Numerical methods – 23rd July 2020
Question 1 [7 Marks]
Student Name: «First_Name» «Last_Name»
Student ID: «Student_ID»
(a) Solve the following equations analytically to calculate the true value of
the variables 𝑥𝑥1 and 𝑥𝑥2 [1 Mark]
𝐴𝐴𝑥𝑥1 + 𝐵𝐵𝑥𝑥2 = 𝐶𝐶
𝐷𝐷𝑥𝑥1 + 𝐸𝐸𝑥𝑥2 = 𝐹𝐹
Where:
𝐴𝐴 = «A»
𝐵𝐵 = «B»
𝐶𝐶 = «C»
𝐷𝐷 = «D»
𝐸𝐸 = «E»
𝐹𝐹 = «F»
(b) Use the Gauss-Seidel method using overrelaxation (𝜆𝜆 = 1.25) to estimate
the value of the variables starting from initial values of 𝑥𝑥1 = 0 and 𝑥𝑥2 = 0.
Perform 2 iterations only [3 Marks]
(c) Calculate the relative percent true error at the end of each iteration [3
Marks]
Please use four significant figures in your answers.
General Instructions:
-
You have 5 minutes to read the question
You have 20 minutes to solve the question
You have 5 minutes to submit your solution.
Late submissions will be penalised (up to zero marks).
Write your name and QUID at the top of your answer sheets
After you finish, please take a clear picture of your solution and upload to
Final Exam – Question 1 under Assignments on BB.
Numerical methods – 23rd July 2020
Question 2 [8 Marks]
Student Name: Fatemeh Ahmad Salman
Student ID: 201702833
It is known that the data in the table below can be modelled by the following
equation:
𝐴𝐴√𝑥𝑥
𝑦𝑦 = 𝐵𝐵+
√𝑥𝑥
1. Use a suitable technique to linearize the equation above
2. Find the value of 𝐴𝐴 and 𝐵𝐵
[4 Marks]
3. Find the value of 𝑦𝑦 that corresponds to 𝑥𝑥 = 4.5
𝑥𝑥
𝑦𝑦
2
0.763
[2 Marks]
4
1
[2 Marks]
6
1.16
8
1.282
General Instructions:
-
You have 5 minutes to read the question
You have 25 minutes to solve the question
You have 5 minutes to submit your solution.
Late submissions will be penalised (up to zero marks).
Write your name and QUID at the top of your answer sheets
After you finish, please take a clear picture of your solution and upload to
Final Exam – Question 2 under Assignments on BB.
Numerical methods – 23rd July 2020
Question 3 [10 Marks]
Student Name: Fatemeh Ahmad Salman
Student ID: 201702833
Consider the following velocity data.
𝑡𝑡(𝑠𝑠)
𝑣𝑣(𝑚𝑚/𝑠𝑠)
𝑎𝑎 (𝑚𝑚/𝑠𝑠 2 )
1
5
2
6
3
5.5
4
7
5
8.5
a) Use the trapezoidal rule to find the distance travelled. Note that the distance
is the integration of the velocity (𝑑𝑑 = ∫ 𝑣𝑣𝑣𝑣𝑣𝑣)[3 Marks]
b) Use Simpson 1/3 rule to find the distance travelled during the time listed
above [3 Marks]
c) Calculate the acceleration (which is the derivative of the velocity) at the
indicated times using the standard error formulae 𝑂𝑂(ℎ2 ) [4 Marks]
General Instructions:
-
You have 5 minutes to read the question
You have 25 minutes to solve the question
You have 5 minutes to submit your solution.
Late submissions will be penalised (up to zero marks).
Write your name and QUID at the top of your answer sheets
After you finish, please take a clear picture of your solution and upload to
Final Exam – Question 3 under Assignments on BB.
Numerical methods – 23rd July 2020
Question 4 [10 Marks]
Student Name: «First_Name» «Last_Name»
Student ID: «Student_ID»
Consider the following differential equations:
𝑥𝑥̈ + 5𝑥𝑥̇ + 2𝑥𝑥 − 𝑦𝑦 = 0
𝑦𝑦̇ + 2𝑥𝑥𝑥𝑥 − 𝑥𝑥 = 1
1. Convert the equations into a system of ODE [3 Marks]
2. Using 2nd Order Rung-Kutta method (Heun’s option) what is the value of
𝑥𝑥̈ (0.2) as obtained from the numerical solution? [3 Marks]
3. Using 2nd Order Rung-Kutta method (Heun’s option) what is the value of
𝑦𝑦̇ (0.4) as obtained from the numerical solution? [4 Marks]
Use ℎ = 0.2 and initial conditions:
𝑥𝑥(0) = «x11»
𝑥𝑥̇ (0) = «x21»
𝑦𝑦(0) = «x31»
General Instructions:
-
You have 5 minutes to read the question
You have 25 minutes to solve the question
You have 5 minutes to submit your solution.
Late submissions will be penalised (up to zero marks).
Write your name and QUID at the top of your answer sheets
After you finish, please take a clear picture of your solution and upload to
Final Exam – Question 4 under Assignments on BB.