the following ratio and explain how they are connected to the ratio obtained
in Problem 1.
20.1(1) – u(1) 40.05(1) – u(1)
11
20.05(1) – u(1)’
20.025 (1) u(1)
T2
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